Beitr a-dieresis ge zur Algebra und Geometrie \noindentContributionsB e i tot r Algebra $ \ddot and{ Geometrya} $ ge zur Algebra und Geometrie Volume 45 open parenthesis 2004 closing parenthesis comma No period .. 2 comma 369 hyphen 387 period \noindentBeitr a¨ Contributionsge zur Algebra und to Algebra Geometrie and Geometry CompletenessContributions .. Criteria to Algebra .. and and.. Invariants Geometry .. for OperationVolume .. 45 and ( 2004 .. T-r ) ansformation , No . 2 .. , 369Algebras - 387 . \noindentDanica JakubVolume acute-dotlessi 45 ( 2004 kov acute-a ) , No hyphen . \quad Studenovsk2 , 369 acute-a− 387 to the . power of 1 .. Dragan Ma s-caron ulovi acute-c to the power of 2 .. ReinhardCompleteness P dieresis-o schel Criteria and Invariants for \ centerlineKatedra geometrie{ Completeness a algebry comma\quad PrC acute-dotlessi r i t e r i a \quad rodovedeckand \ a-acutequad I fakulta n v a r i a UPJ n t s caron-S\quad subf o r comma} Ko caron-s ice comma Slovakia InstituteOperation of Mathematics comma and .. University ofT − Novyr ansformation Sad comma .. Yugoslavia Algebras 1 2 \ centerlineInstituteDanica of{ AlgebraOperation Jakub comma´ı kov\quad .. TUa´− DresdenandStudenovsk\quad comma$a´ .. T Germany−rDragan $ ansformation Ma sˇ ulovi c´\quadReinhardAlgebras P o¨} schel ˇ AbstractKatedra period geometrie .. Operation a algebrasalgebry s , erve Pr as´ı representationsrodovedeck a´ offakulta composition UPJ algebrasS, Ko opensˇ ice parenthesis , Slovakia in \ centerlinethe sense of{ LauschDanicaInstitute slash Jakub N of o-dieresis Mathematics $ \acute bauer{\ closing ,imath University parenthesis} $ kov period of $ Novy\acute .. In Sad this{a ,} paper −Yugoslavia they$ are Studenovsk described and $ characterized\acute{a} ˆ{ 1 }$ \quad Dragan Ma $ \bycheck invariant{ s } relations$ u l o v as i GaloisInstitute $ \acute hyphen of{c closed Algebra} ˆ{ sets2 ,} w$ period TU\quad Dresdenr periodReinhard t ,period P Germany a suitable $ \ddot Galois{o} $ connection s c h e l period} .. Fur hyphen ther commaAbstract the completeness . Operation problem algebras in operation s erve algebras as representations is considered and of solved composition for algebras ( in \ centerlineconcretethe cases sense{ Katedra open of Lausch parenthesis geometrie / N e periodo¨ bauer a algebry g period) . ..In , for Prthis transformation paper$ \acute they{\ open areimath described parenthesis} $ rodovedeck and max characterized comma circ $ \ closingacute{ parenthesisa} $ fakulta hyphenUPJ semirings $ \check{S} { , }$ Koclosing $by parenthesis\check invariant{ s } period relations$ ice as, Slovakia Galois - closed} sets w . r . t . a suitable Galois connection . Fur - MSC 2000ther : .. ,8 the A 40 completeness comma .. 1 6 Y problem 60 in operation algebras is considered and solved for \ centerlineIntroduction{ Instituteconcrete casesof Mathematics ( e . g . for , \ transformationquad University ( max of Novy, ◦)− semirings Sad , \quad ) . Yugoslavia } In this paper operation algebras areMSC investigated 2000 comma : 8 Ai period 40 , e period 1 6 Y algebras 60 whose elements are operations \ centerlineopenIntroduction parenthesis{ Institute of fixed arity of kAlgebra closing parenthesis , \quad ....TU on Dresden a set A and , \ whosequad fundamentalGermany } operations are induced by an algebra openIn this parenthesis paper operation from some algebras fixed classare open investigated parenthesis variety , i . e closing . algebras parenthesis whose K elements closing parenthesis are operations on the base set A and also include composition\ centerline( of fixed semicolon{ arityAbstract fork) unary . \onquad a setOperationA and whose algebras fundamental s erve operations as representations are induced ofby an composition algebra algebras ( in } mappings( from some open parenthesis fixed class k ( = variety 1 closing) parenthesisK) on the basesuch algebras set A and will bealso called include transformation composition algebras ; for period unary \ centerlineOnemappings of the motivations{ the(k = sense 1) such to .... of algebras study Lausch operation will / N be algebras $called\ddottransformation ....{o comes} $ .... bauer from algebras the) . observation\quad. In that this .... such paper they are described and characterized } algebrasOne of .... the are motivations .... concrete .... to cases study .... of operation the .... so algebras hyphen called comes .... composition from the .... observation algebras .... introducedthat such .... in .... open square bracket 4 comma\ centerlinealgebras .... Ch period{ areby invariant concrete .... 3 closing cases relations square of bracket the as so - Galois called− compositionclosed sets algebras w . r introduced . t . a suitable in [ 4 , Galois Ch . connection . \quad Fur − } open3 ] parenthesis special examples are near hyphen r ings comma algebras of binary relations and distributive latt ices comma see 4 period 2 closing\ centerline( special parenthesis{ examplesther , the are near completeness - r ings , algebras problem of in binary operation relations algebras and distributive is considered latt ices and, see solved 4 . for } emdash2 ) — the the most most general general approach approach known known to the toauthors the authors generalizing generalizing the classical the Cayley classical theorem Cayley theorem \ centerlinehline { concrete cases ( e . g . \quad for transformation ( max $ , \ circ ) − $ semirings ) . } 1 sub supported by Slovak Grant VEGA 1 slash 423 slash 3 \ centerline2 sub supported{MSC by 2000 the postgraduate : \quad 8 programme A 40 , \ ..quad quotedblleft1 6 Y Specification 60 } of Discrete Processes and Systems of Processes by Operational Models and Logics quotedblright1 by Slovak comma Grant of the VEGA Deutsche 1 / Forschungsgemeinschaft 423 / 3 comma GK 334 period \noindent0 138 hyphenIntroduction 482 1 slash 93 dollar 2supported period 50 circlecopyrt-c 2004 Heldermann Verlag 2supported by the postgraduate programme “ Specification of Discrete Processes and Systems of Processes by \noindentOperationalIn Models this paper and Logics operation ” , of theDeutsche algebras Forschungsgemeinschaft are investigated , GK , i 334 . e . . algebras whose elements are operations 0138 − 4821/93$2.50circlecopyrt − c2004 Heldermann Verlag \noindent ( of fixed arity $k )$ \ h f i l l on a set $ A $ and whose fundamental operations are induced by an algebra

\noindent ( from some fixed class ( variety $ ) K ) $ on the base set $ A $ and also include composition ; for unary

\noindent mappings $ ( k = 1 ) $ such algebras will be called transformation algebras .

\noindent One of the motivations to \ h f i l l study operation algebras \ h f i l l comes \ h f i l l from the observation that \ h f i l l such

\noindent a l g e b r a s \ h f i l l are \ h f i l l c o n c r e t e \ h f i l l c a s e s \ h f i l l o f the \ h f i l l so − c a l l e d \ h f i l l composition \ h f i l l a l g e b r a s \ h f i l l introduced \ h f i l l in \ h f i l l [ 4 , \ h f i l l Ch . \ h f i l l 3 ]

\noindent ( special examples are near − r ings , algebras of binary relations and distributive latt ices , see 4 . 2 ) −−− the most general approach known to the authors generalizing the classical Cayley theorem

\ begin { a l i g n ∗} \ r u l e {3em}{0.4 pt } \end{ a l i g n ∗}

\ centerline { $ 1 { supported }$ by Slovak Grant VEGA 1 / 423 / 3 }

$ 2 { supported }$ by the postgraduate programme \quad ‘‘ Specification of Discrete Processes and Systems of Processes by Operational Models and Logics ’’ , of the Deutsche Forschungsgemeinschaft , GK 334 .

\noindent $ 0 138 − 482 1 / 93 \$ 2 . 50 circlecopyrt −c 2004 $ Heldermann Verlag 370 .. R period P o-dieresis schel et al period : .. Completeness Citeria and Invariants period period period \noindentthat every370 group\quad i s i somorphicR . P to $ a\ groupddot{ ofo} permutations$ schel where et al composition . : \quad playsCompleteness the role Citeria and Invariants . . . of the group multiplication open parenthesis s ee 4 period 4 closing parenthesis period .. It shows that algebras open parenthesis from some fixed 370 R . P o¨ schel et al . : Completeness Citeria and Invariants . . . class\noindent open parenthesisthat every variety group closing i parenthesis s i somorphic to a group of permutations where composition plays the role ofthat the every group group multiplication i s i somorphic ( s to ee a group 4 . 4 of ) permutations . \quad It shows where thatcomposition algebras plays ( fromthe role some of fixed class ( variety ) Kthe closing group parenthesis multiplication with an (additional s ee 4 . ..4 )open . parenthesis It shows thatk plus algebras 1 closing ( parenthesis from some hyphen fixed ary class operation ( variety kappa satisfying .. quotedblleft composition$K ) hyphen $ with like anquotedblright additional .. properties\quad can$ ( k + 1 ) − $ ary operation $ \kappa $ satisfying \quad ‘‘ composition − l i k e ’ ’ \quad properties can ) K) with an additional (k +1)− ary operation κ satisfying “ composition - like ” properties bebe represented represented as concrete as concrete operation operation algebras where algebras kappa is where given by $ composition\kappa $ open is parenthesis given by s ee composition 1 period 6 closing ( s parenthesis ee 1 . 6 period ) . ..\quad The can be represented as concrete operation algebras where κ is given by composition ( s ee 1 . 6 ) Theconnection between composition algebras ( in a setting s l ightly more general than in [ 4 ] ) and operation. The algebras connection will between be reported composition in algebrasSection ( 1 in . a setting s l ightly more general than in [ connection4 ] ) and between operation composition algebras algebras will be open reported parenthesis inSection in a setting 1 . s l ightly more general than in open square bracket 4 closing square bracket closing parenthesis and \noindentIn SectionIn 2Section we show 2 wehow show operation how operation algebras can algebras be described can be and described characterized and characterized via invariant via invariant operationrelations algebras ; they will beare reported the Galois in Section closed 1 period elements with respect to a suitable Galois connection ( r eIn l a Section t i o n s 2 ; we\quad show howthey operation are the algebras Galois can be closed described elements and characterized with respect via invariant to a suitable Galois connection ( seesee 2 2 . . 4 4 ) ) . . The completeness problem in operation algebras i s considered relationsin Section semicolon .. 3 they , arei . thee . Galois we closed ask elements for systems with respect of operations to a suitable generating Galois connection the full operation Theopen\quad parenthesiscompleteness see 2 period\ 4quad closingproblem parenthesis\quad periodin \quad operation \quad a l g e b r a s \quad i s \quad c o n s i d e r e d \quad in \quad Sec tion \quad 3 , \quad i . e . \quad we \quad ask forK− systemsalgebra of . operations For near - generating r ings this was the investigated full operation in [ 1 $ ] , K here− we$ present algebra a much . \quad moreFor near − r ings this was Thegeneral .. completeness approach .. which problem can .. in be .. applied operation not .. algebras only to .. arbitrary i s .. considered operation .. in .. algebras Section .. but 3 comma also to .. iother period e period .. we .. ask investigatedfor systems of operations in [ 1 generating] , here the we full present operation a K much hyphen more algebra general period approach .. For near hyphen which r can ings bethis applied was not structures . Finally , in Section 4 we shall consider the case where κ i s binary ( onlyinvestigated to arbitrary in open square operation bracket 1 algebras closing square but bracket also comma to other here westructures present a much . more general approach which can be applied not F i nthis a l l y ,means\quad ink \=quad 1, Sectioni . e . we 4 we deal shall with representations consider the case of algebras where by\quad unary$ operations\kappa $ ) i s binary \quad ( t h i s \quad means \quad only. to Here arbitrary we discuss operation some algebras prob but -also lems to connected other structures with period the concrete representation of composition $ kFinally = comma 1 .. , in $ ..\ Sectionquad 4i we . shall e . consider the case where .. kappa i s binary .. open parenthesis this .. means .. k = 1 comma .. i period algebras ( introduced in Sec - t ion 1 ) by operations on some base set A, in particular we ask for ewe period deal with representations of algebras by unary operations ) . \quad Here we discuss some prob − lemsminimal connected representations with the concrete ( a representation representation i s called of composition minimal algebras i f A (has introduced minimal in Sec − wes izdeal e )with and representations give of several algebras by examples unary operations for transformation closing parenthesis algebras period .. . Here The we discuss completeness some prob hyphen tlems ion connected 1 ) by with operations the concrete on representation some base ofset composition $ A ,algebras $ in open particular parenthesis we introduced ask for in Sec minimal hyphen representations ( atheorem representation will be demonstrated\quad i s for\quad transformationc a l l e d \quad ( maxminimal, ◦)− semirings\quad i f over $ A{1 $, ..., n\}quad. has \quad minimal \quad s i z e ) \quad and \quad g i v e \quad s e v e r a l \quad examples \quad f o r tAcknowledgements ion 1 closing parenthesis by . operationsWe are on somegrateful base for set stA comma imulating in particular dicussions we ask with for F minimal . Binder representations , P transformationopen parenthesis a algebras representation . \ ..quad i s ..The called completeness .. minimal .. i theorem f A .. has will .. minimal be demonstrated .. s iz e closing for parenthesis transformation .. and .. give .. several .. ( max. Mayr $ , and\ circ G .) Pilz− $ on semirings near - r ings over ( $ 200\{ 11 during , the . . AAA . 62 , conference n \} . $ examplesin .. Linz for ) , which finally led to the plan for this paper . Our thanks are also due to G . transformation algebras period .. The completeness theorem will be demonstrated for transformation \noindentEigenthalerAcknowledgements for his helpful hints . \quad We are grateful for st imulating dicussions with F . \quad Binder , \quad P. \quad Mayr openconcerning parenthesis composition max comma algebras circ closing . parenthesis hyphen semirings over open brace 1 comma period period period comma n closing brace periodand \quad G. \quad P i l z \quad on near − r i n g s \quad ( 200 1 \quad during the \quad AAA 62 \quad c o n f e r e n c e \quad in \quad Linz ) , \quad which finally led 1. K− composition algebras and their representation by operation K− algebras toAcknowledgements the plan for period this.. paper We are . grateful\quad forOur st imulating thanks aredicussions also with due F to period\quad .. BinderG. comma\quad ..Eigenthaler P period .. Mayr for his helpful hints Let Alg (τ) denote the class of all algebras of a given type τ. Throughout the paper K denotes and .. G period .. Pilz .. on near hyphen r ings .. open parenthesis 200 1 .. during the .. AAA 62 .. conference .. in .. Linz closing parenthesis a quasivariety of type τ, i . e . a subclass of Alg (τ) which i s closed with respect to i somorphic comma\noindent .. whichconcerning finally led composition algebras . copies , subalgebras and direct powers ; thus K may be any variety . Moreover to the plan for this paper period .. Our thanks are also due to .. G period .. Eigenthaler for his helpful hints , we assume that K is nontrivial , i . e . it does not consist of one - element \noindentconcerning composition$ 1 . algebras K − period$ composition algebras and their representation by operation $ K − $ a l g e b r a s algebras only . The algebras in A ∈ K must be of type τ. However , often the type is not 1 period K hyphen composition algebras and their representation by operation K hyphen algebras very essential ; then we also use the non - indexed form A = hA; Ui; it has to be understood \noindentLet Alg openLet parenthesis Alg $ (tau closing\tau parenthesis) $ denote denote the the class class of all of algebras all algebrasof a given type of tau a given period .. type Throughout $ \tau the paper. $ K\ denotesquad Throughout the paper as the algebra hA;(fi)i ∈ Ii ∈ K with U = {fi | i ∈ I}. Further , k denotes a positive integer . $ Ka quasivariety $ denotes of type tau comma i period e period .. a subclass of Alg open parenthesis tau closing parenthesis which i s closed with respect to i 1.1. K− composition algebras . Let hB;(fi)i ∈ Ii ∈ Alg (τ) and suppose that κ is a (k + 1)− somorphica quasivariety of type $ \tau , $ i . e . \quad a subclass of Alg $ ( \tau ) $ which i s closed with respect to i somorphic c o pary i e s operation , \quad subalgebras on the set B.\quadThenandhB d i;( rfi e c) ti ∈\quadI, κi willpowers be called ; \quad a thus(k− dimensional\quad $ K $ )K−\quad may be any variety . \quad Moreover , \quad we assume copiescomposition comma .. subalgebras algebra ( .. [ and 4 , p direct . 73 .. ] powers ) , i f semicolon .. thus .. K .. may be any variety period .. Moreover comma .. we assume thatthat\ ..quad K is .. nontrivial$ K $ i comma s \quad .. i periodnontrivial e period , ..\ itquad .. doesi not . econsist . \quad .. of onei t hyphen\quad elementdoesnot .. algebras consist .. only\quad periodo .. f The one algebras− element in \quad a l g e b r a s \quad only . \quad The algebras in $A A underbar{\underline in K must{\}}\ be of type tauin periodK$ .. However must be comma of type often the $ type\tau is not. very $ essential\quad However semicolon then , often we also the use type the is not very essential ; then we also use the (i) hB;(fi)i ∈ Ii ∈ K, nonnon− hyphenindexed indexed form form $ A A underbar{\underline = angbracketleft{\}} A= semicolon\ langle U rightA;U angbracket semicolon\rangle .. it has; to $ be\quad understoodit has as the to algebra be understood as the algebra angbracketleft$ \ langle AA semicolon ; (open( fparenthesis i i ) iκ i ) fs isuperassociative closing i \ parenthesisin I i,\ in irangle . I eright . angbracket we\ in have inK K $ with with $U U = open = brace\{ f i barf i in i I closing\mid bracei period\ in .. FurtherI \} comma. k $ denotes\quad a positiveFurther integer $ period , k $ denotes a positive integer . 1 period 1 period K hyphenκ(κ(x0 composition, x1, ..., xk), y1 algebras, ..., yk) period= κ(x0, .. κ( Letx1, y angbracketleft1, ..., yk), ..., κ( Bxk semicolon, y1, ..., yk)) open parenthesis f i closing parenthesis i in I right \noindent $ 1 . 1 . K − $ composition algebras . \quad Let $ \ langle B ; ( f i ) angbracket in Alg open parenthesis tau closing parenthesis and supposeforall thatx0, kappa..., xk, yis1, a ..., open yk ∈ parenthesisB. k plus 1 closing parenthesis hyphen ary i operation\ in onI the\rangle set B period\ ..in Then$ Algangbracketleft $ ( \ Btau semicolon) $ open and parenthesis suppose f i that closing $parenthesis\kappa i$ in I i scomma a $ kappa ( right k +angbracket 1 will) be− called$ ary a .. open parenthesis k hyphen dimensional closing parenthesis K hyphen composition operationalgebra open on parenthesis the set open $B square . bracket $ \quad 4 commaThen p period $ \ langle 73 closing squareB ; bracket ( closing f iparenthesis ) i comma\ in i f I, \kappa \rangleopen parenthesis$ will i beclosing called parenthesis a \quad angbracketleft$ ( kB semicolon− $ dimensional open parenthesis $ f i ) closing K parenthesis− $ composition i in I right angbracket in K comma algebraopen parenthesis ( [ 4 i,p. i closing 73 parenthesis ] ) , i kappa f i s superassociative comma i period e period .. we have Line 1 kappa open parenthesis kappa open parenthesis x sub 0 comma x sub 1 comma period period period comma x sub k closing parenthesis comma\ [ ( y i1 comma ) period\ langle periodB period ; comma ( y f k closing i parenthesis ) i \ =in kappaI open\rangle parenthesis x\ in sub 0K, comma kappa\ ] open parenthesis x sub 1 comma y 1 comma period period period comma y k closing parenthesis comma period period period comma kappa open parenthesis x sub k comma y 1 comma period period period comma y k closing parenthesis closing parenthesis Line 2 for all x sub 0 comma period period period comma x sub k\ centerline comma y 1 comma{( i period i $ period ) \kappa period comma$ i ys k superassociative in B period , i . e . \quad we have } \ [ \ begin { a l i g n e d }\kappa ( \kappa ( x { 0 } , x { 1 } , . . . , x { k } ) , y 1 , . . . , y k ) = \kappa ( x { 0 } , \kappa ( x { 1 } , y 1 , . ..,yk),..., \kappa ( x { k } ,y1,...,yk) ) \\ f o r a l l x { 0 } , . . . , x { k } , y 1 , . . . , y k \ in B. \end{ a l i g n e d }\ ] R period P o-dieresis schel et al period : .. Completeness Citeria and Invariants period period period .. 371 R .open P parenthesis $ \ddot{o ii} i closing$ schel parenthesis et al kappa . : i\ squad right hyphenCompleteness distributive Citeria with respect and to Invariants every f i comma . . i period . \quad e period371 (for i i every i $ i in ) I and\kappa all x sub$ 1 i comma s r i g h period t − distributive period period comma with x respect sub n comma to every y 1 comma $ f period i period , $ period i . comma e . y k in B open R.P o¨ schel et al . : Completeness Citeria and Invariants . . . 371 ( ii i ) κ i s right - parenthesis n being the arity of f i closing parenthesis we have distributive with respect to every fi, i . e . \ centerlineLine 1 kappa{ f open o r everyparenthesis $ f i i comma\ in y 1I comma $ and period a l periodl $ x period{ 1 comma} , y k closing . . parenthesis . , = f x i for{ nullaryn } f, i comma y Line 1 2 , kappa . for every i ∈ I and all x , ..., x , y1, ..., yk ∈ B(n being the arity of fi) we have open. parenthesis . , y f i open k parenthesis\ in B x sub ( 1 comma1 n$n period beingthearityof period period comma $f x sub n closing i )$ parenthesis wehave comma} y 1 comma period period period comma y k closing parenthesis = Line 3 f i open parenthesisκ(fi, y1, kappa ..., yk) open = fi parenthesisfornullaryfi, x sub 1 comma y 1 comma period period period comma y k closing\ [ \ begin parenthesis{ a l i g n e comma d }\kappa period period( period f icomma , kappa y open 1 parenthesis , . x . subn . comma , y y 1 comma k period ) = period f period i comma for y nullary k closing κ(fi(x , ..., x ), y1, ..., yk) = parenthesisf i , closing\\ parenthesis for n hyphen ary f i period 1 n \kappa ( f i ( x { 1 } , . . . , x { n } ),y1,...,yk 1 period 2 period .. Selector systemsfi(κ(x period1, y1, ..., .. yk A) s, ...,elector κ(xn system, y1, ..., ..yk open)) for parenthesisn − aryfi. open square bracket 4 comma p period .. 73 closing square bracket) = closing\\ parenthesis for an operation kappa : B to the power of k plus 1 right arrow B i s a f1 . i 2 . ( \ Selectorkappa systems( x { .1 } A,y1,...,yk),...,s elector system ( [ 4 , p . 73 ] ) for an operation \kappa familyκ : Bk s+1 sub→ 1B commai s a family period periods , ..., period s of elements comma s sub of B k ofsuch elements that of for B all suchy0 that, y1, for ..., yk all∈ yB 0 commaand i ∈ y 1{1 comma, ..., k} period period period comma y( k in x B and{ n i in} open,y1,... brace 1 comma1 periodk period period ,yk))forn comma k closing brace − ary f i . \end{ a l i g n e d }\ ]

Line 1 kappa open parenthesis s sub i commaκ(s yi, 1 y1 comma, ..., yk) period = yi periodand period comma y k closing parenthesis = y i and Line 2 kappa open parenthesis y 0 comma s sub 1 comma period period period comma s sub k closing parenthesis = y 0 period κ(y0, s , ..., s ) = y0. \noindentA K hyphen1 composition . 2 . \quad algebraSelector angbracketleft systems B semicolon . \quad1 openkA parenthesis s elector f i system closing parenthesis\quad ( i [in 4I comma , p . kappa\quad right73 angbracket ] ) for is an called operation K$ hyphen\kappaA K− compositioncomposition: Balgebra ˆ{ k algebra with + selectorh 1B;(}\fi system)irightarrow∈ I, κi is calledBK− $composition i s a algebra with selector system familyi fi there f there exists $ exists s a{ selector1 a selector} system, system . for kappa . for . periodκ. , s { k }$ of elements of $B$ such that for all $y 0 , y11 period1 . 3 , 3 . period . .. . Remarks Remarks . , period . y .. ak closinga )\ in parenthesis TheB $ class and .. The $ .. of i classK−\ ..incomposition of K\{ hyphen1 composition algebras , . algebras .shares . .. shares , k .. many\} algebraic$ .. propertiesmany algebraic properties with the underlying class K, e . g . closedness with respect to \ [ \withbeginsubalgebras the{ underlyinga l i g n e , d products}\ classkappa K comma or homo( e period -s morphisms{ gi period} ,y1,...... closedness In particular with respect , i f toK subalgebrasi s a variety ,yk)=yiand comma then products the class or homo hyphen \\ \morphismskappaof all K− period(composition y .. In 0 particular algebras , s comma{ i s1 ia} f K i, s a variety . . then . the class , of s all{ Kk hyphen} ) composition = y algebras 0 . i\ send a { a l i g n e d }\ ] varietyvariety comma , too too . bperiod ) If there exists a selector system for a K− composition algebra then it is unique b( closing [ 4 , Ch parenthesis . 3 , If there exists a selector system for a K hyphen composition algebra then it is unique open parenthesis open square bracket 4\noindent comma1 . 1 Ch 1 period]A ) . $ c K3 ) comma For− $k = composition 1 superassociativity algebra of κ $ reduces\ langle to ordinaryB ; associativity ( f i and ) a selector i \ in I, \kappa \rangle1system period$ 1 1i is sclosing j called ust square an identity $K bracket for− closing$ the composition parenthesisbinary operation period algebraκ. with selector system ic f1 closing there . 4 . parenthesis exists NotationFor a selector k . = 1Let superassociativityA system= h forA;( offi kappa)$i ∈\kappaIi reduces ∈ K to.. For ordinary $ any associativity set B let andAB adenote selector the system set i sof j allust an identity for the binary operation kappa period \noindent1 period 41 period . 3 .... . \ Notationquad Remarks period .... . Let\quad .... Aa underbar ) \quad =The angbracketleft\quad c lA a ssemicolon s \quad openo f parenthesis $ K − f$ i closing composition parenthesis algebras i in I right\quad s h a r e s \quad many algebraic \quad p r o p e r t i e s angbracketwith the in underlying K period .... For class .... any $K .... set .... , $ B .... e let . g .... . A\ toquad the powerclosedness of B .... denote with.... respect the .... setto .... subalgebras of all , products or homo − . Leti∈I n of i. Then i morphismsmappings . \quadf :InB particular→ AA ,and i flet $K$denote i s athe varietyarity thenB f the class(f .induces of all $Kan∈n−AB− $aryby composition algebras i s a mappings operationoperation fiˆ f subon f-circumflexB as i : B: onFor rightevery arrow A Ah B1,..., as toh then power∈ A of perioddefine Letfiˆ sub1h, follows sub..,hn :) to the power of i in to the power of I and For let every to the power of n denote sub h sub 1 comma period period period comma the sub h sub n arity in A to the power of B to the power\noindentsetting of of f fordefinevariety every to the ,b ∈ power tooB . of i period Then sub circumflex-f i open parenthesis f 1 h comma to the power of i period induces period period commab ) If h sub there n closing exists parenthesis a selector an in n hyphen system A forB ary a by $ K − $ composition algebra then it is unique ( [ 4 , Ch . 3 ,

setting for every b in B fˆ \noindentEquation:1 open . 1 parenthesis 1 ] ) . 1 period 4( periodi(h1, ..., 1 hclosingn))(b) :=parenthesisfi(h1(b), ....., open hn(b)) parenthesis to the power of f-circumflex (1.4.1) i open parenthesis h sub 1 commac ) For period $ period k = period 1 comma $ superassociativity h sub n closing parenthesis of closing $ \kappa parenthesis$ reduces open parenthesis to ordinary b closing associativity parenthesis : = f i and open a parenthesis selector h system ˆ B A subi1 s( open note j ust parenthesis that an identityfi i b s closing j ust for the parenthesis operationthe binary commafi operation periodin the period Cartesian period $ \kappa power commaA h. sub $of n the open algebra parenthesis b closing). If parenthesis closing parenthesis ˆ openthere parenthesis i s no danger note that of f-circumflex confusion i we i s shallj ust the omit operation the hat f i and in the write Cartesianf instead power A of underbarf, in particular to the power if of B of the algebra hline to for \noindent 1 . 4 . \ h f iigns l l Notationk like .+B\ h f i l l Let \ h f i l l $ A {\underline {\}} = \ langle A ; ( f the powerwe useInspecial of A closingcase parenthesisBs= A periodfthe( ..set IfA thereor∧)gives.the so - called ( full ) Menger algebra i )k i \ in I \rangle \ in K . $ \ h f i l l For \ h f i l l any \ h f i l l s e t \ h f i l l $ B $ \ h f i l l l e t \ h f i l l i s noA danger of confusion we shall omit the hat and write f instead of f-circumflex sub comma in particular if we hA ; κAi of k− ary operations , where κA is the composition ( superposition ) defined by $ Ause ˆ{ InB special}$ \ caseh f iB l l s =denote to the power\ h f i l of l ignsthe A\ toh f the i l l powers e t of\ h k f toi l l theo f power a l l of for f the open parenthesis set to the power of like A to the power of plus B or and closing parenthesis gives period sub the .. so hyphen called .. open parenthesis full closing parenthesis .. Menger .. algebra .. \ [ mappings { operation } f {\hat{ f } i } :B { on }\rightarrow{ A } A { B as }ˆ{ . Let } {\succ }ˆ{ i angbracketleft A to the power of A to the powerκA(f of0, kf1 semicolon, ..., fk) := kappaf0(f1 sub, ..., A fk right) angbracket .. of k hyphen ary (1.4.2) \ inoperations} { : } commaˆ{ I } whereand kappa{ subFor A} is thel e t composition{ every open}ˆ{ parenthesisn } denote superposition{ h closing{ 1 } parenthesis,..., defined by } the { h { n }} a r i t y {\ in A ˆ{ B }}ˆ{ o f } f { de f i ne }ˆ{ i . Then } {\hat{ f } i } ( f { 1 { h } , }ˆ{ i } Equation:( here f open0 : Ak parenthesis→ A acts 1 as periodfˆ0 according 4 period 2 to closing 1 . 4 parenthesis . 1 ) . .. For kappak = sub 1 the A open operation parenthesisκ fi 0 s comma j ust the f 1 comma period period period . induces { . . , h { n } ) } an{\ in } n − { A } B ary { byA}\ ] commausual f k closing composition parenthesis◦ of : = unary f 0 open mappings parenthesis:( f 1f comma0 ◦ f1)( periodb) = f0( periodf1(b)) period. comma f k closing parenthesis open parenthesis here f 0 : A to the power of k right arrow A acts as f-circumflex 0 according to 1 period 4 period 1 closing parenthesis period .. For k = 1 the operation kappa sub A i s j ust the usual \noindent.5 setting for every $ b \ .in B $ Given algebra A;(fi)i∈I i it that composition1h .AkOperation circ of unaryfˆ mappingsK −algebras : open parenthesisa f 0 circ f 1an closingK parenthesis openA = parenthesish b∈ closing K iseasythistocheckalgebrathe parenthesis = f 0 open parenthesisfull A ;( i)i∈I,κAi is (k−dimensional) −composition algebra ; we call f 1 open parenthesis b closing parenthesis closing parenthesis period \ begin1 angbracketleft{ a l i g n ∗} sub A to the power of period 5 period A k Operation semicolon open parenthesis to the power of f-circumflex i closing parenthesis i in( Iˆ{\ commahat{ kappaf }} subi A right ( angbracket h { 1 } K is, hyphen . algebras . . a sub , open h parenthesis{ n } ))(b):=fi(h k hyphen dimensional to the power of period sub closing{ 1 } parenthesis( b ) to the , power . of . Given . an K , sub h hyphen{ n composition} ( b to the) power ) \ tag of algebra∗{$ ( to the 1 power . of 4 hline . A = 1 angbracketleft ) $} sub algebra sub semicolon\end{ a l i tog n the∗} power of A semicolon open parenthesis f i closing parenthesis i in I sub we to the power of right angbracket in K call to the power of it i s easy this to check algebra the to the power of that full \noindent ( note that $ \hat{ f } i $ i s j ust the operation $ f i $ in the Cartesian power $A {\underline {\}}ˆ{ B }$ of the algebra $ \ r u l e {3em}{0.4 pt } ˆ{ A } ) . $ \quad I f there i s no danger of confusion we shall omit the hat and write $ f $ instead of $ \hat{ f } { , }$ in particular if we $ use { In } s p e c i a l { case B } s { = }ˆ{ i g n s } A ˆ{ k ˆ{ f o r }} f { the } ( { s e t }ˆ{ l i k e } A ˆ{ + { B }} or \wedge ) { g i v e s } . { the }$ \quad so − c a l l e d \quad ( f u l l ) \quad Menger \quad algebra \quad $ \ langle A ˆ{ A ˆ{ k }} ; \kappa { A }\rangle $ \quad o f $ k − $ ary operations , where $ \kappa { A }$ is the composition ( superposition ) defined by

\ begin { a l i g n ∗} \kappa { A } (f0,f1,...,fk):=f0(f1,. . . , f k ) \ tag ∗{$ ( 1 . 4 . 2 ) $} \end{ a l i g n ∗}

\noindent (here $f 0 : Aˆ{ k }\rightarrow A $ a c t s as $ \hat{ f } 0$ according to 1 . 4 . 1 ) . \quad For $k = 1$ the operation $ \kappa { A }$ i s j ust the usual composition $ \ circ $ ofunarymappings $: ( f 0 \ circ f 1 ) ( b ) = f 0 ( f 1 ( b ) ) . $

\[1{\ langle }ˆ{ . 5 } { A } . { A } k Operation { ; ( ˆ{\hat{ f }} i ) i \ in I, \kappa { A } \rangle } K { i s } − a l g e b r a s { a }ˆ{ . } { ( k − dimensional }ˆ{ Given } { ) } an{ K }ˆ{ algebra } { − composition }ˆ{\ r u l e {3em}{0.4 pt }} A = \ langle { algebra }ˆ{ A ; ( f i ) i \ in I } { ; }ˆ{\rangle } { we } \ in K { c a l l }ˆ{ i t } i s easy { t h i s } to check { algebra } the ˆ{ that } f u l l \ ] 372 .. R period P o-dieresis schel et al period : .. Completeness Citeria and Invariants period period period \noindentopen parenthesis372 \quad k hyphenR . dimensional P $ \ddot closing{o} parenthesis$ schel operation et al . K : hyphen\quad algebraCompleteness over A underbar Citeria sub period and Invariants .. It has a selector . . system . namely$ ( the k proj− ections$ dimensional ) operation $ K − $ algebra over $A {\underline {\}} { . }$ \quad It has a selector system namely the proj ections 372 R . P o¨ schel et al . : Completeness Citeria and Invariants . . . (k− dimensional ) k e sub 1 comma period period period comma k e to the power of k defined by i e to the power of k open parenthesis a sub 1 comma period operation K− algebra over A It has a selector system namely the proj ections period\ begin period{ a l i g comma n ∗} a sub k closing parenthesis. = a sub i open parenthesis i in open brace 1 comma period period period comma k closing brace closingk { parenthesise { 1 }} period, . . . , k { e }ˆ{ k } de f i ned by i { e }ˆ{ k } ( a { 1 } ,. . . , a { k } ) = a { i } ( i \ in \{ 1 , . . . , k \} ). By a .... open parenthesis k hyphen dimensionalk closingk parenthesis operation K hyphen algebra we understand any subalgebra of a full open ke , ..., k definedbyi (a1, ..., ak) = ai(i ∈ {1, ..., k}). parenthesis\end{ a l i g kn ∗} hyphen dimen hyphen 1 e e sBy ional a closing( parenthesisk− dimensional operation ) operation algebra over someK− algebra A underbarwe in understand K open parenthesis any subalgebra 1 hyphen dimensional of a full ( operationk− algebras are also called \noindenttransformationBy a ..\ Kh hyphen f i l l $ algebras ( k comma− $ .. see dimensional .. open square ) bracket operation 3 closing $ square K − bracket$ algebra closing parenthesis we understand period .. Sany ince subalgebra .. K .. i s of a full $ (dimen k -− $ dimen − .. closeds ional .. under ) operation .. subalgebras algebra comma over .. somesuch ..A operation∈ K(1− dimensional operation algebras are also called algebras are K hyphen composition algebras in the s ense of Definition .. 1 period 1 period .. The following theorem \noindenttransformations ional )K− operationalgebras , algebra seeover [ 3 ] ) . some S ince $ A {\K underlinei s closed{\}}\ underin subalgebrasK ( 1 − $ dimensional operation algebras are also called shows, such that the operationconverse is also algebras true up are to isomorphismK− composition period algebras in the s ense of Definition 1 . 1 . 1 period 6 period .... Representation Theorem period .... open parenthesis open square bracket 4 comma Ch period 3 comma Thm period 1 \noindentThe followingtransformation theorem shows\quad that$ Kthe converse− $ a l gis e balso r a s true , \quad up tosee isomorphism\quad [ . 3 ] ) . \quad S i n c e \quad $ K $ \quad i s \quad c l o s e d \quad under \quad subalgebras , \quad such \quad operation period1 5 . 1 6 closing . square Representation bracket closing parenthesis Theorem Let . K be a quasivariety( [ 4 , Ch period . 3 , Thm .... Then . 1 ev . 5hyphen 1 ] ) Let K be a algebrasery open parenthesis are $ K k hyphen− $ dimensional composition closing algebras parenthesis in K thehyphen s compositionense of Definition algebra is is omorphic\quad 1 to . s ome1 . open\quad parenthesisThe following k hyphen theorem showsquasivariety that the . converse is also true up to isomorphism . Then ev - dimensionalery ( closingk− dimensional parenthesis operation)K− composition algebra is is omorphic to s ome (k− dimensional ) Koperation hyphen algebra period \noindent1 period 71 period . 6 .. . Remark\ h f i l l periodRepresentation .. The proof of Theorem .. . 1\ periodh f i l l 6 ..( is [ .. 4 constructive ,Ch . 3 and ,Thm. generalizes 1 . the 51 proof ] of ) the Let $K$ bea quasivariety . \ h f i l l Then ev − Cayley representation theorem for .. open parenthesis semi hyphen closing parenthesis groups period .. It .. was given in .. open square bracket \noindent ery $ ( k − $ dimensional $ ) K − $ composition algebra is is omorphic to s ome $ ( 4 closing square bracket .. for varieties but this K − algebra. k proof− $ also dimensional works for quasivarieties ) operation period .. The proof is particularly easy in case of algebras with a selector1 . 7 system . : Remark .. If B underbar . =The angbracketleft proof of TheoremA semicolon open 1 . parenthesis 6 is constructivef i closing parenthesis and generalizes i in I comma kappa right angbracket i s a\ begin k hyphenthe{ proofa l dimensional i g n of∗} the K Cayley hyphen representation composition algebra theorem with s for elector ( semi - ) groups . It was given in [ K system4− ] comma foralgebra varieties then the .but mapping this proof also works for quasivarieties . The proof is particularly easy \endain arrowright-mapsto{ a case l i g n ∗} of algebras phi with a comma a selector where phi system a : A to : the power If B of= k righthA;(fi arrow)i ∈ I, A κ :i openi s aparenthesisk− dimensional a sub 1 commaK− period period period comma a subcomposition k closing parenthesis algebra arrowright-mapsto with s elector system kappa open , then parenthesis the mapping a comma a sub 1 period period period comma a sub k closing parenthesis comma\noindent 1 . 7 . \quad Remark . \quad The proof of Theorem \quad 1 . 6 \quad i s \quad constructive and generalizes the proof of the Cayley representation theorem for \quad k( semi − ) groups . \quad I t \quad was given in \quad [ 4 ] \quad for varieties but this i s an embedding of B underbara 7→ intoϕa, thewhere fullϕa k hyphen: A → dimensionalA :(a1, ..., a operationk) 7→ κ(a,a K1..., hyphen ak), algebra over A underbar = angbracketleft A semicolon openproof parenthesis also works f i closing for parenthesis quasivarieties i in I right angbracket . \quad The period proof is particularly easy in case of algebras with a selectori s an embedding system : of\quadB intoI f the $ B full{\k−underlinedimensional{\}} operation= K−\ langlealgebra overA ;A = (hA;(fi f)i ∈ iIi. ) i \ in I, ExamplesExamples will will be considered be considered in Section in Section 4 period 4 . \kappa2 period ..\rangle Characterization$ i s of a operation $ k K− hyphen$ dimensional algebras by invariant $ K relations− $ composition algebra with s elector 2 . Characterization of operation K− algebras by invariant relations systemIn the .. , previous then the .. section mapping we .. have .. seen .. that .. K hyphen composition .. algebras .. can .. be .. represented by In the previous section we have seen that K− composition algebras can be operation K hyphen algebras period .. In this s ection we describe and characterize operation K hyphen algebras by represented by operation K− algebras . In this s ection we describe and characterize operation \ [invariant a \mapsto relations period\varphi .. Howevera comma , where now we shall\varphi restrict toa K hyphen : Acomposition ˆ{ k }\ algebrasrightarrow with s electorA : ( a { 1 } , K− algebras by invariant relations . However , now we shall restrict to K− composition algebras .system . period . , .. This a means{ k } that) we shall\mapsto consider\ operationkappa K( hyphen a algebras , a which{ 1 always} . contain . the . , a { k } ), \ ] with s elector system . This means that we shall consider operation K− algebras which always projcontain ections the k e sub 1 comma period period period comma k e to the power of k period .... In most .... cases this i s not .... a real restriction s ince one can j ust .... add a k \noindentproj ectionsi s anke , embedding ..., k . In most of $ cases B {\ thisunderline i s not a{\}} real restriction$ into the s ince full one can $k j ust− $ add dimensional a operation $ K selector system period1 e − $selector algebra system over . $A {\underline {\}} = \ langle A ; ( f i ) i \ in I \rangle . $ 22 period . 1 .1 period Some .. Some notions notions and notation notation period . .. ForFor operations operations and relations and relations on a fixed on base a fixed set A base we set introduce the following notation : A we introduce the following notation : \noindentLine 1 OpExamples to the power will of open be parenthesis considered k closing in Section parenthesis 4 open . parenthesis A closing parenthesis : = A to the power of A to the power of k = open brace f bar f : A to the power of k rightk arrow A closing brace open parenthesis k hyphen ary operations closing parenthesis comma Line Op(k)(A) := AA = {f | f : Ak → A} (k − aryoperations), 2\noindent infinity 2 . \quad Characterization of operation $ K − $ algebras by invariant relations Op open parenthesis A closing parenthesis : = union of Op to the power of open parenthesis∞ k closing parenthesis open parenthesis A closing parenthesis\noindent openIn parenthesis the \quad finitaryprevious operations\quad closings e c parenthesis t i o n we \ commaquad have \quad seen \quad that \quad $ K − $ composition \quad a l g e b r a s \quad can \quad be \quad represented by Op (A) := S Op(k)(A)( finitary operations ) , operationLine 1 k = 1 $ Line K 2 Rel− $ to the a l g power e b r a s of . open\quad parenthesisIn this m closing s ection parenthesis we describe open parenthesis and characterize A closing parenthesis operation : = open $ brace K rho− $ bar rhoa l g subsete b r a s equal by A to the power of m closing brace open parenthesis m hyphen ary relations closing parenthesis comma Line 3 infinity invariant relations . \quad However , now we shall restrictk = to 1 $ K − $ composition algebras with s elector Rel open parenthesis A closing parenthesis(m) : = union of Relm to the power of open parenthesis m closing parenthesis open parenthesis A closing parenthesissystem . open\quad parenthesisThis means finitary thatrelationsRel we(A closing)shall := {% parenthesis| % consider⊆ A } period(m operation− aryrelations) $, K − $ algebras which always contain the m = 1 ∞ \noindent proj ections $ k { e { 1 }} , . . . , k { e }ˆ{ k } . $ \ h f i l l In most \ h f i l l cases this i s not \ h f i l l a real restriction s ince one can j ust \ h f i l l add a An m hyphen tuple r in A to the power of mS may( bem) regarded as a mapping r : m underbar right arrow A open parenthesis with m underbar : = open brace 1 comma period periodRel period(A) := commaRel m closing(A)( bracefinitary closing relations parenthesis ) . comma and \noindentit s componentsselector are given system by r = . open parenthesis r open parenthesis 1 closing parenthesis comma period period period comma r open parenthesis m closing parenthesis closing parenthesis period m = 1 \noindent 2 . 1 . \quad Some notions and notation . \quad For operations and relations on a fixed base set An m− tuple r ∈ Am may be regarded as a mapping r : m → A( with m := {1, ..., m}), and it s $ A $ we components are given by r = (r(1), ..., r(m)). introduce the following notation :

\ [ \ begin { a l i g n e d } Op ˆ{ ( k ) } ( A ) : = A ˆ{ A ˆ{ k }} = \{ f \mid f : A ˆ{ k } \rightarrow A \} ( k − ary operations ) , \\ \ infty \end{ a l i g n e d }\ ]

\ centerline {Op $ ( A ) : = \bigcup Op ˆ{ ( k ) } ( A ) ( $ finitary operations ) , }

\ [ \ begin { a l i g n e d } k = 1 \\ Rel ˆ{ ( m ) } ( A ) : = \{\varrho \mid \varrho \subseteq A ˆ{ m }\} ( m − ary relations ) , \\ \ infty \end{ a l i g n e d }\ ]

\ centerline { Rel $ ( A ) : = \bigcup Rel ˆ{ ( m ) } ( A ) ($ finitary relations ) . }

\ [ m = 1 \ ]

\noindent An $ m − $ tu pl e $ r \ in A ˆ{ m }$ may be regarded as a mapping $ r : m {\underline {\}} \rightarrow A ( $ with $ m {\underline {\}} : = \{ 1 , . . . , m \} ) , $ and itscomponentsaregivenby $r = ( r ( 1 ) , . . . , r ( m ) ) .$ R period P o-dieresis schel et al period : .. Completeness Citeria and Invariants period period period .. 373 \ hspaceA relation∗{\ rhof i l inl }R Rel . to P the $ power\ddot of{ openo} $ parenthesis schel m et closing al . parenthesis : \quad openCompleteness parenthesis A Citeria closing parenthesis and Invariants .. i s invariant . .for . an\ operationquad 373 f in Op to the power of open parenthesis k closing parenthesis open parenthesis A closing parenthesis open parenthesis also f preserves .. rho comma R.P o¨ schel et al . : Completeness Citeria and Invariants . . . 373 or\noindent A r e l a t i o n $ \varrho \ in Rel ˆ{ ( m ) } ( A ) $ \quad i s invariant for an operation % ∈ Rel(m)(A) f ∈ Op(k)(A)( f %, $ ff isA a relation\ polymorphismin Op ˆ{ of( rho k closing )i parenthesiss} invariant( A i f forfor ) all an r($ sub operation 1 also comma $f$ period period preserves periodalso comma\quadpreserves r sub$ k\varrho in rho we have, $ f open or square bracket f %) r , ..., r ∈ % f[r , ..., r ] ∈ %. m− r sub$or f1 comma $is is a periodpolymorphism a polymorphism period periodof comma ofi $ f forr\ subvarrho all k closing1 )$ squarek ifforallwe bracket have in rho1 period $rk ..{ Here1 }Here the, m the hyphen . .tupletuple . , r { k }\ in f[r , ..., r ] f[r , ..., r ](i) := f(r (i), ..., r (i)) (i ∈ m). F, S, U ⊆ \varrhof open1 square$k wehavei sbracket defined r$f sub component 1 comma [ r period- wise{ 1 period by} , period1 .k comma . r . sub1 ,k closingk r { squarek } bracket] \ iin sFor defined\varrho component. hyphen $ \quad wise byHere f open the (A) Q ⊆ (A) square$ m Op bracket− $and r tu sub pl 1e commaRel periodwe period define period comma r sub k closing square bracket open parenthesis i closing parenthesis : = f open parenthesis Q := {f ∈ (A) | % ∈ Q f}, r sub$ f1 open [ parenthesis r { 1 i} closingPol, parenthesis . .Op comma . , periodevery r period{ k } period]i s $ comma invariant i s r defined sub for k open component parenthesis− i closingwise parenthesis by $ f closing [ parenthesis r { 1 } open, parenthesis . . . i in hline , to r the{ powerk } of] m closing ( i parenthesis ) : period = f ( r { 1 } ( i ) , . . . , r { k } (k) Q := Op(k)(A) ∩ PolQ, (For i F comma ) ) S comma ( U i subset\ in equal\ Opr u l opene {3emPol parenthesis}{0.4 pt } Aˆ closing{ m } parenthesis) . $ and Q subset equal Rel open parenthesis A closing parenthesis weFor define $ F , S , U \subseteq $ OpEnd $(Q := (1) APolQ, )$ and $Q \subseteq $ Rel $( A )$ wedefine Pol Q : = open brace f in Op open parenthesis A closingSPol parenthesisQ := S ∩ Pol barQ, every rho in Q i s invariant for f closing brace comma \ centerlineLine 1 open{ parenthesisPol $ Q k closing : =parenthesis\{ subf Pol\ in Q :$ = Op Op to the$ ( power A of open ) parenthesis\mid $ kevery closing parenthesis$ \varrho open\ parenthesisin Q$ A closing i s invariant for Inv F := {% ∈ Rel (A) | % i s invariant for every f ∈ F }, parenthesis$ f \} cap, Pol $ Q} comma Line 2 End Q : = open parenthesis 1 closing parenthesis sub Pol Q comma Line 3 S sub Pol Q : = S cap Pol Q comma Inv F : = open brace rho in Rel open parenthesis A closing parenthesis bar rho i s invariant for every f in F closing brace comma U F := InvU ∩ InvF. \ [ \Ubegin sub Inv{ a Fl i g : n = e Invd } U( cap k Inv F ) period{ Pol } QInv : = Op ˆ{ ( k ) } (A) \cap Pol Q , \\ EndNoteNote that Q that the : the notation =notation Pol ( stands Pol 1 stands for ) polymorphisms{ Pol for polymorphisms} Q, not for\\ polynomials not for polynomials ! ! S22 period{ .Pol 2 .2 period} Q Galois .. Galois : = .. connections S \cap .. derivedPol derived .. from Q .. , Pol\ fromend minus{ a l Inv i g n Pol period e d }\ ]− ..Inv The operators . The .. Pol operators and Inv form a GaloisPol and connection Inv form between a Galois sets of connection mappings and between sets of finitary sets of relations mappings on a and base setsset .. of A finitaryperiod relations on Froma base this set one canA. derive the Galois connections open parenthesis k closing parenthesis sub Pol minus Inv and End minus Inv or comma more \ centerline { Inv $ F : = \{\varrho \ in $ Rel $ ( A ) \mid \varrho $ i s invariant for every generalFrom comma this one can derive the Galois connections (k)Pol− Inv and End − Inv or , more general , $ fLine\ 1in S subF Pol minus\} to, the $ power} of U Inv period Line 2 The For Galois F subset equal Op sub open parenthesis A closing parenthesis comma to the power of closed subsets H subset equal Op of 1 open parenthesis closing parenthesis open parenthesis sub A subU closing parenthesis to the power\ [U of{ OpInv sub} andF to the : power = of Inv open Uparenthesis\cap A closingInv parenthesis F . \ ] and Q sub subset equal of Rel RelSPol sub− openInv. parenthesis A closing parenthesis to theclosed power of open parenthesisOp A(A closing) parenthesis sub(A) with are finitewell− toknown the powerseee of.g. well[9, hyphen1.2.1, known.2.3]). A we open parenthesis TheForGaloisF ⊆Op(A),subsetsH⊆Opof1()(A ) andandQ⊆ofRelRel(A)witharefinite Awe(have(for infinite 1A some have to the power of see e sub open parenthesis for to the power of period g period to the power of open square bracket 9 sub infinite to the power \noindent Note that the notation Pol stands for polymorphisms not for polynomials ! of commamodifications to the power are of necessary 1 period 2 period , [ 6 1 ]comma , [ 17 A ] ) to : the power of period 2 sub some to the power of period 3 closing square bracket closing parenthesis period F = Pol Inv F ⇐⇒ F is a clone , \noindentmodifications2 . are 2 necessary . \quad commaGalois .. open\quad squareconnections bracket 6 closing\quad squarederived bracket comma\quad ..from open\ squarequad bracketPol $7 closing− $ square Inv . bracket\quad closingThe operators \quad Pol and Inv form a Galois connectionF = between (k)Pol Inv setsF of mappings⇐⇒ F is and a Menger sets ofalgebra finitary on A of relations order k, on a base set \quad $ A . $ parenthesis : H = End Inv H ⇐⇒ H is a transformation monoid , F = .. Pol Inv F arrowdblleft-arrowdblrightQ = Inv Pol FQ is a clone⇐⇒ commaQ i s a relational algebra , \noindentF = openFrom parenthesis this k oneclosing can parenthesis derive sub the Pol Galois Inv F arrowdblleft-arrowdblright connections $ ( F k is a ) Menger{ Pol algebra} − on$ A of Inv order and k comma End $ − $ Inv or , more general , Q = Inv (k)PolQ ⇐⇒ Q i s a k− locally closed relational algebra , H = .. End Inv H arrowdblleft-arrowdblright H is a transformation monoid comma \ [ \ begin { a l i g n e d } S Q { =Pol Inv} End− ˆ{ QU } ⇐⇒InvQ i . s\\ a weak Krasner algebra . QAll = .. necessary Inv Pol Q notionsarrowdblleft-arrowdblright will be defined belowQ i s a relational. In general algebra , comma for F, S, U ⊆ Op (A) and finite A we TheQ = ..{ InvFor open} parenthesisGalois k{ closingF \ parenthesissubseteq subOp Pol Q}ˆ arrowdblleft-arrowdblright{ c l o s e d } { (A), Q i s a k hyphen} s u blocally s e t s closed{ H relational\subseteq algebra commaOp } o f { 1 { ( } )( } { A }ˆ{ Op } { ) }ˆ{ (A) } { and } and{ Q } {\subseteq } o f Rel { Rel }ˆ{ ( Q = .. Inv End Q arrowdblleft-arrowdblright Q i s a weak Krasner algebrahave period : A)All necessary} { (A) notions will} be{ with defined} beloware period f i .... n i In t e general ˆ{ w ecomma l l − for Fknown comma} S commaA we U subset ( { equalhave Op} openˆ{ see parenthesis e }ˆ A{ closing. SPolUInvF = S ∩ clone(F ∪ U). (2.2.1) parenthesisg . } { and( finite f o rA we}ˆ{ [ 9 }ˆ{ , } { in f i n i t e }ˆ{ 1 . 2 . 1 , } 1{ A }ˆ{ . 2 }ˆ{ . 3 ] ). } { some }\end{ a l i g n e d }\ ] LineIn 1fact have,S : Line∩ clone 2 S sub(F Pol∪ U) sub = InvS∩ FPol = S Inv cap clone(F ∪ openU) = parenthesisS∩ Pol ( F Inv cup UF closing∩ Inv parenthesisU) = SPol periodUInvF. openFor parenthesis 2 period 2 period 1 closingan parenthesis arbitrary S there i s no general procedure how to characterize the Incorresponding fact comma S cap Galois clone open closed parenthesis sets of relations F cup U closing . However parenthesis , = Sif cap we Pol assume Inv open that parenthesisS i s a F clone cup U closing parenthesis = S cap Pol\noindent openthen parenthesis theremodifications exists Inv aF cap set Inv areQ0 U closing necessary⊆ Rel parenthesis(A) , such\quad = that S sub[ 6 PolS = ] UPol , sub\quadQ Inv0 F ([ periode 7 .g ] . ) one : can take Q0 = Inv ForS) .. anand .. arbitrary for any such .. S ..Q there0 .. i s .. no .. general .. procedure .. how .. to .. characterize .. the .. corresponding \ centerlineGaloiswe have closed for{ sets$ finite F of relations =A( $see\ periodquad [ 8 , Thm ..Pol However . Inv 3 . comma 2 $ ] F ) : ..\ ifi we f f assumeF$ that is .. a S i clone s a clone , then} there exists a set Q 0 subset equal Rel open parenthesis A closing parenthesis .. such that S = Pol Q 0 open parenthesis e period g period one can take Q 0 = Inv\ centerline S closing parenthesis{ $ F .. = and ( for any k such ) Q{ 0 Pol }$ Inv $ F \ i f f F $ is a Menger algebra on $A$ of order U S Q = InvU ∩ [Q0 ∪ Q]RA (2.2.2) $ kwe have , $ for} finite A open parenthesis see open square bracket 8 comma Thm period 3 period 2 closingInv Pol square bracket closing parenthesis : notationbelow). Recall is on A Line( 1 U sub Inv S sub Pol Q =that Invk U capa opencloneOp( squarek)(on bracketA),A Q 0a cup.composition Q closing square bracket RAclosedsetofby open parenthesisoperations 2 period 2 period 2 closing containing allprojections ie ∈ k∈ N Theclonegenerated a set F ⊆ Op(A) parenthesis\ centerline Line{ 2$ open H parenthesis = $ \quad containingEnd Inv to the $ power H of\ i notation f f H below $ is sub a all transformation proj ections to the monoid power of closing , } parenthesis period to the powerwill of Recall be denoted that i e to by the clone power(F of). kNote in a clone that Op the openk− parenthesisary operations k closingF parenthesis(k) of a clone open parenthesisF ⊆ Op on(A A) closing parenthesis comma A k\ centerline in to the power{ of$ Qi s a N = period $ \quad compositionInv Pol The clone $ Q generated\ i f f closedQ$ set iof sby a sub relational a operations s algebra et F subset , equal} Op sub open parenthesis A to the power of on sub closing parenthesis to the power of A \ centerlinewill be .... denoted{ $ Q by = clone $ open\quad parenthesisInv $ F ( closing k parenthesis ) { Pol period} Q .... Note\ i f thatf ....Q $ the ....i s k a hyphen $ k ary− operations$ locally .... F to closed the power relational algebra , } of open parenthesis k closing parenthesis .... of a clone .... F subset equal .... Op open parenthesis A closing parenthesis \ centerline { $ Q = $ \quad Inv End $ Q \ i f f Q $ i s a weak Krasner algebra . }

\noindent All necessary notions will be defined below . \ h f i l l Ingeneral,for $F , S , U \subseteq $ Op $( A )$ andfinite $A$ we

\ [ \ begin { a l i g n e d } have : \\ S { Pol } U { Inv } F = S \cap c lo n e ( F \cup U ) . ( 2 . 2 . 1 ) \end{ a l i g n e d }\ ]

\noindent In f a c t $ , S \cap $ cl o ne $ ( F \cup U ) = S \cap $ Pol Inv $ ( F \cup U ) = S \cap $ Pol ( Inv $ F \cap $ Inv $ U ) = S { Pol } U { Inv } F . $ For \quad an \quad a r b i t r a r y \quad $ S $ \quad ther e \quad i s \quad no \quad g e n e r a l \quad procedure \quad how \quad to \quad characterize \quad the \quad corresponding Galois closed sets of relations . \quad However , \quad if we assume that \quad $ S $ i s a clone then there exists a s e t $ Q 0 \subseteq $ Rel $ ( A ) $ \quad suchthat $S =$ Pol $Q 0 ($ e.g.onecantake $Q 0 =$ Inv $S )$ \quad and for any such $Q 0 $

\noindent wehave for finite $A ($ see [ 8 ,Thm. 3 . 2 ] ) :

\ [ \ begin { a l i g n e d } U { Inv } S { Pol } Q = Inv U \cap [ Q 0 \cup Q ] RA ( 2 . 2 . 2 ) \\ ( { c o n t a i n i n g }ˆ{ notation below }ˆ{ ). } { all proj ections }ˆ{ R e c a l l } that { i { e }ˆ{ k } \ in } a cl o ne { Op } ( k ) ( on{ A } ),A { k \ in }ˆ{ i s } a{ N } . composition { The clone generated } closed set of { by } { a } o p e r a t i o n s { s et F \subseteq Op }ˆ{ on } { ( A }ˆ{ A } { ) }\end{ a l i g n e d }\ ]

\noindent w i l l be \ h f i l l denotedbyclone $( F ) .$ \ h f i l l Note that \ h f i l l the \ h f i l l $ k − $ ary operations \ h f i l l $ F ˆ{ ( k ) }$ \ h f i l l o f a cl o ne \ h f i l l $ F \subseteq $ \ h f i l l Op $ ( A ) $ 374 .. R period P o-dieresis schel et al period : .. Completeness Citeria and Invariants period period period \noindentalways form374 a open\quad parenthesisR . P concrete $ \ddot closing{o} parenthesis$ schel Menger et al algebra . : \ angbracketleftquad Completeness F to the power Citeria of open and parenthesis Invariants k closing . parenthesis . . semicolon kappa sub A right angbracket comma i period e period .. a subset of Op to the power of open parenthesis k closing parenthesis open 374 R . P o¨ schel et al . : Completeness Citeria and Invariants . . . parenthesis\noindent Aalways closing parenthesis form a ( which concrete i s closed ) Menger algebra $ \ langle F ˆ{ ( k ) } ; \kappa { A }\rangle always form a ( concrete ) Menger algebra hF (k); κ i, i . e . a subset of Op(k)(A) which i s , $under i kappa. e . sub\quad A opena parenthesis subset of see $ 1 period Op ˆ{ 4( closing k parenthesis )A } ( period A )$ which i s closed closed under κ ( see 1 . 4 ) . The notion of relational clone is les s common . For finite underThe notion $ \kappa of relationalA{ A clone} is( les $ s common see 1 . period 4 ) .. . For finite A it coincides with the following TheA notionit coincides of relational with the following clone is notion les of s commona relational . \quad algebraFor finite ( in the sense$ A $ of ite . coincides g . [ 9 with the following notion] ; of however a relational it algebra i s different .. open parenthesisfrom Tarski in the’ s sense relation of e algebraperiod g periodof binary .. open relations square bracket) : A 9 closingre square bracket semicolon .. howevernotion it ..of i s a different relational from Tarski algebra quoteright\quad s ( in the sense of e . g . \quad [ 9 ] ; \quad however i t \quad i s different from Tarski ’ s relationlati onal algebra algebra ofQ binaryis a set relations of relations ) : in\ Relquad(A) \whichquad isre closed lati under onal algebra the following $ Q ( $ set\ -quad is a set of relations in Rel relationtheoretical algebra ) ofoperations binary relations : closing parenthesis : .. A .. re lati onal algebra Q .. is a set of relations in Rel open parenthesis A closing parenthesis$ ( A ) $ • ∆ ( nullary operation : Q must contain the diagonal ( or equality ) whichwhich is closed closedA under under the following the following open parenthesis ( set set− hyphentheoretical theoretical ) closing operations parenthesis : operations : relation ∆ := bullet Capital DeltaA sub A open parenthesis nullary .. operation : Q .. must .. contain .. the .. diagonal .. open parenthesis or .. equality closing parenthesis\ hspace ∗{\ .. relationf i l l } $ ..\ Capitalbullet Delta\ subDelta A : = { A } ( $ n u l l a r y \quad operation $ : Q$ \quad must \quad contain \quad the \quad diagonal \quad ( or \quad e q u a l i t y ) \quad r e l a t i o n \quad {(a, a) | a ∈ A}), $ \openDelta brace{ openA } parenthesis: = $ a comma a closing parenthesis bar a in A closing brace closing parenthesis comma bullet• ∩ cap( inters open parenthesis ection of inters relations ection of of the relations same of arity the same ) , • arity × closing( product parenthesis: for commam− ary % and s− \ [ bullet\{ary timesσ(let open a parenthesis , a ) product\mid : .. fora m hyphen\ in aryA rho\} and s), hyphen ary\ ] sigma let rho times sigma = open brace open parenthesis a sub 1 comma period period period comma a sub m comma b sub 1 comma period period period m+s comma b sub s closing parenthesis% × σ = {(a1 in, ..., A a tom, the b1, ...,power bs) ∈ ofA m plus| s( bara1, ...,open am parenthesis) ∈ %, (b1, a ..., sub bs) 1∈ commaσ}), period period period comma a sub m closing parenthesis\noindent in rho$ \ commabullet open\ parenthesiscap ( $ b sub inters 1 comma ection period of period relations period comma of the b sub same s closing arity parenthesis ) , in sigma closing brace closing • pr ( projection onto a subset I of coordinates : for m− ary % and I = {i , ..., i } with parenthesis$ \ bullet commaI \times ( $ product : \quad f o r $ m − $ ary $ \varrho $1 andt $ s − $ ary $ \sigma $ l e tbullet pr sub I open parenthesis projection onto a subset I of coordinates : .. for m hyphen ary rho and I = open brace i sub 1 comma period 1 ≤ i1 < i2 < ··· < it ≤ mwedefine period period comma i sub t closing brace with pr (%) := {(a , ..., a ) | ∃a (j ∈ {1, ..., m}\ I):(a , ..., a ) ∈ %}), \ [ Line\varrho 1 1 less or\times equal i subI\ 1sigma less i subi1= 2 lessi\{t times times(j a times{ less1 } i sub, t less .1 or .equalm . m we , define a Line{ m 2 pr} sub, I open b { parenthesis1 } ,.. rho closing . , b { s } ) \ in A ˆ{ m + s }\mid ( a { 1 } , . . . , a { m } ) \ in \varrho parenthesis : = open• π brace( permutation open parenthesis aof sub coordinates i sub 1 comma : period for m period− ary period% and comma a permutation a sub i sub t closingα of parenthesis bar exists a sub j , ( b { 1 } α , . . . , b { s } ) \ in \sigma \} ), \ ] open{ parenthesis1, ..., m} let j in open brace 1 comma period period period comma m closing brace backslash I closing parenthesis : open parenthesis a sub 1 comma period period period comma a sub m closing parenthesis in rho closing brace closing parenthesis comma

bullet pi sub alpha open parenthesisπα permutation(%) := {(aα(1) .., ..., of coordinatesaα(m)) | (a :1 .., ..., for am m) hyphen∈ %}). ary .. rho and a permutation alpha .. of open brace 1 comma period\ hspace period∗{\ periodf i l l } comma$ \ bullet m closingpr brace{ ..I let} ( $ projection onto a subset $ I $ of coordinates : \quad f o r $ m ,Q − $piIf sub ary , in alpha addition $ open\varrho parenthesisi$ s also and rho closed closing $ I with parenthesis = respect\{ : =i to open{ 1 brace} open, parenthesis . . . a sub , alpha i open{ t parenthesis}\} $ 1 closing with parenthesis comma • ∪ ( Q period periodunion periodof comma relations a sub of alpha the opensame parenthesis arity ) , then m closingi parenthesis s called a closingweak Krasner parenthesis algebra bar open. parenthesis a sub 1 comma period k Q k− k− Q = Q period\ [ \ beginFor period a{ positivea lcomma i g n e d } ainteger sub1 m\ closingleqa relational parenthesisi { 1 algebra} in rho< closingi is{ bracecalled2 } closing< local parenthesis\cdot ly clos\ periodcdot ed if \cdotLOC < i { t }\ leq m we deIf fwhere commai ne \\ in addition comma Q i s also closed with respect to prbullet{ cupI } open( parenthesis\varrho union) of relations : = of the\{ same( arity a closing{ i parenthesis{ 1 }} comma, . . . , a { i { t }} ) \mid k − LOCQ := {% ∈ Rel(A) | ∀r , ..., r ∈ %∃σ ∈ Q : {r , ..., r } ⊆ σ ⊆ %}. \ existsthen Q i sa called{ aj weak} ( Krasner j algebra\ in period\{ 11 ,k . . .1 ,k m \}\setminus I ) : ( a { 1 } , . . . , a { m } ) \ in \varrho \} ), \end{ a l i g n e d }\ ] ForThe a positive relational integer algebra k a relational ( weak algebra Krasner Q is algebra called k ) hyphen generated local byly clos a set ed ifQ k⊆ hyphenRel ( LOCA) will Q = be Q denoted where kby hyphen[Q] RA LOC Q([ :Q =] WKA open brace, resp rho . )in . Rel open parenthesis A closing parenthesis bar forall r sub 1 comma period period period comma r sub k in rho exists sigma in Q :. open⊆ braceAk r sub 1 comma period period period−algebra comma r sub k closing brace subset equal sigma subset equal rho closing 2.3.andLemmaonlyifF F=OpA A)is∩anoperationclone(F ∪ U)K. with s e lector system over brace\ hspace period∗{\ f i l l } $ \ bullet k()(\ pi {\alpha } ( $ permutation \quad of coordinates : \quad f o r $ m − $ ary \quad fˆ $ \ThevarrhoA relational= h$A; U andalgebrai if a open permutation Proof parenthesis . weak $ Let\alpha KrasnerhF ;($ algebrai)i \∈quadI, closing κAio fbe parenthesis $ an\{ operation1 generated ,K− by .algebra a set . Q subset .over , equalA m= Rel open\} $ parenthesis\quad Al eclosing t parenthesishA;(fi) willi ∈ I bei,U denoted:= {fi | i ∈ I}. \ [ by\ pi open{\ squarealpha bracket} Q( closing\varrho square bracket) :RA .. = open\{ parenthesis( a open{\ squarealpha bracket( Q closing 1 ) square} , bracket . WKA . comma . , resp a period{\alpha ( m ) } ) \mid ( a { 1 } , . . . , a { m } ) \ in \varrho \} ). \ ] closing parenthesis period1.5 itcanbe characterized hF ; Ai of full Byh Definition k respecttoasasubalgebraeach ..κ ) the forMenger algebra 2 periodA 3 periodA and;κAi Lemmawhichisclosed onlywith sub if F to the power of period F =fi,ˆ Opi.e. tofiˆ the(g1, power., gni of subset∈F equal Ag k1,...,gn openi parenthesis∈F, closing parenthesis open parenthesis sub A closing parenthesis to the power of A to the power of k is cap an operation clone open parenthesis Fi cup∈ I. U closing parenthesis K\noindent period to theIf power , in of addition hyphen algebra $ ,with Q$ s e lector i ssystem also .. closed over A underbar with respect = angbracketleft to A semicolon U right angbracket .. if ProofBut period these .. properties Let angbracketleft can be F reformulated semicolon open: parenthesisfiˆ (g1, ..., to gn the) is power j ust of a f-circumflex composition i closing of the parenthesis opera - i t in I comma kappa sub A right \noindent $ \ bullet \cup ( $ union of relationsi of the same arity ) , angbracketions fi, be g an1, ..., operation gn ( see K 1 hyphen . 4 ) and algebra thus over belongs A underbar to the = clone angbracketleft generated A by semicolonF and openfi, and parenthesis so also f i to closing parenthesis i in I right then $ Q $ i si called a weak Krasner algebra . angbracketclone comma(F ∪ U U). : =Conversely open brace f, i every bar i ink− I closingary operation brace period in clone (F ∪ U) must belong to F. Line 1 By angbracketleft sub A Definition A to the power of k semicolon kappa subk A right angbracket which to the power of 1 period 5 sub i s To conlude the proof we mention that the proj ections ke1 , ..., ke belong to every clone to\noindent theand power therefore ofFor it sub a only positive closed algebras to the integer power with of selector can $ k be $system sub a with relational are to the characterized power algebra of characterized by $Q$ the condition sub is respect called of to the as $ a subalgebrak − $ each local sub lyf-circumflex clos ed i if $ k − $ LOC $Q = Q$ where commalemma i period. e line period− line circumflex-f i open parenthesis g sub 1 comma to the power of angbracketleft F sub period to the power of semicolon sub period kappa period comma to the power of A sub g n to the power of right angbracket sub i to the power of closing parenthesis to the power of of the\ [ in k F to− theLOC power of Q full for : Menger = \{\ g 1 commavarrho period period\ in periodRel comma ( g A n sub ) i algebra\mid in F\ commaf o r a l l Liner 2 i in{ I1 period} ,... ,But r these{ k properties}\ in can\ bevarrho reformulated\ exists : f-circumflex\sigma i open parenthesis\ in Q: g 1 comma\{ periodr { period1 } period, comma . . g nsub . i closing, r parenthesis{ k } is\}\ j ust asubseteq composition of\sigma the opera hyphen\subseteq \varrho \} . \ ] t ions f i comma g 1 comma period period period comma g n sub i open parenthesis see 1 period 4 closing parenthesis and thus belongs to the clone generated by F and f i comma and so also \noindentto clone openThe parenthesis relational F cup algebra U closing ( parenthesis weak Krasner period .. algebra Conversely ) comma generated every k by hyphen a set ary operation $ Q \ insubseteq clone open$ parenthesis Rel $ F ( cup AU closing ) $ parenthesis will be must denoted belong to F period To .... conlude .... the .... proof .... we .... mention .... that .... the .... proj ections .... k e sub 1 comma period period period comma k e to the power\noindent of k ....by belong $ ....[ to Q .... every ] $ .... RA clone\quad $( [ Q ]$ WKA,resp.). and therefore only algebras with selector system are characterized by the condition of the \noindentlemma period$ line-line2 . 3 . { and } Lemma{ only }ˆ{ . } { i f F } F { = Op }ˆ{\subseteq } A { k { ( } )( }ˆ{ A ˆ{ k }} { A) } i s {\cap } an operation { c lo n e } (F \cup U)K { . }ˆ{ − algebra }$ with s e lector system \quad over $ A {\underline {\}} = \ langle A;U \rangle $ \quad i f Proof . \quad Let $ \ langle F ; ( ˆ{\hat{ f }} i ) i \ in I, \kappa { A }\rangle $ be an operation $ K − $ algebra over $A {\underline {\}} = \ langle A ; ( f i ) i \ in I \rangle , U : = \{ f i \mid i \ in I \} . $

\ [ \ begin { a l i g n e d }By{\ langle } { A } De f i n i t i o n { A ˆ{ k } ; \kappa { A }\rangle which }ˆ{ 1 . 5 } { i s }ˆ{ i t } { c l o s e d }ˆ{ can be } { with }ˆ{ characterized } { r e s p e c t to } as a subalgebra { each } {\hat{ f } i , i . e . \hat{ f } i ( g }ˆ{\ langle F } { 1 , }ˆ{ ; } { . } { . }\kappa { ., }ˆ{ A }ˆ{\rangle } { g n }ˆ{ o f } { i ˆ{ ) }} the {\ in F }ˆ{ f u l l } f o r Menger { g 1 , . . . , g n { i }} algebra {\ in F, }\\ i \ in I. \end{ a l i g n e d }\ ]

\noindent But these properties can be reformulated $ : \hat{ f } i ( g 1 , . . . , g n { i } ) $ is j ust a composition of the opera − tions$f i , g 1 , . . . , gn { i } ( $ see 1 . 4 ) and thus belongs to the clone generated by $F$ and $f i ,$ andsoalso to c lo n e $ ( F \cup U ) . $ \quad Conversely , every $ k − $ ary operation in clone $ ( F \cup U )$ mustbelongto $F .$

\noindent To \ h f i l l conlude \ h f i l l the \ h f i l l proof \ h f i l l we \ h f i l l mention \ h f i l l that \ h f i l l the \ h f i l l proj ections \ h f i l l $ k { e { 1 }} , . . . , k { e }ˆ{ k }$ \ h f i l l belong \ h f i l l to \ h f i l l every \ h f i l l c lo n e

\noindent and therefore only algebras with selector system are characterized by the condition of the

\noindent lemma $ . line −l i n e $ R period P o-dieresis schel et al period : .. Completeness Citeria and Invariants period period period .. 375 \ hspaceNote that∗{\ byf i l 2 l period}R . P2 every $ \ Mengerddot{o algebra} $ schel F subset et equal al A . to : the\quad powerCompleteness of A to the power Citeria of k can beand characterized Invariants by invariant . . . \ relationsquad 375 : F = open parenthesis k closing parenthesis sub Pol Inv F period .. Therefore it makes sense to ask how to characterize those Menger algebras \noindent Note that byR.P 2 . 2o¨ everyschel et Menger al . : algebra Completeness $ F Citeria\subseteq and InvariantsA ˆ{ A ˆ .{ .k .}} 375$ can be characterized by invariant relations : which are at the same t ime operation K hyphen algebrasAk with respect to a given algebra A underbar = angbracketleft A semicolon U right angbracketNote that in by 2 . 2 every Menger algebra F ⊆ A can be characterized by invariant relations : \noindentKF period= (k) ..Pol$ FollowingInv FF. = a generalTherefore ( k approach ) it makes{ describedPol sense}$ already Inv to ask in $ open F how square . to $ characterize bracket\quad 7Therefore comma those 1 5 Menger period it makes 1 algebras comma sense page to 84 closing ask how square to bracket characterize those Menger algebras wewhich getwhich those are are Menger at at the the same t ime ime operation operationK− $algebras K − $ with algebras respect to with a given respect algebra to aA given= hA; U algebrai ∈ $ A {\underline {\}} = algebrasK\.langleFollowing as GaloisA;U closeda general elements approach\rangle of the restricteddescribed\ in Galois$ already connection in [ 7 , open 1 5 . parenthesis 1 , page 84 k closing ] we get parenthesis those Menger sub Pol minus to the power of U Inv U $algebras K . $ as\ Galoisquad Following closed elements a general of the restrictedapproach Galois described connection already(k) inPol− [ 7Inv , . 1 5 . 1 , page 84 ] we get those Menger period k 22 period . 4 4. period Theorem .. Theorem . periodLet .. Let AA underbar= hA; Ui = angbracketleftbe s ome finite A semicolon algebra U in rightK. angbracketThen ..F be⊆ sA omeA finite algebra in K period .. Then\noindentis F an subset operationalgebras equal A toK− as thealgebra Galoispower of over A closed to theA power elementsif and of only k .. of is if an the operation restricted Galois connection $ ( k ) { Pol } − ˆ{ U }$ InvK .hyphen algebra over A underbar if and only if F = open parenthesis k closing parenthesis sub PolF U= sub (k)Pol InvUInv FF. period \noindent 2 . 4 . \quad Theorem . \quad Let $ A {\underline {\}} = \ langle A;U \rangle $ \quad be s ome finite algebra in In particular open parenthesis for k = 1 closing parenthesisA comma .... a s et H subset equal A to the power of A .... is a transformation K hyphen $ KIn particular. $ \quad ( forThenk = $ 1) F, a s\subseteq et H ⊆ A isA a ˆ{ transformationA ˆ{ k }}$ \quadK− algebrais an over operationA if and only algebraif over A underbar if and only if $Line K 1 H− =$ End algebra U sub Inv over H period $A Line{\ 2underline Proof Now take{\}} to the$ power if and of period only By if Lemma into account to the power of 2 period 3 comma F i s that an A operation k A cap clone open parenthesis F to the power of K sub cup to the power of hyphen algebra sub U closing parenthesis = \ [ F = ( k ) { Pol } U { Inv } F. \ ] open parenthesis closing parenthesis k sub Pol to the power of i f to the power of and only Inv U F if open parenthesisH = sub End byUInv FH. sub 2 sub period 2 period to the. powerBy of = A2.3 1, closing parenthesis sub periodK to the− poweralgebra of A toif and the power of k cap= cloneAk open parenthesis F cup U line-line closing P roof Lemma F isthatanAoperation () only if( F .2.A . ∩ clone(F ∪ Uline−line). parenthesisNowtake period intoaccount kA∩clone(F ∪ U)= k Pol InvU F by 2 1) \noindent2 period 5Inparticular(for period .... Remark period .... $k Let .... = us 1 consider ) the ,$ case\ ofh transformationf i l l a s et K $ hyphen H \ algebrassubseteq .... openA ˆ parenthesis{ A }$ k\ h = f 1i l closing l is a transformation $ K 2 .− 5$ . algebra Remark over . $ALet{\ usunderline consider{\}} the case$ of if transformation and only if K− algebras (k = 1). parenthesisAccording period .... According to 2 period 2 we have open square bracket Q closing square bracket WKA = Inv End Q open parenthesis for finite A closing parenthesis period to 2 . 2 we have [Q] WKA = Inv End Q( for finite A). Consequently , from 2 . 4 we get ..\ [ Consequently\ begin { a l i g comma n e d } fromH 2 = period End 4 we U get { Inv } H. \\ Inv H = [U Inv H] WKA ProofInv H ={ openNow square take bracket}ˆ{ to. the power By } of ULemma Inv H closing{ into square account bracket WKA}ˆ{ 2 . 3 , } F i s { that } an{ A } for a transformation K− algebra H over hA; Ui. The U− invariants U H always form a operationfor .... a ....{ transformationk { A }\ ....cap K hyphenc lo n algebra e ( .... H F ....}ˆ{ overK ....}ˆ{ angbracketleft − algebra A semicolon} Inv{\cup U right} { angbracketU )= period} ....() The ....{ Uk hyphen}ˆ{ i f } relational{ Pol }ˆ{ algebraand } (only s ince{ itInv is the{ U intersection} F } i f { of( relational} { by } algebrasF { 2 , see}ˆ{ 2= .} 1{ and. 2 2 . 2 ) . } A { 1 ) }ˆ{ A ˆ{ k }} { . } invariants. .... U sub Inv H .... always .... form .... a It \caprelationalc lo ....n e algebra ( .... F open\cup parenthesisU { s incel i n e ....−l it i n .... e is} ....). the ....\ intersectionend{ a l i g n .... e d of}\ relational] .... algebras comma .... see .... 2 period 1 .... follows that every invariant relation % ∈ Inv H is a union of U− invariant relations . and ....3 . 2 period Completeness 2 closing parenthesis period .... It follows that every invariant relation rho in Inv H is a union of U hyphen invariant relations period In this s ection we want to characterize generating sets F of the full operation K− algebras over \noindent3 period ..2 Completeness . 5 . \ h f i l l Remark . \ h f i l l Let \ h f i l l us consider the case of transformation $ K − $ a l g e b r a s \ h f i l l $ (In this k s ection = we1 want ) to . characterize $ \ h f i l generating l According sets F of the full operation K hyphen algebras over k of angbracketleftAk ˆ A to the powergeneralizethisproblemusing of A to the power of k semicolonAk U-circumflex sub comma kappa sub A right angbracketsetSofoperationsinsteadof generatedA byA . fromThusweare \noindentof hA to2.2wehave; U,κAi generatedbysomegivenfinitealgebra $[ Q ]$A =equals WKAA $=$∩clone( InvEndF ∪U) = $Q (k)Pol ($UInvF. forWe finite slightlyF $Aan; )Ui∈K. .$ItfollowsfromTheorem2\quad Consequently.4thatthesubalgebra , from 2 . 4 we get generalize this problem using to some given finite algebra A underbar = equals A to the power of A to the power of k caphAarbitrary clone open parenthesis F cup U closing parenthesis = open parenthesis k closing parenthesis sub Pol U sub Inv F period We s l ightly from set S of operations instead of A \ centerline { Inv $ H = [ ˆ{ U }$ Inv $H ]$ WKA } 00 to thefaced power with of A a to typical the power completeness of k period Thus problem we are to F from “ does an toS angbracketleft∩ clone (F ∪ AU) sub = S arbitrary hold and to thewe power shall of semicolon to the power of U right angbracket in K period It fo llows from Theorem 2 period 4 that the subalgebra ? \noindentfacedconnect withfit oa r typical with\ h f imethods completenessl l a \ h knownf i l l problemtransformation from .... clone quotedblleft theory\ h f does . i l l 3 S .$ cap K 1 clone .− $ open Notions algebra parenthesis and\ h F f i cupNotationl l U$ closing H $ . \ parenthesish f i l l over = S\ holdRowh f i l l 1$ \ langle A;UquotedblrightLet S, U Row⊆\Oprangle 2 ?( holdA) be we. arbitrary shall $ \ h f i fixed l l The s ets\ h of f i loperations l $ U − .$ For invariants \ h f i l l $ U { Inv } H $ \ h f i l l always \ h f i l l form \ h f i l l a connect it with methods known from clone theory period \noindent3 period 1r period e l a t i ..o n Notions a l \ h andf i l l Notationalgebra period\ h f .. i l Let l ( S scomma i n cF e U⊆\ subsethSset f i l l equali t \ Oph f open i l l parenthesisi s \ h f i l l Athe closing\ h parenthesis f i l l intersection be arbitrary\ fixedh f i sl l of relational \ h f i l l a l g e b r a s , \ h f i l l see \ h f i l l 2 . 1 \ h f i l l and \ h f i l l 2 . 2 ) . \ h f i l l I t ets of operations period .. For \noindent follows that every invariant relation $ \varrho \ in $ Inv $H$ is aunion of $U − $ invariant relations . Line 1 F subset equal S set Line 2 hline LineF 3:= FS : =∩ clone( S capF clone∪ U) open (3. parenthesis1.1) F cup U closing parenthesis open parenthesis 3 period 1 period 1 closing parenthesis Line 4 hline \noindentopen parenthesis3 . \quad see FigCompleteness period .. 1 closing parenthesis period .. We shall say that .. F subset equal S i s a U hyphen s et .. open parenthesis with( respect see Fig to ... S closing 1 ) . parenthesis We shall .. ifsay F = that F periodF ..⊆ A ..S Ui hyphen s a U set− s et ( with respect to S) if \noindentNF negationslash-equal= F.InA thisU− s Sset ection i s saidN 6= to weS bei swant maximal said to to i bef characterize formaximal every U hypheni f generating for set every P commaU− setsset fromP, N $from subset F $N P of⊂ subsetP the⊆ equalS fullit fo Sit operation fo llows that $ P K= S period− $ algebrasAllows s et F that subset over P equal= S. S i s called U hyphen complete .. open parenthesis or F i s a generating system closing parenthesis if overbar F = S comma i periodA s e et periodF ⊆ togetherS i s called with U− complete ( or F i s a generating system ) if F = S, i . e . together \ [U owith fit generates\Ulangleit generates all elementsA ˆ all{ ofA elements S ˆ period{ k }} of ;S. \hat{U} { , }\kappa { A }\rangle generated by ˆ{ g e n e r a l i z e thishline problem using } { some given fi nite algebra A {\underline {\}} = } equals A ˆ{ A ˆ{ k }} \cap3 periodc lo 2 n period e ( .. Lemma F \ periodcup ..U 1 period ) .. = The operator( k F ) mapsto-arrowright{ Pol } U F{ isInv an algebraic} F c . losure We operator s on l P open ightlyˆ parenthesis{ s eS t closingS of parenthesis operations period instead of Aˆ{ A ˆ{ k }} . Thus we are } { F ˆ{ an } {\ langle A }ˆ{ ; } { a r b i t r a r y }ˆ{ U 3 . 2 . Lemma . 1 . The operator F 7→ F is an algebraic c losure operator on P(S). \rangle2 period ..\ Ifin C is aK. clon e containing} It Ufo comma llows .. then fromS cap C is Theorem a U hyphen 2s et period . 4 that the subalgebra }\ ] 2 . If C is a clon e containing U, then S ∩ C is a U− s et .

\noindent faced with a typical completeness problem \ h f i l l ‘ ‘ does $ S \cap $ cl o ne $ ( F \cup U ) = S \ l e f t . hold \ begin { array }{ c} ’’ \\ ? \end{ array }and\ right . $ we s h a l l

\noindent connect it with methods known from clone theory . 3 . 1 . \quad Notions and Notation . \quad Let $ S , U \subseteq $ Op $ ( A ) $ be arbitrary fixed s ets of operations . \quad For

\ [ \ begin { a l i g n e d } F \subseteq S s e t \\ \ r u l e {3em}{0.4 pt }\\ F : = S \cap c lo n e ( F \cup U ) ( 3 . 1 . 1 ) \\ \ r u l e {3em}{0.4 pt }\end{ a l i g n e d }\ ]

\noindent ( see Fig . \quad 1 ) . \quad We shall say that \quad $ F \subseteq S $ i s a $ U − $ s et \quad ( with respect to \quad $ S ) $ \quad i f $ F = F . $ \quad A \quad $ U − $ s e t $ N \not= S$ i s said to be maximal i f for every $U − $ set $P ,$ from $N \subset P \subseteq S$ itfollowsthat $P = S .$

\noindent A s et $ F \subseteq S$ i s called $U − $ complete \quad ( or $F$ i s a generating system ) if $\ overline {\}{ F } = S ,$ i . e . togetherwith $U$ it generates all elements of $ S . $

\ [ \ r u l e {3em}{0.4 pt }\ ]

\noindent 3 . 2 . \quad Lemma . \quad 1 . \quad The operator $ F \mapsto F $ is an algebraic c losure operator on $ P ( S ) . $ 2 . \quad If $C$ is a clon e containing $U , $ \quad then $ S \cap C $ i s a $ U − $ s et . 376 .. R period P o-dieresis schel et al period : .. Completeness Citeria and Invariants period period period \noindenthline 376 \quad R . P $ \ddot{o} $ schel et al . : \quad Completeness Citeria and Invariants . . . Figure 1 period The closure operator F arrowright-mapsto F and a U hyphen maximal set N 376 R . P o¨ schel et al . : Completeness Citeria and Invariants . . . \ [ 3\ periodr u l e {3em overbar}{0.4 F is pt the}\ least] U hyphen s e t containing F open parenthesis for F subset equal S closing parenthesis period Proof period 1 period overbar F i s a Galois closure for the Galois connection S sub Pol minus to the power of U Inv open parenthesis see 2 period 2 period 1 closing parenthesis comma thus F arrowright-mapsto overbar F \ centerlinei s a closure{ operatorFigureFigure period 1 . 1 The .... . The It closure i s closure also algebraic operator operator s inceF $ it7→ iF sF anand\ intersectionmapsto a U− maximal openF $ parenthesis and set aN $ofU S closing− $ parenthesis maximal with set an algebraic $N$ } closure operator open parenthesis3. F is namely the least F arrowright-mapstoU− s e t containing clone openF parenthesis( for F ⊆ F cupS). U closing parenthesis closing parenthesis period \ centerline { $ 3 . \ overline {\}{ F }$ is the least $U U− $ set containing $F ($ for $F 2Proof period ... Assume1. F Ui subset s a Galois equal closureC period for .. From the Galois open parenthesis connection S capSPol C− closingInv parenthesis ( see 2 . cup2 . U1 )subset , thus equal C it follows immediately that\subseteqF clone7→ openF parenthesisS ) . open $ parenthesis} S cap C closing parenthesis cup U closing parenthesis subset equal C period hlinei s a closure operator . It i s also algebraic s ince it i s an intersection ( of S) with an algebraic \noindentNowclosure comma operatorProof taking $intersection ( . namely 1 ofF . both7→\ overlineclone s ides with(F {\}{∪ SU gives)). 2F S .} cap$ CAssume i subset s a GaloisequalU ⊆ SC. cap closureFrom C period( forS ∩ C the) ∪ U Galois⊆ C it connection $ S { Pol } − ˆhline{followsU }$ immediately Inv( see2.2.1) that clone ((S ,thus∩ C) ∪ U) $F⊆ C. \mapsto \ overline {\}{ F }$ 3 period .... From .... 2 period .... it .... fo llows .... that .... F .... i s .... a .... U hyphen set period .... Let .... H .... be .... a .... U hyphen set ....\noindent such .... thati s .... a F closure subset equal operator H period .....\ Thenh f i l l It i s also algebraic s ince it i s an intersection ( of $ S ) $ withLine an 1 hline algebraic hline Line 2 F subset equal H = H period line-line Now , taking intersection of both s ides with S gives S ∩ C ⊆ S ∩ C. In clone theory there are several completeness criteria via maximal clones which can easily \noindentbe adaptedclosure to .. U hyphen operator completeness ( namely .. open $ parenthesis F \mapsto here the$ maximal cl o ne subclones $ ( F of a clone\cup B willU be ) called ) B hyphen . $ 2maximal . \quad closingAssume parenthesis $ U : \subseteq C . $ \quad From $ ( S \cap C) \cup U \subseteq C $ it3 follows3 period . From 3 period immediately 2 . U it hyphen fo llows that completeness that cloneF criterioni $ s ( a periodU (− set S .... . Suppose Let\capH BbeC) : = a cloneU− openset\cup parenthesissuchU) that SF cup\subseteq U⊆ closingH. parenthesisC . ....$ is a finitely generatedThen clone period \ [ Then\ r u lwe e {3em have}{ : 0.4 pt }\ ] open parenthesis 1 closing parenthesis .. A .. s et F subset equal S is U hyphen complete if and only if for every B hyphen maximal clone M with U subset equal M F ⊆ H = H. line − line \noindentwe have FNow reflexsubset-negationslash , taking intersection M period of both s ides with $ S $ gives $ S \cap C \subseteq S \cap CIn . $clone theory there are several completeness criteria via maximal clones which can easily be adapted to U− completeness ( here the maximal subclones of a clone B will be called B− \ [ \ r u l e {3em}{0.4 pt }\ ]

maximal): \noindent 3 . \ h f i l l From \ h f i l l 2 . \ h f i l l i t \ h f i l l f o l l o w s \ h f i l l that \ h f i l l $ F $ \ h f i l l i s \ h f i l l a \ h f i l l $ U 3.3.− U$− completeness s e t . \ h f i l l Let criterion\ h f i l l . $Suppose H $ \ h f iB l l:=beclone\ h f( iS l l∪ Ua )\ ish f a i l finitely l $ U generated− $ s eclone t \ h f i l l such \ h f i l l that \ h f i l l $ F . \subseteq H . $ \ h f i l l Then Then we have : \ [ \ begin {(a l1 i g ) n e dA}\ r u s l eet{3emF }{⊆ 0.4S is pt }\U− completer u l e {3em if}{ and0.4 only pt }\\ if for every B− maximal clone M F with\subseteqU ⊆ M H = H . l i n e −l i n e \end{ a l i g n e d }\ ] wehaveF reflexsubset − negationslashM. \noindent In clone theory there are several completeness criteria via maximal clones which can easily be adapted to \quad $ U − $ completeness \quad ( here the maximal subclones of a clone $ B $ will be called $ B − $

\ begin { a l i g n ∗} maximal ) : \end{ a l i g n ∗}

\noindent $ 3 . 3 . U − $ completeness criterion . \ h f i l l Suppose $B : =$ clone $( S \cup U ) $ \ h f i l l is a finitely generated clone .

\noindent Then we have :

\ hspace ∗{\ f i l l }( 1 ) \quad A \quad s et $ F \subseteq S $ i s $ U − $ complete if and only if for every $ B − $ maximal clone $M$ with $U \subseteq M $

\ [ we have F reflexsubset −negationslash M . \ ] R period P o-dieresis schel et al period : .. Completeness Citeria and Invariants period period period .. 377 \ hspaceopen parenthesis∗{\ f i l l }R 1 to . the P power $ \ddot of prime{o} closing$ schel parenthesis et al .. . A : s et\quad F subsetCompleteness equal S is U hyphen Citeria complete and if Invariants and only if for . every . . maximal\quad 377 U hyphen s et N we have F negationslash-reflexsubset N period R.P o¨ schel et al . : Completeness Citeria and Invariants . . . 377 \ hspaceThus comma∗{\ f i l for l } the$ completeness ( 1 ˆ{\ problemprime } it becomes) $ \ essentialquad A to s know et the $ F maximal\subseteq U hyphen sS ets $ period i s .. $ There U hyphen− $ complete if and only if for every maximal (10) A s et F ⊆ S is U− complete if and only if for every maximal U− s et N we have $ Ufore comma− $ set before giving $N$ the wehave proof of 3 period $F 3 comma\not\subseteq we shall relateN maximal . $ U hyphen sets to maximal subclones of F 6⊆ N. the clone B = clone open parenthesis S cup U closing parenthesis period Thus , for the completeness problem it becomes essential to know the maximal U− s ets . There \noindentRecall thatThus for a finitely , for generated the completeness clone B there problem are only finitely it becomes many B hyphen essential maximal to clones know the maximal $ U − $ s e t s . \quad There − - fore , before giving the proof of 3 . 3 , we shall relate maximal U− sets to maximal subclones foreand every , before proper givingsubcloneof the B i proof s contained of 3in a . B 3 hyphen , we maximal shall relate clone open maximal parenthesis $ see U e period− $ g sets period to .. open maximal square subclones bracket 9 comma of of the clone B = clone (S ∪ U). Recall that for a finitely generated clone B there are only finitely 4theclone period 1 period $B 2 comma =$ 4 period clone 1 period $( 1 closing S \ squarecup bracketU ) closing . parenthesis $ period Recallmany thatB− maximal for a finitely clones and generated every proper clone subclone $ B $ of thereB i s contained are only in finitely a B− maximal many $clone B − $ maximal clones 3( period see e 4 . period g . .... [ 9 Proposition , 4 . 1 . 2 period , 4 . 1.... . Suppose 1 ] ) . B : = .... clone open parenthesis S cup U closing parenthesis .... is .... a finitely generated ....and clone every .... and proper let M = subclone of $ B $ i s contained in a $ B − $ maximal clone ( see e . g . \quad [9,4.1.2,4.1.1]). 3 . 4 . Proposition . Suppose B := clone (S ∪ U) is a finitely generated clone and open brace M sub 1 comma M sub 2 comma period period period comma M sub s closing brace .. be the s et of al l B hyphen maximal clon es let M = that\noindent contain U3 period . 4 . ..\ Thenh f il we l haveProposition . \ h f i l l Suppose $B : =$ \ h f i l l c lo ne $ ( S \cup U ) $ {M ,M , ... , M } be the s et of al l B− maximal clon es that contain U. Then we have \ hopen f i l l 1 parenthesisi s 2 \ h f i l l2s closinga finitely parenthesis generated S .. is .. a\ finitelyh f i l l generatedc lo n e \ h U f hyphen i l l and s et l e .. t and $ .. M every = .. $ U hyphen s et not .. equal .. to .. S .. is .. (2) S is a finitely generated U− s et and every U− s et not equal to contained in .. s ome S is contained in s ome maximal U− s e t . \noindentmaximal U$ hyphen\{ sM e t period{ 1 } ,M { 2 } ,...,M { s }\} $ \quad be the s et of al l $B ( 3 ) Let N be a maximal U− s e t . Then there exists an − $open maximal parenthesis clon 3 closing es that parenthesis contain .. Let $U .. N .. be . ..$ a ..\quad maximalThen .. U we hyphen have s e t period .. Then .. there .. exists .. an .. M in M .. such M ∈ M such that N = S ∩ M ( s ee Fig . 1 ) . .. that .. N = S cap M ( 4 ) Let N be a maximal U− s et and le t P ⊃ N be a subset of S properly containing $open ( parenthesis 2 ) S s ee $ Fig\quad periodi .. s 1\ closingquad a parenthesis finitely period generated $ U − $ s et \quad and \quad every \quad $ U N. Then − $open s parenthesis et not \quad 4 closingequal parenthesis\quad ..to Let\ Nquad be a maximal$ S $ U\ hyphenquad si s et\ andquad le tcontained P supset N be in a subset\quad ofs S ome properly containing N period .. Thenmaximal $ U − $ s e t . clone(P ∪ U) = B. clone open parenthesis P cup U closing parenthesis = B period ( 3 ) \quad Let \quad $ N $ \quad be \quad a \quad maximal \quad $ U − $ s e t . \quad Then \quad the re \quad e x i s t s \quad an \quad open( parenthesis 5 ) Let 5 closingN be theparenthesis partially .. Let ordered N be the s et partially formed ordered by S s∩ etM formed1,S ∩ M by2, S ..., cap S ∩ MM subs under 1 comma inclusion S cap M sub 2 comma period period period$ M . comma\ in SM cap $ M\ subquad s undersuch inclusion\quad periodthat \quad $ N = S \cap M $ (A s U eeA hyphen FigU− s . s et\ etquad N is maximalN1is ) maximal . if and only if and if i t only is a maximal if i t is element a maximal of N period element .. In of particularN . commaIn particular .. there , arethere only finitely are only many finitely maximal many U hyphen maximal s e ts periodU− s e ts . \ hspaceProofProof of∗{\ of 3 periodf 3 i l . l } 3( 3 and and4 ) 3 3\ . periodquad 4 . 4Let( period 1 ) $N$ “ ....⇐ open” :be parenthesis Let a maximalC := 1 closingclone $U parenthesis(F −∪ U$). Clearly setandle .... quotedblleft,C it sdouble contained $P stroke\supset left arrow quotedblrightN$ be a : subset .... of Let$ S ....in $ C no properly: = .... clone containing open parenthesis $N F cup . U $ closing\quad parenthesisThen period .... Clearly comma C i s .... containedB in− no .... B hyphen maximalmaximal clone clone open ( parenthesis for every forB− everymaximal B hyphen clone maximalM we clone have M either we haveF either6⊆ M, For negationslash-reflexsubsetU 6⊆ M). S ince B M comma or U negationslash- reflexsubset\ [ c l on e M closing( P parenthesis\cup periodU ) .... S = ince B . \ ] hline i s finitely generated comma it fo llows that C = B whence F = S cap C = S cap B = S period i s finitely generated , it fo llows that C = B whence F = S ∩ C = S ∩ B = S. \ hspacequotedblleft∗{\ f i double l l }( 5 stroke ) \quad right arrowLet quotedblright $ N $ be : the .. Suppose partially overbar ordered F = S but s et F subset formed equal by M $ for S some\cap B hyphenM maximal{ 1 } clone,S M “ ⇒ ” : Suppose F = S but F ⊆ M for some B− maximal clone M ⊇ U. Then clone supseteq\cap UM period{ 2 ..} Then,...,S clone open parenthesis F cup\ Ucap closingM parenthesis{ s }$ subset under equal inclusion M . (F ∪ U) ⊆ M s ince U subset equal M comma so S = overbar F = S cap clone open parenthesis F cup U closing parenthesis subset equal M period .. Therefore s ince U ⊆ M, so S = F = S∩ clone (F ∪ U) ⊆ M. Therefore ,S ∪ U ⊆ M whence B ⊆ M which Acomma $ U S cup− U$ subset s et equal $N$ M whence is B maximal subset equal if M and only if i t is a maximal element of $N . $ \quad In particular , \quad the re contradicts the fact that M i s B− maximal . arewhich only contradicts finitely the fact many that maximal M i s B hyphen $ U maximal− $ speriod e t s . ( 2 ) : It is easy to see that S 6⊆ M for all M ∈ M( i f S ⊆ M then U ∪ S ⊆ M whence B = M, open parenthesis 2 closing parenthesis : .... It is easy to see that S negationslash-reflexsubset M for all M in M open parenthesis i f S subset equal which contradicts the fact that M i s a maximal B− clone ) . Take arbitrary f1 ∈ S \ M , ... , M\noindent then U cupProof S subset of equal 3 . M 3 whence and 3 B . = 4 M .comma\ h f i l l ( 1 ) \ h f i l l ‘ ‘ $ \Leftarrow $ ’ ’ : 1\ h f i l l Let \ h f i l l $ C : = $which\ h contradicts f i l l c lo ne the fact $ ( that F M i s\ acup maximalU B hyphen ) . clone $ \ closingh f i l l parenthesisClearly period $, .... C$ Take arbitrary is \ h f f i 1 l l incontained S backslash M in sub no 1 comma\ h f i l l period$ B period− $ period comma hlinefs ∈ S \ Ms. Then by 3 . 3 ( 1 ) we have {f1, ... , fs} = S. By Lemma 3 . 2 ( 1 ) \noindentf, sub s the in Smaximal latt backslash ice of clone MU− subsets( s period for is algebraic every .. Then .$by B 3 Speriod ince− $ 3S open maximalis finitely parenthesis clone generated 1 closing $M$ , parenthesis it we fo have llows .. we either now have that .. $ open F brace\not f 1\subseteq comma period Mperiodmaximal , period $ or comma $U U− fsets sub\not s exist closing\subseteq and brace every = S periodMU− set ) .. By6= .S Lemma $i s\ containedh 3 f iperiod l l S 2 ini n open c a e maximal parenthesis $ B $ one 1 closing . parenthesis comma .. the latt ice of U( hyphen 3 ) : sets Let is algebraicC := clone period(N ..∪ U S) ince. S S ince is finitelyN is generated a maximal commaU− ..set it fo,S llows∩ C = nowN that⊂ S, maximaland thus .. UC hyphen6= B. sets exist \ [ and\Sr u ince every l e {3emB Ui hyphen}{ s0.4 finitely pt set}\ equal-negationslash generated] , there S exists i s contained a B− inmaximal a maximal clone one periodM such that C ⊆ M. S openince parenthesisU ⊆ C ⊆ 3M closingwe have parenthesisM ∈ M :. .. LetWe C show : = clone that openM parenthesisis the B− Nmaximal cup U closing clone parenthesis we are looking period .. for S ince N is a maximal U hyphen set comma, S cap C = N subset S comma and thus C negationslash-equal B period \noindentS ince B i si finitely s finitely generated generated comma there , exists it fo a B llows hyphen that maximal $C clone = M such B$ that whence .. C subset $F equal M = period S ..\ Scap ince C = S \capU subsetB equal = C S subset . equal $ M we have M in M period .. We show that M is the B hyphen maximal clone we are looking for comma \ hspace ∗{\ f i l l } ‘ ‘ $ \Rightarrow $ ’ ’ : \quad Suppose $\ overline {\}{ F } = S $ but $ F \subseteq M$ for some $B − $ maximal clone $M \supseteq U . $ \quad Then clone $ ( F \cup U) \subseteq M $

\noindent s i n c e $ U \subseteq M , $ so $ S = \ overline {\}{ F } = S \cap $ cl o ne $ ( F \cup U) \subseteq M . $ \quad Therefore $ , S \cup U \subseteq M $ whence $ B \subseteq M $ which contradicts the fact that $M$ i s $ B − $ maximal .

\noindent ( 2 ) : \ h f i l l It is easy to see that $ S \not\subseteq M $ f o r a l l $ M \ in M ( $ i f $ S \subseteq M $ then $ U \cup S \subseteq M$ whence $B = M ,$

\noindent which contradicts the fact that $M$ i s a maximal $ B − $ cl o ne ) . \ h f i l l Take arbitrary $ f 1 \ in S \setminus M { 1 } , . . . , $

\ [ \ r u l e {3em}{0.4 pt }\ ]

\noindent $ f { s }\ in S \setminus M { s } . $ \quad Thenby3 . 3 ( 1 ) \quad we have \quad $ \{ f 1 , . . . , f { s }\} = S . $ \quad By Lemma 3 . 2 ( 1 ) , \quad the latt ice of $ U − $ sets is algebraic . \quad S ince $ S $ is finitely generated , \quad it fo llows now that maximal \quad $ U − $ s e t s e x i s t and every $ U − $ s e t $ \ne S $ i s contained in a maximal one .

\noindent ( 3 ) : \quad Let $C : =$ clone $( N \cup U ) . $ \quad S ince $N$ is a maximal $ U − $ s e t $ , S \cap C = N \subset S ,$ andthus $C \not= B . $ S ince $B$ i s finitely generated , there exists a $B − $ maximal clone $M$ such that \quad $ C \subseteq M . $ \quad S i n c e $ U \subseteq C \subseteq M $ we have $ M \ in M . $ \quad Weshow that $M$ is the $B − $ maximal clone we are looking for , 378 .. R period P o-dieresis schel et al period : .. Completeness Citeria and Invariants period period period \noindentopen parenthesis378 \quad 5 closingR .parenthesis P $ \ddot .. quotedblleft{o} $ schel double et stroke al right . : \ arrowquad quotedblrightCompleteness : .. Suppose Citeria that and N is Invariants a maximal U . hyphen . . set period .. Then by open parenthesis 3 closing parenthesis there i s a j in open brace 1 comma period period period comma s closing brace such that 378 R . P o¨ schel et al . : Completeness Citeria and Invariants . . . \noindentN = S cap( M 5 sub ) j\quad period ..‘ ‘ If S $ cap\Rightarrow M sub j i s not$ a maximal ’ ’ : \quad elementSuppose of M comma that there $ N i s $ a k inis open a maximal brace 1 comma $ U period− $ period s e t period . \quad Then by ( 3 ) there i s a ( 5 ) “ ⇒ ” : Suppose that N is a maximal U− set . Then by ( 3 ) there i s a j ∈ {1, ... , s} comma$ j s\ closingin \{ brace such1 that , . . . , s \} $ such that such that N = S∩M . If S∩M i s not a maximal element of M, there i s a k ∈ {1, ... , s} such that $S Ncap M = sub Sj subset\cap S capj M sub{ kj periodj} . .. $ S ince\quad the UI f hyphen $ S set N\cap is maximalM { commaj }$ by openi s parenthesisnot a maximal 4 closing element parenthesis of comma $M we S ∩ M ⊂ S ∩ M . S ince the U− set N is maximal , by ( 4 ) , we obtain clone ((S ∩ M ) ∪ U) = B. obtain,$ clonetherej open isa parenthesisk $k open\ in parenthesis\{ S1 cap ,M sub . k closing . . parenthesis , s cup\} U closing$ such parenthesis thatk = B period $hline S \cap M { j }\subset S \cap M { k } . $ \quad S i n c e the $ U − $ set $N$ is maximal , by ( 4 ) , we obtain clone $ (Now (S cap S M sub\cap k = S capM M{ subk k} =) S cap\ clonecup openU parenthesis ) = open B parenthesis . $ S cap M sub k closing parenthesis cup U closing parenthesis = S capNow B =S S∩ periodMk = S ..∩ ThereforeMk = S∩ commaclone S(( cupS ∩ UM subsetk) ∪ U equal) = MS ∩ subB k= S. Therefore ,S ∪ U ⊆ Mk which \ [ \ r u l e {3em}{0.4 pt }\ ] whichimplies impliesMk M= subB – k contradiction = B endash contradiction with the factwith thatthe factM thatk i s MB sub− maximal k i s B hyphen . maximal period quotedblleft“ ⇐ ” double: Take strokej ∈ left {1 arrow, ... , s quotedblright} such that :S ..∩ TakeMj i sjin a maximal open brace element 1 comma of periodN and period let periodN = S comma∩ Mj. s closing brace such that S cap MSuppose sub j i s further a maximal that element the of NU− andset let NN =i S s cap not M sub maximal j period . Then there is a maximal U− \noindent Now $ S \cap M { k } = S \cap M { k } = S \cap $ cl o ne $ ( ( S \cap M { k } Supposeset P furthersuch that theP ..⊃ UN. hyphenAccording set .. N i to s not ( 3 .. ) maximal there i period s a k ..such Then thatthere isP a= maximalS ∩ Mk ... UBut hyphen then set P such ) \cup U ) = S \cap B = S . $ \quad Therefore $ , S \cup U \subseteq M { k }$ thatS ∩ PM supsetk ⊃ S ∩ NM periodj, which .. According is impossible to open due parenthesis to the 3 choice closing of parenthesisj. there i s a k such that P = S cap M sub k period .. But then S cap Mwhich sub(10 k) supseti implies s a direct S cap $M consequence M sub{ j commak } of= ( 1 B ) $ and−− ( 5contradiction ) , but it follows with also the from fact(2). that line − line $ M { k }$ i s $ B − $ maximal . whichThe is impossible maximal due to clones the choice in of the j period latt ice of all clones ( i . e . the Op (A)− ‘ ‘openmaximal $ \ parenthesisLeftarrow clones 1 to$ the ) ’ power ’ : are\ ofquad prime wellTake closing known $ parenthesis j .\ in They i s a\{ direct can consequencebe1 described , . of open as . Pol parenthesis .% where , 1 s closing%\}is parenthesis$ a such and that open $S parenthesis\cap M5 closing{relationj parenthesis}$ from i s amaximal one comma of s but ix it finite element follows classes also of from described $N$ open parenthesis and by I let . Rosenberg 2 closing $N parenthesis = ( [ 1 S 2 ] , period\ seecap also line-lineM [ 9{ , Chj } .. 4 $ SupposeThe. 3 .. ] ) maximal –further we shall .. clones that call these.. the in the\ relationsquad .. latt$ iceRosenberg U ..− of all$ .. s eclones relations t \quad .. open.$ parenthesis N Thus $ i combining s i notperiod\quad e3 period.3(1maximal0) with .. the 3 .. . Op 4\quad openThen parenthesis there A closing is a maximal \quad parenthesis$ U ( 5− ) we$ hyphen immediately set maximal $P$ .. get clonessuch the closing following parenthesis theorem .. are . .. well thatknown3 . $ 5 period P . ..\supset They Theorem can beN described . . $ Suppose as\quad Pol rhoAccordingclone where( ..S rho∪ U to is) a ( relation= 3 )Op there from(A) one. iLet sof as ixQ $k$finite0 classesdenote such that the $P = S \cap M described{sk e t} by. of $I period al\quad l Rosenberg RosenbergBut then open relations parenthesis $ S which\cap open are squareM contained{ bracketk }\ in1supset 2 closingInv U, squareS and\ bracketcap le t commaMQ1 { be seej } the also, s open $ square bracket 9 comma Chwhich periodet of is .. al 4 impossible l period% ∈ 3Q closing0 such due square that to thebracketS∩ choicePol closing% is of parenthesis maximal $ j endash in . $the we partially shall call ordered these relations s et RosenbergN := h{S∩ Pol relations{%} | % ∈ periodQ0}; ⊆i ... ThusThen combining : 3 period 3 open parenthesis 1 to the power of prime closing parenthesis with 3 period 4 open parenthesis 5 closing\noindent( 1 ) parenthesisA $U− ( wes e immediately 1 t ˆ{\N isprime maximal get the} following) if $ and i only theorem s a if direct there period exists consequence a % ∈ Q1 ofsuch ( 1 )that andN (= 5S )∩ Pol , but{%} it. follows also from $ (3 period 2 5 )period( 2 ) . .. TheoremA l i ns e e− tlperiod i nF e⊆ $ ..S Supposeis U− clonecomplete open parenthesis if and only S cupif for U closing every parenthesis% ∈ Q1 =there .. Op exists open parenthesis an A closing parenthesis periodf ∈ ..F Letsuch Q 0 .. denote .. the .. s e t .. of al l Rosenberg \noindentrelations whichThe are\quad containedmaximal in Inv\quad U commac l o .. n e and s \ lequad t Q 1in .. be the the\ squad et of all a l t rho t i in c e Q\ 0quad such thato f S a l cap l \ Polquad rhoc l o n e s \quad ( i . e . \quad the \quad Op $ (is maximal A ) in the− $ partially maximal ordered\quad s et Nc lthatfelement : o = n e angbracketleft s ) \quad− slashare openPol\ brace{quad%}. Sw cap e l l Pol open brace rho closing brace bar rho in Q 0 closing brace semicolonknown . subset\quad equalThey right can angbracket be described period .. Then as Pol : $ \varrho $ where \quad $ \varrho $ is a relation from one of s ix finite classes ( 3 ) Suppose that al l U− s e ts S∩ Pol {%},% ∈ Q0, are mutually in comparable . describedopen parenthesis by I 1 . closing Rosenberg parenthesis ( [ .. 1 A 2U hyphen ] , see s e alsot N is maximal [ 9 , Ch if and . \ onlyquad if there4 . 3exists ] ) a−− rhowe in Q shall 1 .. such call that theseN = S cap relations Pol open Rosenberg Then a U− s et bracer e l a rho t i o closing n s . brace\quad periodThuscombining $3 . 3 ( 1ˆ{\prime } ) $ with 3 . 4 ( 5 ) we immediately get the following theorem . N is maximal if and only if N = S∩ Pol {%} for s ome % ∈ Q0. open parenthesis 2 closing parenthesis .. A s e t F subset equal S is U hyphen complete if and only if for every rho in Q 1 .. there exists an f in F \noindentExamples3 . how 5 . to\ applyquad Theorem the completeness . \quad criteriaSuppose will clone be given $ ( in the S next\cup sectionU . ) = $ \quad Op $ ( A ) such 4 . 1 - dimensional composition algebras : Examples and problems . $that\quad f element-slashLet $ Pol Q open 0 $ brace\quad rho closingdenote brace\quad periodthe \quad s e t \quad of al l Rosenberg relationsIn this section which we are present contained examples in for Inv the $ main U results , $ \ ofquad theand paper l e and t discuss $ Q some 1 $ prob\quad - lemsbe the s et of al l $ \varrho open. parenthesis We restrict 3 closing mainly parenthesis to 1 - dimensional .. Suppose that composition al l U hyphen algebras s e ts S and cap correspondingPol open brace rho transfor closing - brace comma rho in Q 0 comma .. are\ in mutuallyQ in 0$ comparable suchthat period .. $S Then a\cap U hyphen$ Pol s et $ \varrho $ ismation maximal algebras in the ( seepartially 1 . 5 ) , ordered i . e . s et we $N deal with : the = representation\ langle \{ by unaryS \ mappingscap $ Pol ( $ \{\varrho \} Nalthough is maximal if and only if N = S cap Pol open brace rho closing brace for s ome rho in Q 0 period \midExamples\varrho how to apply\ in theQ completeness 0 \} criteria; will\subseteq be given in the\rangle next section. period $ \quad Then : the problems can be formulated for arbitrary k− dimensional composition algebras as well ) . 4 period .. 1 hyphen dimensional composition algebras : .. Examples and problems \ centerline4 . 1 . {( 1 )Specializing\quad A $1 .U 1 to− the$ case s e tk = $N$ 1 we get is that maximal a 1 - dimensional if and onlyK− ifcomposition there exists a $ \varrho \ in Inalgebra this section is we present examples for the main results of the paper and discuss some prob hyphen Qlems 1 $ period\quad .. Wesuchthat restrict mainly $N to 1 hyphen = S dimensional\cap $ composition Pol $ algebras\{\varrho and corresponding\} . transfor $ } hyphen an algebra B = hB;(fi)i ∈ I, ·i satisfying mation algebras open parenthesis see 1 period 5 closing parenthesis comma i period e period .... we deal with the representation by unary mappings\ hspace ∗{\ openf i parenthesis l l }( 2 ) although\quad A s e t $ F \subseteq S $ i s $ U − $ complete if and only if for every (i) hB;(fi)i ∈ Ii ∈ K, $ \thevarrho problems\ canin be formulatedQ 1 $ for\quad arbitrarythere k hyphen exists dimensional an $ fcomposition\ in algebrasF $ such as well closing parenthesis period 4 period 1 period .... Specializing 1 period( i 1 i to) thehB case; ·i i k s = a 1semigroup we get that , a 1 hyphen dimensional K hyphen composition algebra is \ [an that algebra f B underbar element( ii i = )− angbracketlefts l a The s h operation Pol B semicolon\{\· i s right openvarrho parenthesisdistributive\} f i. closingover\ ] every parenthesisfi(i ∈ iI in), I comma times right angbracket satisfying open parenthesis i closing parenthesisi . e angbracketleft . , we have B for semicolon all b, c1 open, c2, ..., parenthesis cn ∈ B : f i closing parenthesis i in I right angbracket in K comma open parenthesis i i closing parenthesisfi · angbracketleftb = fi for every B semicolon nullary timesfi, rightand angbracket i s a semigroup comma \ hspace ∗{\ f i l l }( 3 ) \quad Suppose that al l $ U − $ s e t s $ S \cap $ Pol $ \{\varrho \} , open parenthesis ii i closingfi(c1, c parenthesis2, ..., cn) · b = ..fi The(c1 operation· b, c2 · b, ..., times cn · b i) s rightfor every distributiven− ary overfi. every f i open parenthesis i in I closing parenthesis comma\varrhoFurther\ in,B hasQ a selector 0 , $ system\quad ( seeare 1 mutually. 3 c ) iff h inB; ·i comparableis a monoid . . \quad Then a $ U − $ s et i period e period comma we have for all b comma c sub 1 comma c sub 2 comma period period period comma c sub n in B : \ centerlinef i times b ={ f$N$ i .. for every is nullarymaximal f i comma if and and only if $N = S \cap $ Pol $ \{\varrho \} $ f o r s ome $ \fvarrho i open parenthesis\ in cQ sub 1 0 comma . $ c sub} 2 comma period period period comma c sub n closing parenthesis times b = f i open parenthesis c sub 1 times b comma c sub 2 times b comma period period period comma c sub n times b closing parenthesis .. for every n hyphen ary f i period \noindentFurther commaExamples B underbar how hasto aapply selector the system completeness open parenthesis criteria see 1 period will 3 c be closing given parenthesis in the iff next angbracketleft section B .semicolon times right angbracket is a monoid period \noindent 4 . \quad 1 − dimensional composition algebras : \quad Examples and problems

\noindent In this section we present examples for the main results of the paper and discuss some prob − lems . \quad We restrict mainly to 1 − dimensional composition algebras and corresponding transfor −

\noindent mation algebras ( see 1 . 5 ) , i . e . \ h f i l l we deal with the representation by unary mappings ( although

\noindent the problems can be formulated for arbitrary $ k − $ dimensional composition algebras as well ) .

\noindent 4 . 1 . \ h f i l l Specializing 1 . 1 to the case $ k = 1 $ we get that a 1 − dimensional $ K − $ composition algebra is

\noindent an algebra $ B {\underline {\}} = \ langle B ; ( f i ) i \ in I, \cdot \rangle $ satisfying

\ [ ( i ) \ langle B ; ( f i ) i \ in I \rangle \ in K, \ ]

\ centerline {( i i $ ) \ langle B; \cdot \rangle $ i s a semigroup , }

\ centerline {( i i i ) \quad The operation $ \cdot $ i s right distributive over every $ f i ( i \ in I ) , $ }

\ centerline { i.e. ,wehavefor all $b , c { 1 } , c { 2 } , . . . , c { n }\ in B : $ }

\ centerline { $ f i \cdot b = f i $ \quad for every nullary $ f i , $ and }

\ centerline { $ f i ( c { 1 } , c { 2 } , . . . , c { n } ) \cdot b = f i ( c { 1 }\cdot b , c { 2 }\cdot b , . . . , c { n }\cdot b ) $ \quad f o r every $ n − $ ary $ f i . $ }

\noindent Further $ , B {\underline {\}}$ has a selector system ( see 1 . 3 c ) iff $ \ langle B ; \cdot \rangle $ is a monoid . R period P o-dieresis schel et al period : .. Completeness Citeria and Invariants period period period .. 379 \ hspace4 period∗{\ 2f period i l l }R .. .Examples P $ \ ofddot 1 hyphen{o} $ dimensional schel et K hyphen al . : composition\quad Completeness algebras period Citeria and Invariants . . . \quad 379 open parenthesis A closing parenthesis .. Near hyphen rings period .. An .. algebra .. angbracketleft N semicolon plus comma times right R.P o¨ schel et al . : Completeness Citeria and Invariants . . . 379 angbracket\noindent ..4 is .. . called 2 . \ ..quad a .. nearExamples hyphen ring of ..1 open− dimensional parenthesis see $comma K − .. e$ period composition g period comma algebras .. open square. bracket 5 closing square 4 . 2 . Examples of 1 - dimensional K− composition algebras . ( A ) Near - bracket(A) closing\quad parenthesisNear − commar i n g s .. . if\ ..quad angbracketleftAn \quad Nalgebra semicolon plus\quad right$ angbracket\ langle .. isN .. a ; + , \cdot \rangle $ \quad i s \quad c a l l e d \quad a \quad near − r i n g \quad ( see , \quad e . g . , \quad [ 5 ] ) , \quad i f \quad rings . An algebra hN; +, ·i is called a near - ring ( see , e . g . , [ 5 ] $ \grouplangle commaN angbracketleft ; + \ Nrangle semicolon$ times\quad righti s angbracket\quad a .. is a semigroup and times .. i s r ight .. distributive over .. plus period .. ) , if hN; +i is a Obviously comma .. near hyphen r ings group , hN; ·i is a semigroup and · i s r ight distributive over +. Obviously , near groupangbracketleft $ , \ Nlangle semicolon plusN; comma\ timescdot right\rangle angbracket$ are\quad 1 hyphenis dimensional a semigroup K hyphen and composition $ \cdot $ algebras\quad fori K s being r i theg h t class\quad of distributive over \quad - r ings hN; +, ·i are 1 - dimensional K− composition algebras for K being the class of all groups . all$ + groups . period $ \quad Obviously , \quad near − r i n g s B := Rel(2)(Y ) Y. $open(\ langle B parenthesis ) BinaryN B closing ; relations + parenthesis , . \cdot .. BinaryLet \ relationsrangle period$be are .. the Let 1 − set Bdimensional :of = all Rel binary to the powerrelations $ K of open− on$ parenthesis compositionIt i 2 closing algebras parenthesis for open $ K $ parenthesisbeings easy the Y class closing parenthesis of all groups be the set . of all binary relations on Y period .. It i s easy hB; ∪, ◦i K− K (B)to checkto\quad check that ..Binary that angbracketleft relations B semicolonis . a\quad 1 cup - dimensional commaLet $B circ right : angbracketcomposition = Relˆ .. is{ algebra a( .. 1 hyphen 2 for ) the dimensional} variety( Y K hyphenof ) $ be composition the set algebra of all for binary relations on ◦ the$ Y varietysemilattices . $ K of\quad .IHere t i s easydenotes the relational product semilattices period .. Here circ denotes the relational product % ◦ σ = {(x, y) ∈ Y 2 | ∃z ∈ Y :(x, z) ∈ %&(z, y) ∈ σ} torho check circ sigma that =\quad open brace$ \ openlangle parenthesisB; x comma\cup y closing, parenthesis\ circ in\ Yrangle to the power$ \quad of 2 bari sexists a \ zquad in Y :1 open− dimensional parenthesis x comma $ K − $ composition algebra for the variety $ K $ of z closing parenthesis in rho ampersandfor %, σ open∈ B, parenthesisand ∪ is z the comma set - y theoretical closing parenthesis union in . sigma closing brace semilattices . \quad Here $ \ circ $ denotes the relational product for( rho C ) comma Distributive sigma in B comma lattices and cup . is theObviously set hyphen ,theoretical any distributive union period latt ice L = hL; ∧, ∨i can be openconsid parenthesis - ered C as closing a 1 - parenthesis dimensional .. DistributiveK− composition lattices period algebra .. Obviously over the comma variety any distributiveK of semilattices latt ice L , underbar = angbracketleft L \ [ \varrho \ circ \sigma = \{ ( x , y ) \ in Y ˆ{ 2 }\mid \ exists z \ in Y: semicolonwhere and∨ commaplaysthe or right role angbracket of · . can be consid hyphen (ered xNote as a , 1 thathyphen z e ) .dimensional g .\ in for K\varrho Boolean hyphen composition latt\& ices ( we algebra zcannot over , add the y variety complementation ) K\ in of semilattices\sigma to the comma\}\ s ignature where] , orbecause plays the∨ rolei s of not times r ight period distributive over complementation . Note further that , according to Note4 . that 1 ( ii e period i ) , g theperiod largest .. for Boolean element latt ices 1 we cannot i s r ight add complementation distributive with to the respect s ignature to comma∨, but the \ centerlinebecauseleast or i{ sf noto r r $ight\varrho distributive, over complementation\sigma \ in periodB .. , Note $ further and $ that\cup comma$ according i s the s e t − theoretical union . } to 4 period 1 open parenthesis ii i closing parenthesiselement comma 0 is not .. the . largest .. element .. 1 .. i s r ight distributive with respect to .. or comma \noindent (C) \quad Distributive lattices . \quad Obviously , any distributive latt ice $ L {\underline {\}} but the( D least ) Semirings . Semirings S = hS; +, ·i are r ings with the usual axioms except that hS; +i = element\ langleis 0 is a notL; period commutative\wedge semigroup, \vee and\rangle not$ necessarily can be consid a− group ( s ee , e . eredopeng . as ,parenthesis a [ 12− ] )dimensional .D closing Therefore parenthesis $ K ....− Semirings$ composition period .... Semiringsalgebra S over underbar the = variety angbracketleft $ K S $ semicolon of semilattices plus comma times , where right $ \vee $ plays the role of $ \cdot . $ angbracketsemirings are r ings with are the usual 1 - dimensional axioms except thatK− ....composition angbracketleft S algebras semicolon plus for right the angbracket variety of iscommutative .. a .. commutative .. semigroup .. and .. not .. necessarily .. a .. group .. open parenthesis s ee comma .. e period g period comma .. open squareNote thatbracket e 2 . closing g . square\quad bracketfor Boolean closing parenthesis lattsemigroups ices period we ... cannot Therefore add complementation to the s ignature , becausesemirings .. $ are\vee .. 1$ hyphen i s dimensional not r ight .. K distributive hyphen composition1.6provides over .. algebras complementation .. for .. the .. . variety\quad .. ofNote commutative further that , according gebras B4.3. TheRepresentationbytransformationTheoremK−algebras ,i.e. subalgebras us with a representationof hAA;(fˆi)i∈I ,of◦ifor some Acompositional=− to4.1(iisemigroups period i), \quad the l a r g e s t \quad element \quad 1 \quad i s r ight distributive with respect to \quad hA;(fi)i ∈ Ii ∈ K, where ◦ is the composition of ( unary ) mappings . It can be checked easily $ \gebrasvee B, from $ hline but to the 4 period least 3 period The Representation by sub transformation Theorem K sub hyphen algebras sub comma to the power of ( see also [ 3 ] ) that the representation sketched in 1 . 7 works here not only for K− 1 periodcomposition 6 provides sub i period e period subalgebras us with a representation of angbracketleft A to the power of A semicolon open parenthesis to the\ centerline power of f-circumflex{ element i closing 0 is notparenthesis . } i in I sub comma of circ right angbracket for some A from hline to composition al = hyphen algebras B with selector system but for any K− composition algebra sat- angbracketleftisfying one A semicolon of the open following parenthesis conditions f i closing : parenthesis i in I right angbracket in K comma where circ is the composition of open parenthesis\noindent unary(D) closing\ h f parenthesis i l l Semirings .... mappings . \ h fperiod i l l Semirings .... It can be checked $ S {\ easilyunderline {\}} = \ langle S ; + , (ii ) all left translations are distinct , i . e . ([∀c ∈ B : a · c = b · c] =⇒ a = b) for all \cdotopen parenthesis\rangle0 see$ also are .... r open ings square with bracket the 3 usual closing axiomssquare bracket except closing that parenthesis\ h f i l l that$ the\ langle representationS sketched ; + in 1\rangle period 7 works$ a, b ∈ B, here not only for K hyphen composition (ii ) hB; ·i i s a monoid , \ hspacealgebras∗{\ ..f B i lunderbar l } i s \quad with ..a selector\quad ..commutative system1 .. but ..\ forquad .. anysemigroup .. K hyphen\quad compositionand \ ..quad algebranot .. satisfying\quad necessarily .. one .. of the \quad a \quad group \quad ( s ee , \quad e . g . , \quad [ 2 ] ) . \quad Therefore (ii ) hB; ·i i s a semigroup with a r ight unit , following conditions : 2 (ii ) hB; ·i i s a r ight cancellative semigroup ( i . e . x · y = z · y implies x = z). \ hspaceopen parenthesis∗{\ f i l3 l } iis e sub m i r 0 i n closing g s \quad parenthesisare \ ..quad all left1 translations− dimensional are distinct\quad comma$ iK period− e$ period composition parenleftbig\ openquad squarea l g e bracketb r a s \ forallquad c f o r \quad the \quad v a r i e t y \quad of commutative Note ( i i2) =⇒ (ii1) =⇒ (ii0) ⇐= ( i i3), in particular (ii0) is satisfied i f any of the other three in Bconditions : a times c = holds b times . c closing square bracket equal-arrowdblright a = b parenrightbig .. for all a comma b in B comma \ centerlineopen parenthesis{ semigroups ii sub 1 closing . } parenthesis angbracketleft B semicolon times right angbracket i s a monoid comma Consequently , given a K− composition algebra B = hB;(fi)i ∈ I, ·i, under any of these condi - opent ions parenthesis , the mapping ii sub 2 closing parenthesis angbracketleft B semicolon times right angbracket i s a semigroup with a r ight unit comma \noindentopen parenthesis$ gebras ii sub 3 closing B ˆ{\ parenthesisr u l e {3em angbracketleft}{0.4 pt }} B{ semicolon4 . times 3 right . angbracket The } Representation i s a r ight cancellative{ by semigroup} { transformation open parenthesis } Theorem{ K } { − ia period l g e b ra e s period}ˆ{ x1 times . y = 6 z times provides y implies} x{ = z, closing} { i parenthesis . e period . subalgebras }$ us with a $ representation { o f \ langle A ˆ{ A } ; ( ˆ{\hat{ f }} i ) i \ in I } { , } o f {\ circ \rangle } f o r some A ˆ{\ r u l e {3em}{0.4 pt }} { composition } Note open parenthesis i i sub 2 closingb parenthesis7→ ϕb where equal-arrowdblrightϕb : B → B : x 7→ b open· x parenthesis ii sub 1 closing parenthesis(4.3.1) arrowdblright-equal open parenthesisa l { = } ii sub − $ 0 closing parenthesis arrowdblleft-equal open parenthesis i i sub 3 closing parenthesis comma in particular open parenthesis ii sub 0 closing parenthesis is satisfied i f any of the other three B fˆ \noindenti s an embedding$ \ langle of BAinto ; the ( full f transformation i ) i algebra\ in hBI ;(\irangle)i ∈ I, ◦i over\ in hB;(Kfi)i ∈ ,I $i. 4 where $ \ circ $ is the composition of ( unary ) \ h f i l l mappings . \ h f i l l It can be checked easily conditions. 4 . holds Remark period . The representation theorem for 1 - dimensional K− composition Consequently comma given a K hyphen composition algebra B underbar = angbracketleft B semicolon open parenthesis f i closing parenthesis i \noindentalgebras( see ( 1 a . l6 s o , \ h f i l l [ 3 ] ) that the representation sketched in 1 . 7 works here not only for $ K in I commak = 1) generalizes times right angbracket various Cayley comma ....- type under theorems any of these for condi special hyphen structures ; we l ist some of them : − $t ions composition comma the mapping Equation: open parenthesis 4 period 3 period 1 closing parenthesis .. b arrowright-mapsto phi b where phi b : B right arrow B : x arrowright-mapsto b\noindent times x a l g e b r a s \quad $ B {\underline {\}}$ with \quad s e l e c t o r \quad system \quad but \quad f o r \quad any \quad $ Ki s an− embedding$ composition of B underbar\quad intoalgebra the full transformation\quad s a t i algebra s f y i n g angbracketleft\quad one B\quad to the powero f the of B semicolon open parenthesis to the power offollowing f-circumflex i conditions closing parenthesis : i in I comma circ right angbracket over angbracketleft B semicolon open parenthesis f i closing parenthesis i in I right angbracket period \ hspace4 period∗{\ 4 periodf i l l } ..$ Remark ( i iperiod{ 0 .. The} ) representation $ \quad theoremall left for .. translations 1 hyphen dimensional are distinct K hyphen composition , i . e algebras $ . ..( open [ parenthesis\ f o r a l l 1c period\ in 6 commaB : a \cdot c = b \cdot c ] \Longrightarrow a = b ) $ \quad f o r a l l $ ak = 1 , closing b parenthesis\ in B generalizes , $ various Cayley hyphen type theorems for special structures semicolon we l ist some of them : \ centerline { $ ( i i { 1 } ) \ langle B; \cdot \rangle $ i s a monoid , }

\ centerline { $ ( i i { 2 } ) \ langle B; \cdot \rangle $ i s a semigroup with a r ight unit , }

\ centerline { $ ( i i { 3 } ) \ langle B; \cdot \rangle $ i s a r ight cancellative semigroup ( i . e $ . x \cdot y = z \cdot y$ implies $x = z ) .$ }

\noindent Note ( i $ i { 2 } ) \Longrightarrow ( i i { 1 } ) \Longrightarrow ( i i { 0 } ) \Longleftarrow ( $ i $ i { 3 } ) ,$ in particular $( ii { 0 } ) $ is satisfied i f any of the other three

\noindent conditions holds .

\noindent Consequently , given a $ K − $ composition algebra $ B {\underline {\}} = \ langle B ; ( f i ) i \ in I, \cdot \rangle , $ \ h f i l l under any of these condi −

\noindent t ions , the mapping

\ begin { a l i g n ∗} b \mapsto \varphi b where \varphi b : B \rightarrow B : x \mapsto b \cdot x \ tag ∗{$ ( 4 . 3 . 1 ) $} \end{ a l i g n ∗}

\noindent i s an embedding of $ B {\underline {\}}$ into the full transformation algebra $ \ langle B ˆ{ B } ; ( ˆ{\hat{ f }} i ) i \ in I, \ circ \rangle $ over $ \ langle B ; ( f i ) i \ in I \rangle . $ 4 . 4 . \quad Remark . \quad The representation theorem for \quad 1 − dimensional $ K − $ composition algebras \quad ( 1 . 6 ,

\noindent $ k = 1 ) $ generalizes various Cayley − type theorems for special structures ; we l ist some of them : 380 .. R period P o-dieresis schel et al period : .. Completeness Citeria and Invariants period period period \noindentbullet .. For380 the\ classquad KR of .all P sets $ open\ddot parenthesis{o} $ without schel any et algebraic al . : \ structurequad Completeness closing parenthesis Citeria and for anda group Invariants operation times . . . we obtain the classical Cayley theorem : .. every group is is omorphic to s ome permutation 380 R . P o¨ schel et al . : Completeness Citeria and Invariants . . . $ group\ bullet period$ \quad For the class $ K $ of all sets ( without any algebraic structure ) and for a group operation • For the class K of all sets ( without any algebraic structure ) and for a group operation · $ \bulletcdot ..$ Likewise comma if K is the class of all sets and times i s a semigroup operation comma then we obtain that wewe obtain obtain the the classical classical Cayley Cayley theorem theorem : : every\quad groupevery is isgroup omorphic is is to omorphic s ome permutation to s ome permutation every s emigroup is is omorphic to s ome transformationgroup s emigroup . period bullet .. If we consider ordinary r ings as .. 1 hyphen dimensional K hyphen composition algebras .. open parenthesis whereby K is \ centerline• Likewise{ group , . if }K is the class of all sets and · i s a semigroup operation , then we obtain that the class .. of commutativeevery s emigroup groups closing is is omorphic parenthesis to comma s ome .. thentransformation .. 1 period 6 ..semigroup shows that ... every r ing i s i somorphic to .. a r ing of transformations period .. Moreover comma .. every r ing R can be .. considered .. as .. an R hyphen module • If we consider ordinary r ings as 1 - dimensional K− composition algebras ( whereby \ hspaceopen parenthesis∗{\ f i l l } over$ \ itbullet self comma$ \ jquad ust byLikewise left multiplication , if closing $K$ parenthesis is the period class .. of Therefore all sets we have and more $ precisely\cdot $ open i parenthesis s a semigroup operation , then we obtain that K is due to thethe class of commutative groups ) , then 1 . 6 shows that every r ing i s i somorphic \ centerlineproperty 4 period{ every 1 open s emigroup parenthesis is ii i closingis omorphic parenthesis to closing s ome parenthesis transformation the known s result emigroup .. open parenthesis . } see e period g period .. open to a r ing of transformations . Moreover , every r ing R can be considered as an square bracket 1 1 comma S 38 comma Satz 66 closing square bracket closing parenthesis : .. every ring R with unit R− module ( over it self , j ust by left multiplication ) . Therefore we have more precisely ( \ hspaceelement∗{\ is isf i omorphic l l } $ \ bullet t o a subring$ \ ofquad the fullIf endomorphism we consider ring ordinary of the module r ings R period as \quad 1 − dimensional $ K − $ composition algebras \quad ( whereby $ Kdue $ toi s the The unitproperty element 4 .of 1 R ( hereby ii i ) ) ensures the known property result 4 period ( see3 open e . parenthesis g . [ 1 1 ii ,sub§38 1 closing, Satz 66parenthesis ] ) : periodevery ..ring The weaker property 4 period 3 open parenthesis ii sub 0 closing parenthesis R with unit element is is omorphic t o a subring of the full endomorphism ring of the module R. thesays c l ina s the s \ casequad of rof ings commutative that 0 is the only groups left annullator ) , \quad in R periodthen ..\quad In open1 square . 6 \ bracketquad shows 1 1 comma that Satz\quad 66 closingevery square r bracketing i .. s it i was somorphic to \quad a The unit element of R hereby ensures property 4.3(ii ). The weaker property 4.3(ii ) rproved ing of that transformations this is a necessary and . \ sufficientquad Moreover condition that , \quad the Cayleyevery representation r1 ing $R$ can be \quad c0 o n s i d e r e d \quad as \quad an $ R −says$ in module the case of r ings that 0 is the only left annullator in R. In [ 1 1 , Satz 66 ] it was Capitalproved Phi that : R this right is arrow a necessary R to the powerand sufficient of R : b mapsto-arrowright condition that the phi b Cayley open parenthesis representation as given in open parenthesis 4 period 3 period 1 closing( over parenthesis it self with , B j : ust = R by closing left parenthesis multiplication is an isomorphism ) . \quad period Therefore we have more precisely ( due to the Φ: R → RR : b 7→ ϕb( as given in ( 4 . 3 . 1 ) with B := R) is an isomorphism . 4 period 5 period .. The minimal representation problem period .. With Theorem 1 period 6 in hand we are faced with property4 . 5 4 . . 1 (The ii minimal i ) ) the representation known result \ problemquad ( see . e .With g . \ Theoremquad [ 1 1 1 .6 , in\S hand38 , we Satz 66 ] ) : \quad every r i n g theare minimal faced repres with entation the minimal prob lem repres : .. Find entation a minimal prob representation lem : ofFind a given a minimal 1 hyphen representation dimensional of a $R$K hyphen with composition unit algebra hline to the power of B sub comma i period e period comma find a representation of B underbar as a transformation given 1 - dimensional K− composition algebra B i . e . , find a representation of B as a Kelement hyphen algebra is is over omorphic t o a subring of the full, endomorphism ring of the module $ R . $ K− Y = hY ;(fi)i ∈ Ii ∈ K | Y | . sometransformation Y underbar = angbracketleftalgebra over Y semicolon some open parenthesis f i closing parenthesisof minimal i in I cardinality right angbracket in K of minimal cardinality .. bar Y\ hspace barFrom period∗{\ the ..f i Fromconstruction l l }The the construction unit ( element 4 . 3 open . 1 of )parenthesis above$R$ 4 period hereby 3 period ensures 1 closing property parenthesis $ .. 4 above . 3 ( ii { 1 } ) . $ \quad The weaker property $ 4wewe conclude . conclude 3 that ( that such i such ai minimal{ a0 minimal} Y satisfies) $ Y ....satisfies bar Y bar| Y less| ≤ or equal | B | bari f BB barsatisfies .... i f B 4 underbar . 3 ( i i satisfies0); otherwise 4 period 3 open parenthesis i i sub 0 closingit parenthesisis semicolon .... otherwise it is 2 saysknownknown in that the that e case period e . g of g. period r| ingsY |≤| barB that Y| bar 0( lesswhich is or the equal can only barbe further B left bar to annullator improved the power of, see in2 open [ 3$R ] parenthesis ) . . $ which\quad canIn be [ further 11 , improved Satz 66 comma ] \quad see i t was openprovedThe square that bracket fo llowing this 3 closing is three a square necessary examples bracket andclosing i sufficient llustrate parenthesis period the condition representation that the Cayley theorem representation and the Theminimization .. fo llowing three problem .. examples . .. i llustrate .. the .. representation .. theorem .. and .. the .. minimization \ centerlineproblem4 . 6 period.{ $ Example\Phi :R ( Semirings\rightarrow ) . RLet ˆ{ RS }= :h{s1 b, s2, s3\}mapsto; +, ·i be\ thevarphi algebrab with ($ as given in (4.3 .1)with $B4the period : following 6 = period R operation .. Example )$ isanisomorphism tables .. open : parenthesis Semirings . } closing parenthesis period .. Let S underbar = angbracketleft open brace s sub 1 comma s sub 2 comma s sub 3 closing brace semicolon plus comma times right angbracket .. be the algebra with the following \noindentoperation tables4 . 5 : . \quad The minimal representationTable ignored! problem . \quad With Theorem 1 . 6 in hand we are faced with theTable minimal ignored! repres entation prob lem : \quad Find a minimal representation of a given 1 − dimensional $ K − $ composition algebra $ \ r u l e {3em}{0.4 pt } ˆ{ B } { , }$ i . e . , find a representation of $B {\underline {\}}$ ThisThis .. i s .. i s a s emiring a s emiring .. open parenthesis ( it can it .. be can be found .. found in in .. [ open 2 , square page bracket 24 2] ,comma however .. page .. 24 here closing square bracket comma .. howeveras a transformation .. here we .. used the $ transposed K − $ algebra over we used the transposed multiplication table in order to ensure condition 4 . 3 ( i i2) so that somemultiplication $ Y {\ tableunderline in order to{\}} ensure condition= \ langle 4 period 3 openY parenthesis ; ( fi i sub i 2 closing ) parenthesis i \ in ..I so that\rangle we can use\ ..in open parenthesisK $ of minimal cardinality \quad $ \midwe canY use\mid ( 4 . 3. . $ 1 )\quad ) .From Thus theS is construction a 1 - dimensional ( 4 . 3 . 1K− ) composition\quad above algebra for 4 periodthe 3 variety period 1K closingof commutative parenthesis closing s emigroups parenthesis (period see 4 .. . Thus 2 ( D ) ) . Using the notation S underbar is .. a .. 1 hyphen dimensional .. K hyphen composition algebra for the variety K .. of commutative s emigroups \noindentopen parenthesiswe conclude see 4 period that 2 open such parenthesis a minimal D closing $ Y parenthesis $ satisfies closing parenthesis\ h f i l l $ period\mid UsingY the\ notationmid \ leq \mid B \mid $ ϕ = (ϕ(s1) ϕ(s2) ϕ(s3)) \ hphi f i l l = parenleftbigi f $ B phi{\ openunderline parenthesis{\}} s sub$ 1 closing satisfies parenthesis 4 . phi 3( open i parenthesis $ i { 0 s} sub) 2 closing ; $ parenthesis\ h f i l l phiotherwise open parenthesis it is s sub 3 closingfor parenthesis mappings parenrightbigϕ : S → S we have \noindentfor mappingsknown phi : that S right e arrow . g S $ we . have\mid Y \mid \ leq \mid B \mid ˆ{ 2 } ( $ which can be further improved , see [ 3 ] ) . phi s sub 1 = parenleftbigϕs s1 sub= 1 (s s1 subs1 1 ss3 sub), ϕs 32 parenrightbig= (s2 s2 s3 comma), ϕs3 = phi (ss sub3 s 23 =s parenleftbig3) . s sub 2 s sub 2 s sub 3 parenrightbig comma phi\noindent s sub 3 =The parenleftbig\quad sfo sub llowing 3 s sub 3 s three sub 3 parenrightbig\quad examples period \quad i l l u s t r a t e \quad the \quad representation \quad theorem \quad and \quad the \quad minimization problemPut C .= {ϕs1, ϕs2, ϕs3}. By ( 4 . 3 . 1 ) , the algebra hC; +, ◦i is i somorphic to S ( according to Put1 . C 4 = we open write brace+ phiinstead s sub 1 of comma+)ˆ . phiE . s g sub . , 2 we comma have phi s sub 3 closing brace period .. By open parenthesis 4 period 3 period 1 closing parenthesis comma the algebra angbracketleft C semicolon plus comma circ right angbracket is i somorphic to S underbar open parenthesis according \noindent 4 . 6 . \quad Example \quad ( Semirings ) . \quad Let $ S {\underline {\}} = \ langle \{ to 1 period 4 ϕs1 + ϕs2 = (s1 + s2 s1 + s2 s3 + s3) = (s2 s2 s3) = ϕs2 = ϕs1 + s2, s we{ write1 } plus, instead s { of2 plus-circumflex} , s { closing3 }\} parenthesis; period + .., E period\cdot g period\rangle comma$ we\ havequad be the algebra with the following ϕs1 ◦ ϕs2 = (s1 s1 s3) = ϕs1 = ϕs1 · s2. operationLine 1 phi s tables sub 1 plus : phi s sub 2 = parenleftbig s sub 1 plus s sub 2 s sub 1 plus s sub 2 s sub 3 plus s sub 3 parenrightbig = parenleftbig s sub 2 s sub 2 s sub 3 parenrightbig = phi s sub 2 = phi s sub 1 plus s sub 2 comma Line 2 phi s sub 1 circ phi s sub 2 = parenleftbig s sub 1 s sub 1 s sub 3 parenrightbig = phi s sub 1 = phi s sub 1 times s sub 2 period \ vspace ∗{3 ex }\ centerline {\ framebox[.5 \ l i n e w i d t h ] {\ t e x t b f { Table ignored!}}}\ vspace ∗{3 ex}

\noindent This \quad i s \quad a s emiring \quad ( i t \quad can be \quad found in \quad [ 2 , \quad page \quad 24 ] , \quad however \quad here we \quad used the transposed multiplication table in order to ensure condition 4 . 3 ( i $ i { 2 } ) $ \quad so that we can use \quad ( 4 . 3 . 1 ) ) . \quad Thus $ S {\underline {\}}$ i s \quad a \quad 1 − dimensional \quad $ K − $ composition algebra for the variety $ K $ \quad of commutative s emigroups ( see 4 . 2 ( D ) ) . Using the notation

\ [ \varphi = ( \varphi ( s { 1 } ) \varphi ( s { 2 } ) \varphi ( s { 3 } )) \ ]

\noindent for mappings $ \varphi :S \rightarrow S $ we have

\ [ \varphi s { 1 } = ( s { 1 } s { 1 } s { 3 } ), \varphi s { 2 } = ( s { 2 } s { 2 } s { 3 } ), \varphi s { 3 } = ( s { 3 } s { 3 } s { 3 } ). \ ]

\noindent Put $ C = \{\varphi s { 1 } , \varphi s { 2 } , \varphi s { 3 }\} . $ \quad By( 4 . 3 . 1 ) , the algebra $ \ langle C ; + , \ circ \rangle $ is i somorphic to $ S {\underline {\}} ( $ according to 1 . 4 we write $+$ instead of $ \hat{+} ) . $ \quad E . g . , wehave

\ [ \ begin { a l i g n e d }\varphi s { 1 } + \varphi s { 2 } = ( s { 1 } + s { 2 } s { 1 } + s { 2 } s { 3 } + s { 3 } ) = ( s { 2 } s { 2 } s { 3 } ) = \varphi s { 2 } = \varphi s { 1 } + s { 2 } , \\ \varphi s { 1 }\ circ \varphi s { 2 } = ( s { 1 } s { 1 } s { 3 } ) = \varphi s { 1 } = \varphi s { 1 }\cdot s { 2 } . \end{ a l i g n e d }\ ] R period P o-dieresis schel et al period : .. Completeness Citeria and Invariants period period period .. 38 1 \ hspaceIt is easy∗{\ tof i check l l }R that . P this $ representation\ddot{o} $ is minimal schel periodet al . : \quad Completeness Citeria and Invariants . . . \quad 38 1 4 period 7 period .. Example .. open parenthesis Lattices closing parenthesis period .. Let L underbar = angbracketleft open brace 0 comma u \noindent It is easy toR.P checko¨ schel that et this al . : representation Completeness is Citeria minimal and . Invariants . . . 38 1 commaIt isv comma easy to 1 closingcheck that brace this semicolon representation and comma is or minimal right angbracket . .. be the 4 hyphen element .. free boolean lat hyphen t ice .. open parenthesis s ee Fig period .. 2 open parenthesis i closing parenthesis closing parenthesis period .. Thus L underbar i s a .. 1 hyphen 4 . 7 . Example ( Lattices ) . Let L = h{0, u, v, 1}; ∧, ∨i be the 4 - element dimensional\noindent K4 hyphen . 7 . composition\quad Example algebra for\quad the variety( Lattices K of ) . \quad Let $ L {\underline {\}} = \ langle \{ L K− 0semilatticesfree , boolean u .. , open lat v parenthesis - t , ice 1 or ( ..\} s plays ee Fig; .. the .\wedge role 2 .. ( of i multiplication ), ) .\vee Thus comma\rangle ..i s see a$ 4 period\quad 1 - dimensional 2be open the parenthesis 4 − element C closing\quad parenthesisfree closing boolean lat − K (∨ parenthesist icomposition c e \quad period( .. algebra s Moreover ee Fig for comma .the\quad variety .. it ..2 satisfies ( iof ) semilattices .. ) 4 . period\quad 3 openThus parenthesisplays $ L {\ i theiunderline sub role 1 closing of{\}} parenthesis multiplication$ i period s a \quad 1 − dimensional i ). $ KAccording,− see$to 4 composition .4 period 2 ( C 3 ) period ) . algebra 1 Moreover we obtain for phi , the a open it variety parenthesis satisfies $ K x closing$ 4 . of 3 parenthesis ( i 1 According = a or x for to a comma 4 . 3 .x in 1 L we comma and the transformation ϕa(x) = a ∨ x a, x ∈ L, K− D 0 := h{ϕ0, ϕ1, ϕu, ϕv}; ∧, ◦i ( Ksemilattices hyphenobtain algebra \quad for$ ( \veeand$ the\quad transformationplays \quad thealgebra r o l e \quad of multiplication , \quad see4.2(C)) . \quad Moreover , \quad i t \quad s a t i s f i e s \quad 4 . 3 ( i Y = h{0, 1, u, v}; ∧i) L $ iDover underbar{ 1 } 0) : = angbracketleft . $ i s open i somorphic brace phi 0to comma. phi 1 comma phi u comma phi v closing brace semicolon and comma circ right angbracket L ( openAccordingLet parenthesis us to over 4 ask .Y 3underbar for . 1 we a = angbracketleftobtain minimal $ \ openvarphi representation brace 0a comma ( 1 of xcomma ) u commain = a the v closing\vee sense bracex semicolon $ of f the o r and $ a right ,angbracket x \ closingin Lparenthesisminimization , $ i and s i somorphic the transformation problem to L underbar , see 4 sub . 5 $K period ) .− Does$ therealgebra exist a transformation algebra D over some $Letsemilattice D ..{\ us ..underline ask forY ..= ahY{\}} ..; ∧i minimalsuch0 .. that representation :L =∼= D\ ..langlebut of L underbarY has\{\ open lessvarphi parenthesis elements0 in .. the , .. as\ sensevarphi in .. theof the above1 .. minimization , \varphi .. problemu comma , \varphiseerepresentation 4 periodv 5 closing\} parenthesis (; where\wedgeY period= .., DoesL \has therecirc 4 exist\ arangle transformation( $ algebra over D underbar $ Y {\ overunderline some semilattice{\}} Y underbar= \ langle = angbracketleft\{ Y0 semicolon , 1 and , right u angbracket , v such\} ; \wedge \rangle ) $ i s i somorphic to $L {\underline {\}} { . }$ that L underbar thicksim = D underbar but Y .. has .. less .. elements .. as .. in the above representation .. open parenthesis where Y = L .. has\noindent 4 Let \quad us \quad ask f o r \quadelements)?a \quad minimal \quad representation \quad o f $ L {\underline {\}} ( $elements in \quad closingthe parenthesis\quad ?sense \quad o f the \quad minimization \quad problem , Case | Y |= 2 : There i s only one semilattice Y on a 2 - element set , namely the chain Y = {0, 1} seeCase 4 bar . 5 Y )bar . =\quad 2 : ThereDoes i s only there one existsemilattice a transformation Y underbar on a 2 hyphen algebra element $ D set{\ commaunderline namely the{\}} chain$ Y = over open some brace 0 semilattice comma 1 with 0 ∧ 1 = 0. The corresponding full transformation algebra hY Y ; ∧, ◦i i s not i somorphic to closing$ Y {\ braceunderline {\}} = \ langle Y; \wedge \rangle $ such L s ince ◦ i s not commutative but ∨ is . Thus there does not exist a representation of L thatwith 0 $ and L 1{\ = 0underline period .. The{\}}\ correspondingsim full= transformation D {\underline algebra angbracketleft{\}}$ but Y to $ the Y power $ \quad of Y semicolonhas \quad andl comma e s s \quad circ rightelements \quad as \quad in the above representation \quad ( where over a 2 - element Y. angbracket$ Y = .. Li s $ not\ iquad somorphichas 4 to L underbar s ince circ i s not commutative but .. or is period .. Thus there does not exist a representation of L underbar \ beginover a{ a 2 l hyphen i g n ∗} element Y period elements ) ? hlineCase | Y |= 3 : In fact there is a representation with a 3 - element s emilattice Y . \endCaseLet{ a lbar iY g n:= Y∗} bar{a, b,= c 3} :and .. In let factY there= ishY a; representation∧i be the with semilattice a 3 hyphen corresponding element s emilattice to the Y chain underbar subhY period; ≤i Let( as Y shown : = open in brace Fig . a 2 comma ( ii ) ; b as comma usual c closing, ≤ is defined brace and by letx ≤ Yy underbar: ⇐⇒ =x ∧ angbracketlefty = x). Y semicolon and right angbracket .... be the semilattice\noindent correspondingCaseFigure $ \mid to 2 the . The chainY ....\mid latt angbracketleft ice=L and 2 Y the semicolon : $ i There somorphic less or i equal s only right transformation one angbracket semilattice $Y {\underline {\}}$ on a 2 − element set , namely the chain $ Yopen = parenthesis\{ 0 as shown , in 1 Fig\} periodalgebra$ 2 openD parenthesisover the ii semilattice closing parenthesisY semicolon as usual comma less or equal is defined by x less or equal y : arrowdblleft-arrowdblright x and y = x closing parenthesis period \noindentFigure 2 periodwith The $ .. 0 latt\ icewedge L underbar1 and = the 0 .. i somorphic . $ \quad .. transformationThe corresponding full transformation algebra $ \ langle Y ˆ{ Y } ; \wedge , \ circ \rangle $ \quad i s notY i somorphic algebra D underbar over the semilattice Y underbarLetD := {ψ0, ψu, ψv, ψ1} ⊆ Y with to $ L {\underline {\}}$ s i n c e $ \ circ $ i s not commutative but \quad $ \vee $ i s . \quad Thus there does not exist a representation of Let D : = open brace psi 0 commaψ0 = psi (abc sub), u ψcommau = (acc psi), sub ψv v= comma (cbc), psi ψ1 1 = closing (ccc) brace subset equal Y to the power of Y with psi 0 = open parenthesis$ L {\underline abc closing{\}} parenthesis$ comma psi sub u = open parenthesis acc closing parenthesis comma psi sub v = open parenthesis cbc closing over a 2 − element $Y . $ parenthesis( recall comma that psi we 1 represent = open parenthesisψ ∈ cccY Y closingas parenthesis the triple (ψ(a)ψ(b)ψ(c))). Then D := hD; ∧, ◦i openis a parenthesis transformation recall that we algebra represent over psi in Y toY thewhere power of Y the .. as the poset triple .. openhD; ≤i parenthesiscorresponding psi open parenthesis to a closing parenthesis \ [ \ r u l e {3em}{0.4 pt }\ ] , psi openhD; ∧i parenthesis( induced b closing componentwise parenthesis psi by openhY parenthesis; ≤i) i s given c closing in Fig parenthesis . 2 ( i closingii ) . parenthesis Thus closing parenthesis period .. Then D underbar : = angbracketleft D semicolon and comma circ right angbracket .. is a transformation .. algebra over hline to the power of Y sub comma .. where .. the .. poset .. angbracketleft D semicolon less or equal right Ψ: L → D : x 7→ ψx angbracket\noindent ..Case corresponding $ \mid to .. angbracketleftY \mid D= semicolon 3 : and $ right\quad angbracketIn fact open there parenthesis is a induced representation with a 3 − element s emilattice $ Ycomponentwisei s{\ a semilatticeunderline by angbracketleft{\}} is omorphism{ . Y}$ semicolon from hL; less∧i onto or equalh rightD; ∧i. angbracketBut we closing also have parenthesis i s given in Fig period 2 open parenthesis i ii closing parenthesis period .. Thus \noindentCapital PsiLet : L right $ Y arrow : D : = x arrowright-mapsto\{ aΨ( ,x ∨ psiy b) = sub Ψ( , xx) ◦ Ψ( c y) \} $ and l e t $ Y {\underline {\}} = \ langle Y;i s a semilattice\wedge is omorphism\rangle $ from\ h angbracketleft f i l l be the L semicolonsemilattice and right corresponding angbracket onto to .. the angbracketleft chain \ h D f i semicolon l l $ \ langle and right angbracketY; \ leqs ince\rangleψ0◦ψ$x = ψx ◦ψ0 = ψx (ψ0 i s the identity map ), ψx ◦ψ1 = ψ1◦ψx = period .. But we also have D ψ1 and ψu ◦ψv = ψv ◦ψu = ψ1. Therefore Ψ is also an is omorphism from L onto , Capital Psi open parenthesis x or y closing parenthesis = Capital Psi open parenthesis x closing parenthesis circ Capital Psi open parenthesis y and D is a minimal representation of L . closing\noindent parenthesis( as shown in Fig . 2 ( ii ) ; as usual $ , \ leq $ is defined by $ x \ leq y : \ i f f x s ince\wedge .. psi 0 circy psi = sub x x = psi ) sub . x $ circ psi 0 = psi sub x open parenthesis psi 0 .. i s .. the .. identity .. map closing parenthesis comma psi sub x circ psi 1 = psi 1 circ psi sub x = psi 1 .. and \ centerlinepsi sub u circ{ Figure psi sub v 2 = . psi The sub v\quad circ psil asub t t u i = c e psi1 $ period L {\ ..underline Therefore Capital{\}} Psi$ .. isand also the .. an\ isquad omorphismi somorphic from L underbar\quad ontotransformation hline } to the power of D sub comma .. and D underbar is a \ centerlineminimal representation{ algebra of $ L D underbar{\underline sub period{\}}$ over the semilattice $ Y {\underline {\}}$ } \ begin { a l i g n ∗} Let D : = \{\ psi 0 , \ psi { u } , \ psi { v } , \ psi 1 \}\subseteq Y ˆ{ Y } with \\\ psi 0 = ( abc ) , \ psi { u } = ( acc ) , \ psi { v } = ( cbc ) , \ psi 1 = ( ccc ) \end{ a l i g n ∗}

\noindent ( recall that we represent $ \ psi \ in Y ˆ{ Y }$ \quad as the triple \quad $ ( \ psi ( a ) \ psi ( b ) \ psi ( c ) ) ) . $ \quad Then $ D {\underline {\}} : = \ langle D; \wedge , \ circ \rangle $ \quad i s a transformation \quad algebra over $ \ r u l e {3em}{0.4 pt } ˆ{ Y } { , }$ \quad where \quad the \quad poset \quad $ \ langle D; \ leq \rangle $ \quad corresponding to \quad $ \ langle D; \wedge \rangle ( $ induced componentwise by $ \ langle Y; \ leq \rangle )$ i s given inFig .2( i ii ) . \quad Thus

\ [ \Psi :L \rightarrow D : x \mapsto \ psi { x }\ ]

\noindent i s a semilattice is omorphism from $ \ langle L; \wedge \rangle $ onto \quad $ \ langle D; \wedge \rangle . $ \quad But we also have

\ [ \Psi ( x \vee y ) = \Psi ( x ) \ circ \Psi ( y ) \ ]

\noindent s i n c e \quad $ \ psi 0 \ circ \ psi { x } = \ psi { x }\ circ \ psi 0 = \ psi { x } ( \ psi 0 $ \quad i s \quad the \quad i d e n t i t y \quad map $ ) , \ psi { x }\ circ \ psi 1 = \ psi 1 \ circ \ psi { x } = \ psi 1 $ \quad and $ \ psi { u }\ circ \ psi { v } = \ psi { v }\ circ \ psi { u } = \ psi 1 . $ \quad Therefore $ \Psi $ \quad i s a l s o \quad an is omorphism from $ L {\underline {\}}$ onto $ \ r u l e {3em}{0.4 pt } ˆ{ D } { , }$ \quad and $ D {\underline {\}}$ i s a minimal representation of $ L {\underline {\}} { . }$ 382 .. R period P o-dieresis schel et al period : .. Completeness Citeria and Invariants period period period \noindent4 period 8382 period\quad .. ExampleR . P.. open $ \ parenthesisddot{o} $ Lattices schel continued et al closing . : \ parenthesisquad Completeness period .. Let K Citeria be the variety and ofInvariants all s emilattices . . enriched . with 382 R . P o¨ schel et al . : Completeness Citeria and Invariants . . . \noindentan additional4 . unary 8 . operation\quad Example f which i\ squad a semilattice( Lattices endomorphism continued comma ) i . period\quad e periodLet $.. K satisfies $ be f open the parenthesis variety of x and all y closing s emilattices enriched with 4 . 8 . Example ( Lattices continued ) . Let K be the variety of all s emilattices parenthesisan additional = unary operation $ f $ which i s a semilattice endomorphism , i . e . \quad satisfies $ f enriched with an additional unary operation f which i s a semilattice endomorphism , i . e . (f open x parenthesis\wedge xy closing ) parenthesis = $ and f open parenthesis y closing parenthesis period .. Thus comma .. e period g period comma Y underbar $satisfies f (f x(x ∧ y )) = f\wedge(x) ∧ f(y). fThus ( , y e ) . g .., $ Y 0\quad := ThushL; ∧, ,fi\quadwithe .L g= $ .{0, u, , v, 1} Y {\underline {\}} 0 : =from angbracketleft the previous L semicolon example and comma f right angbracket .. with .. L = open brace 0 comma u comma v comma 1 closing brace .. from .. the previous0 : .. = example\ langle L; \wedge , f \rangle $ \quad with \quad $ L = \{ 0 , u , v together with f : L → L given by f(0) = f(v) = v, f(u) = f(1) = 1, belongs to K. S ince this f also ,together 1 \} with$ f :\quad L rightfrom arrow\ Lquad giventhe by f open previous parenthesis\quad 0 closingexample parenthesis = f open parenthesis v closing parenthesis = v comma f open satisfies f(x) ∨ y = f(x ∨ y), i . e ., ∨ i s r ight distributive over f, the algebra parenthesis u closing parenthesis = f open parenthesis 1 closing parenthesis = 1 comma belongs to K period \noindentS ince thistogether f also satisfies with f open $ parenthesis f : L x closing\rightarrow parenthesis orL$givenby$f y = f open parenthesis x or y( closing 0 parenthesis ) = comma f ( i period v e ) period = M := h{0, u, v, 1}; ∧, f, ∨i commav , or i f s r ight ( distributive u ) over = f commaf ( the 1 algebra ) = 1 ,$belongsto$K .$ SinceMi underbar s this a : 1 = $f$ - angbracketleft dimensional also open satisfiesK− bracecomposition 0 comma $f u comma ( algebra x v comma ) .\ 1vee Toclosingy get brace = semicolon a representation f ( and comma x \vee f comma as y or right ) angbracket , $ i . e $ .i stransformation .. ,a .. 1\ hyphenvee $ dimensional iK− s ralgebra ight .. K wedistributive hyphen have composition to extend over .. the algebra $ representation f period , $ .. To the ..D get algebra0 in .. a 4 representation . 7 by the operation .. as .. transformation .. K hyphen ˆ algebraf : D0 we→ D have0 : ϕx to7→ extend the representation D underbar 0 in 4 period 7 by the operation f-circumflex : D sub 0 right arrow D sub 0 : phi x arrowright-mapsto\ [M {\underline {\}} : = \ langle \{ 0 , u , v , 1 \} ; \wedge , f , \vee \ranglephi sub\ f] open parenthesis x closing parenthesis period Now comma contrary to Example 4 period 7 comma thisϕf representation(x). is minimal period To see this comma assume that there is a representation of M underbar as a transformation K hyphen algebra D underbar = \noindentangbracketleftNow , contraryi s D\quad semicolon to Examplea \quad and comma 41 .− 7 g-circumflexdimensional, this representation sub\ commaquad is circ$ minimal K right− angbracket$ . composition .. over a 3\quad hyphenalgebra element .. . semilattice\quad To Y\ underbarquad get = \quad a representation \quad as \quad transformation \quad angbracketleft$ K To− see$ this Y semicolon , assume and that comma there g right is a angbracket representation .. with of a sM emilatticeas a transformation endomorphism g periodK− algebra D = algebra we have to extend the representation $ D {\underline {\}} 0 $ in 4 . 7 by the operation $ \hat{ f } ConsequentlyhD; ∧, gˆ,◦i commaover a D 3 subset- element equal Y semilattice to the powerY of= Y andhY there; ∧, gi wouldwith exist a s an emilattice isomorphism endomorphism comma say Capitalg. Phi : x mapsto-arrowright :D { 0 }\rightarrowY D { 0 } : \varphi x \mapsto $ psi subConsequently x comma from,D M⊆ underbarY and onto there would exist an isomorphism , say Φ: x 7→ ψx, from M onto D ( Dthe underbar mappings open parenthesisψx are st ilthe l notmappings known psi and sub xhave are st nothing il l not toknown do with and have those nothing from to Example do with those 4 . 7 from ) . Example 4 period 7 closing parenthesis\ beginAssume{ a l period i g thatn ∗} Y = {a, b, c} where a denotes the least element ( w . r . t . the corresponding poset \varphiAssume that{ f Y =open ( } bracex a ) comma . b comma c closing brace where a denotes the least element open parenthesis w period r period t period the\end corresponding{ a l i g n ∗} poset angbracketleft Y semicolon less or equal righth angbracketY ; ≤iinducedby inducedhY ; by∧i) angbracketleft. Y semicolon and right angbracket closing parenthesis period \noindentS ince CapitalNow Phi , contrary is an isomorphism to Example comma we 4 .get 7 for , any this x in representation L is minimal . S ince Φ is an isomorphism , we get for any x ∈ L psi sub x = Capital Phi open parenthesis x closing parenthesis = Capital Phi open parenthesis x or x closing parenthesis = Capital Phi open parenthesis\noindent xTo closing see parenthesis this , assumecirc Capital that Phi there open parenthesis is a representation x closing parenthesis of = $M psi sub{\ x circunderline psi sub x{\}} period $ as a transformation ψ = Φ(x) = Φ(x ∨ x) = Φ(x) ◦ Φ(x) = ψ ◦ ψ . $ KTherefore− $ every algebra psi subx $ open D {\ parenthesisunderlinex x in{\}} L closing parenthesis= $ .. belongsx tox the following set .. S subset Y to the power of Y of idempotent $ \ langle D; \wedge , \hat{g} { , }\ circ \rangle $ \quad over a 3 − element \quad semilattice mappingsTherefore every ψ (x ∈ L) belongs to the following set S ⊂ Y Y of idempotent mappings $ Y {\underline {\}}x = \ langle Y; \wedge , g \rangle $ \quad with a s emilattice endomorphism open( where parenthesisψx is where again psi written sub x is as again thewritten triple asψx the= (ψ triplex(a)ψ psix(b sub)ψx( xc))) =open : parenthesis psi sub x open parenthesis a closing parenthesis psi sub$ g x open . $ parenthesis b closing parenthesis psi sub x open parenthesis c closing parenthesis closing parenthesis closing parenthesis : S : = open brace open parenthesisS := {(aaa aaa), (bbb closing), (ccc parenthesis), (aac), (bbc comma), (aba), open(cbc), parenthesis(abb), (acc), bbb(abc closing)}. parenthesis comma open parenthesis ccc closing \noindent Consequently $ , D \subseteq Y ˆ{ Y }$ and there would exist an isomorphism , say $ \Phi parenthesis comma open parenthesis aac closing parenthesis comma open parenthesis bbc closingY parenthesis comma open parenthesis aba closing :Up x to\mapsto i somorphism\ psi { therex } are, $ only from two $ 3 M - element{\underline s emilattices{\}}$ onto, which we shall parenthesisexamine comma open parenthesis cbc closing parenthesis comma open parenthesis abb closing parenthesis comma open parenthesis acc closing parenthesis$ D {\ commaunderline open parenthesis{\}} abc( $ closing the parenthesis mappings closing $ \ psi brace{ periodx }$ are st il l not known and have nothing to do with those from Example 4 . 7 ) . separately . Case 1 : Y i s the s emilattice with b ∧ c = a( see Fig . 3 ) . AssumeUp .... to that .... i somorphism $Y = ....\{ therea .... are , .... b only two , .... c 3 hyphen\} $ element where .... $ s emilatticesa $ denotes hline to the the leastpower of element Y sub comma ( w .... . rwhich . t we . the corresponding poset .... shall .... examine Figure 3 . The semilattice Y ( case 1 ) and the induced poset on S \ beginseparatelyThen{ a l the i g period n induced∗} poset hS; ≤i is as in Fig . 3 . S ince ψu and ψv must be incomparable but \ langleCasemust 1 : have YY; underbar a least i s\upperleq the s emilattice bound\rangle , namelywith binduced andψ1 c = ( as open ee by Fig parenthesis\ .langle 2 ( i ) see ) , FigY; we period get {ψ 3\uwedge closing, ψv} = parenthesis{(aba\rangle), (aac period)}. ). \endFigure{ a l i g3 n period∗} The semilattice Y underbar open parenthesis case 1 closing parenthesis and the induced poset on S Then the induced poset .... angbracketleft S semicolon less or equal right angbracket .... is as in Fig period .... 3 period .... S ince psi sub u and psi\noindent sub v .... mustS i n be c e incomparable $ \Phi $ but is an isomorphism , we get for any $ x \ in L $ must have a least upper bound comma namely psi 1 open parenthesis s ee Fig period 2 open parenthesis i closing parenthesis closing parenthesis comma\ [ \ psi we get{ openx } brace= psi\Phi sub u comma( x psi sub) v = closing\Phi brace =( open x brace\vee open parenthesisx ) aba = closing\Phi parenthesis( x comma ) open\ circ parenthesis\Phi aac( closing x ) parenthesis = \ closingpsi { bracex }\ periodcirc \ psi { x } . \ ]

\noindent Therefore every $ \ psi { x } ( x \ in L ) $ \quad belongs to the following set \quad $ S \subset Y ˆ{ Y }$ of idempotent mappings ( where $ \ psi { x }$ is again written as the triple $ \ psi { x } = ( \ psi { x } ( a ) \ psi { x } ( b ) \ psi { x } ( c ) ) ) : $

\ [ S : = \{ ( aaa ) , ( bbb ) , ( ccc ) , ( aac ) , ( bbc ) , ( aba ) , ( cbc ) , ( abb ) , ( acc ) , ( abc ) \} . \ ]

\noindent Up \ h f i l l to \ h f i l l i somorphism \ h f i l l ther e \ h f i l l are \ h f i l l only two \ h f i l l 3 − element \ h f i l l s emilattices $ \ r u l e {3em}{0.4 pt } ˆ{ Y } { , }$ \ h f i l l which we \ h f i l l s h a l l \ h f i l l examine

\noindent separately . Case $ 1 : Y {\underline {\}}$ i s the s emilattice with $ b \wedge c = a ($ seeFig.3).

\ centerline { Figure 3 . The semilattice $ Y {\underline {\}} ( $ case 1 ) and the induced poset on $ S $ }

\noindent Then the induced poset \ h f i l l $ \ langle S; \ leq \rangle $ \ h f i l l i s as in Fig . \ h f i l l 3 . \ h f i l l S i n c e $ \ psi { u }$ and $ \ psi { v }$ \ h f i l l must be incomparable but

\noindent must have a least upper bound , namely $ \ psi 1 ($ seeFig.2(i)),weget $ \{\ psi { u } , \ psi { v }\} = \{ ( aba ) , ( aac ) \} . $ R period P o-dieresis schel et al period : .. Completeness Citeria and Invariants period period period .. 383 R .But P psi $ 1\ =ddot Capital{o} Phi$ open schel parenthesis et al 1 . closing : \quad parenthesisCompleteness = Capital PhiCiteria open parenthesis and Invariants u or v closing . . parenthesis . \quad =383 psi sub u circ psi sub vBut = open $ parenthesis\ psi 1 aaa = closing\Phi parenthesis( period 1 ) .. Hence = psi\Phi 1 less or( equal u psi\ subvee u commav which ) = is a contradiction\ psi { u to}\ circ \ psi { v } R.P o¨ schel et al . : Completeness Citeria and Invariants . . . 383 But ψ1 = Φ(1) = =D (thicksim aaa = M ) period . $Equation:\quad hlineHence .. hline $ \ psi 1 \ leq \ psi { u } , $ which is a contradiction to Φ(u ∨ v) = ψ ◦ ψ = (aaa). Hence ψ1 ≤ ψ , which is a contradiction to Case 2 : Y underbaru iv s the s emilattice with b and c =u b open parenthesis see Fig period 4 closing parenthesis period \ beginFigure{ a 4 l iperiod g n ∗} The semilattice Y underbar open parenthesis case 2 closing parenthesis and the induced poset on S D \sim = M . \\\ tag ∗{$ \ r u l e {3em}{0.4 pt } $}\ r u l e {3em}{0.4 pt } Then the induced poset .... angbracketleft S semicolonD less∼= orM. equal right angbracket .... is as in Fig period 4 period .... The elements psi sub u comma\end{ a psi l i g sub n ∗} v .... must be incomparable comma thus one of them comma say psi sub u comma must belong to open brace open parenthesis aac closing parenthesis comma open parenthesis abc closing\noindent parenthesisCase comma $ 2 open : parenthesis Y {\underline acc closing parenthesis{\}}$ closing i s the brace s open emilattice parenthesis with see Fig $ period b 4\ closingwedge parenthesisc = period b .... ( We $ Case 2 : Y i s the s emilattice with b ∧ c = b( see Fig . 4 ) . aresee going Fig to . 4 ) . exclude each case byFigure contradiction 4 . The period semilattice Y ( case 2 ) and the induced poset on S \ centerlineopenThen parenthesis the{ inducedFigure 1 closing 4 poset . parenthesisThehS semilattice; ≤i is psi as sub in u Fig = $ .open Y 4 .{\ parenthesis Theunderline elements aac{\}} closingψu, ψ parenthesisv (must $ case be period incomparable 2 ) .... and Then the , psi induced sub v in poset open brace on open $ S $ parenthesis} thus one aba of closing them parenthesis , say ψu, commamust belong open parenthesis to {(aac abb), (abc closing), (acc) parenthesis} ( see Fig comma . 4 ) open .parenthesis We are going bbb toclosing parenthesis closing brace openexclude parenthesis each by caseincomparability by contradiction comma .... . see Fig period .... 4 closing parenthesis comma .... but .... this \noindentgives(1) Capitalψu Then= Phi ( theaac open). induced parenthesisThen ψv poset 1∈ closing\ {(haba parenthesis f i) l, l(abb$), (\bbb =langle) Capital} ( by Phi incomparabilityS; open parenthesis\ leq , u\ seerangle or v Fig closing .$ 4parenthesis\ )h , f i l but l is = psi as sub in u Fig circ .psi 4 sub . \ vh = f i l l The elements open$ \ psithis parenthesis{ u } aaa, closing\ psi parenthesis{ v } which$ \ h i fs i not l l amust least beupper incomparable bound of psi sub , u comma psi sub v in angbracketleft S semicolon less or equal rightgives angbracketΦ(1) period = Φ(u ∨ v) = ψu ◦ ψv = (aaa) which i s not a least upper bound of ψu, ψv in hS; ≤i. \noindentopen(2) parenthesisψu =thus (abc one 2). closing ofThen them parenthesisψv = , say ( psibbb) sub$ \and upsi =ψ openu{◦ ψu parenthesisv }= ψv,gives $ abc must the closing contradiction belong parenthesis to $ period\{ ( analogously ..( Then aac psi to sub ) v = , open ( parenthesis abc bbb) closing, ( parenthesis acc ..) and\} psi sub($ u circ seeFig psi sub v .4) = psi sub . v\ h gives f i l l theWe contradiction are going .. toopen parenthesis analogously to case open parenthesis 1 closing parenthesis closing parenthesis period case(1)). \noindentopen parenthesisexclude 3 closing each parenthesis case by contradiction psi sub u = open . parenthesis acc closing parenthesis period .... Then psi sub v in open brace open parenthesis bbb closing parenthesis comma open parenthesis bbc closing parenthesis comma open parenthesis cbc closing parenthesis closing brace (3) ψ = (acc). Then ψ ∈ {(bbb), (bbc), (cbc)} ( s ince ψ , ψ must be incomparable ) . But open\noindent parenthesisu $ s ( ince 1 psi sub ) u comma\vpsi psi{ subu } v ....= must ( .... aac be incomparable )u . $v closing\ h f iparenthesis l l Then period $ \ psi .... But{ v }\ in \{ ( aba )open , parenthesis ( abb bbb )closing , parenthesis ( bbb circ psi ) sub\} u = open( $ parenthesis by incomparability bbb closing parenthesis , \ h equal-negationslash f i l l see Fig . open\ h f i parenthesis l l 4 ) , ccc\ h closing f i l l but \ h f i l l t h i s parenthesis = psi sub u circ open parenthesis bbb closing parenthesis and open parenthesis bbc closing parenthesis circ psi sub u = open parenthesis (bbb)◦ψu = (bbb) 6= (ccc) = ψu◦ (bbb) and (bbc) ◦ψu = (bcc) element−negationslash S, thus ψv = (cbc). bcc\noindent closing parenthesisg i v e s $element-negationslash\Phi ( 1 S ) comma = thus\Phi psi sub( v = u open\ parenthesisvee v cbc ) closing = parenthesis\ psi { periodu }\ circ \ psi { v } = ( aaa ) $ which i s not a least upper bound of $ \ psi { u } , \ psi { v }$ in $ \ langle S ConsequentlyConsequently .. the .. the unique unique .. remaining remaining .. possibility possibility.. i s .. D = open i s braceD psi= 0 comma{ψ0, ψ psi, ψ sub, ψ u1} commawith psi sub v comma psi 1 closing ; \ leq \rangle . $ u v braceψ ..0 with = .. psi (abc 0), = ψ open1 = parenthesis (ccc) abcas closing in parenthesis 4 . 7 .comma Contrary to 4 . 7 now we $psihave ( 1 = 2 open to ) parenthesis take\ psi ccc{ into closingu } = parenthesis account ( abc .. asthe .. ) in .. additional .4 period $ \quad 7 period unaryThen .. Contraryoperation $ \ psi .. to .{ ..v 4} period From= 7 the .. ( now i bbb .. we .. ) have $ ..\quad to .. takeand .. $ \ psi { u }\ circ \ psi { v } = \ psi { v }$ gives the contradiction \quad ( analogously to into ..somorphism account .. theΦ: .. additionalM → D we get gˆ(Φ(0)) = Φ(f(0)) = Φ(v), i . e ., gˆ(ψ0) = unaryψ . operationThus period(g(a)g( ..b)g From(c)) the = i somorphismg ˆ((abc)) = Capital (cbc) Phi, but : M underbar this rightg arrowi s D not underbar a semilattice we get .. g-circumflex open parenthesis \ beginv { a l i g n ∗} Capitalendomorphism Phi open parenthesis s ince 0g closing(a ∧ b) parenthesis = g(a) = c closing6= b = c parenthesis∧ b = g(a) ∧ =g Capital(b), a contradiction Phi open parenthesis . f open parenthesis 0 closing parenthesis closing case ( 1 ) ) . parenthesisTherefore = Capital no Y Phiwith open 3 parenthesiselements exists v closing . parenthesis comma \endi period{ a l i g e n period∗} comma g-circumflex open parenthesis psi 0 closing parenthesis = psi sub v period .. Thus .. open parenthesis g open parenthesis a closing parenthesis g open parenthesis b closing parenthesis g open parenthesis c closing parenthesis closing parenthesis = circumflex-g open parenthesis\noindent open$ parenthesis ( 3 ) abc closing\ psi parenthesis{ u } = closing ( parenthesis acc ) = open . $ parenthesis\ h f i l l cbcThen closing $ parenthesis\ psi { v comma}\ in .. but ..\{ this ..( g .. bbb i s .. not) ..Finally , a semilattice ( , we bbc shall ) demonstrate , ( how cbc the ) Completeness\} ( $ Theorem s i n c e 3 $ .\ 5psi ( 2 ){ appliesu } to, produce\ psi { comv }$ \ h f i l l must \ h f i l l be incomparable ) . \ h f i l l But endomorphism- pletness criteria s ince g for open concrete parenthesis structures a and b closing . A parenthesis completeness = g open theorem parenthesis for near a closing - r ings parenthesis ( see 4 = . c negationslash-equal b = c and\ [ (b2 = ( gA bbb open ) ) parenthesiscan ) be\ circ found a closing in\ [psi 1 parenthesis ] .{ u Here} and= we g open consider ( parenthesis bbb semirings ) b closing\ne ( see parenthesis( 4 . 2 ccc ( D comma ) ) ) . a = contradiction\ psi { periodu }\ circ ( bbb )Therefore4 and . 9 . no( Y ( bbcwith max 3 elements ), ◦)−\semiringscirc exists period\ psi . { Letu } A ={1, 2 (, ... bcc , n} be ) a finite element at least−negationslash two element set S , thus \ psi { v } =and ( let cbc ) . \ ] hline A FinallyT := commahA ; max we shall, ◦i demonstratebe the full howconcrete the Completeness ( max , ◦)− Theoremsemiring 3 period . Here 5 open max parenthesis denotes the 2 closing maximum parenthesis applies to produce com hyphen \noindentpletness criteriaConsequently for concrete\ structuresquad the period\quad .. Aunique completeness\quad theoremremaining for near\ hyphenquad rpossibility ings open parenthesis\quad seei 4 s period\quad 2 open$ D parenthesis = \{ A\ psi closing0 parenthesis , \ closingpsi { parenthesisu } , \ psi { v } , \ psi 1 \} $ \quad with \quad $ \ psi 0 = ( abc )can , be $ found in open square bracket 1 closing square bracket period .. Here we consider semirings open parenthesis see 4 period 2 open parenthesis D closing$ \ psi parenthesis1 = closing ( parenthesis ccc ) period $ \quad as \quad in \quad 4 . 7 . \quad Contrary \quad to \quad 4 . 7 \quad now \quad we \quad have \quad to \quad take \quad i n t o \quad account \quad the \quad a d d i t i o n a l unary4 period operation 9 period .... . open\quad parenthesisFrom the max i comma somorphism circ closing $ parenthesis\Phi :M hyphen semirings{\underline period{\}}\ .... Let A =rightarrow open brace 1 commaD {\ 2 commaunderline {\}}$ periodwe get period\quad period$ comma\hat{g n} closing( brace\Phi be a finite( at 0 least ) two ) element = set\Phi and let ( f ( 0 ) ) = \Phi ( v ) , $T underbar : = angbracketleft A to the power of A semicolon max comma circ right angbracket .... be the full concrete .... open parenthesis max commai . e circ $ closing . ,parenthesis\hat{ hypheng} ( semiring\ psi period0 ....) Here = max denotes\ psi the{ v maximum} . $ \quad Thus \quad $ ( g ( a ) g ( b ) g ( c ) ) = \hat{g} ( ( abc ) ) = ( cbc ) ,$ \quad but \quad t h i s \quad $ g $ \quad i s \quad not \quad a semilattice endomorphism s ince $ g ( a \wedge b ) = g ( a ) = c \not= b = c \wedge b = g ( a ) \wedge g ( b ) ,$ acontradiction .

\noindent Therefore no $ Y $ with 3 elements exists .

\ [ \ r u l e {3em}{0.4 pt }\ ]

\noindent Finally , we shall demonstrate how the Completeness Theorem 3 . 5 ( 2 ) applies to produce com − pletness criteria for concrete structures . \quad A completeness theorem for near − r ings ( see 4 . 2(A) ) can be found in [ 1 ] . \quad Here we consider semirings ( see 4 . 2 ( D ) ) .

\noindent 4 . 9 . \ h f i l l ( max $ , \ circ ) − $ semirings . \ h f i l l Let $ A = \{ 1 , 2 , . . . , n \} $ be a finite at least two element set and let

\noindent $ T {\underline {\}} : = \ langle A ˆ{ A } ; $ max $ , \ circ \rangle $ \ h f i l l be the full concrete \ h f i l l ( max $ , \ circ ) − $ semiring . \ h f i l l Here max denotes the maximum 384 .. R period P o-dieresis schel et al period : .. Completeness Citeria and Invariants period period period \noindentopen parenthesis384 \quad actingR as . m-hatwidest P $ \ddot ax{ pointwiseo} $ schel on A to et the al power . : of\quad A commaCompleteness .. see .. 1 period Citeria 4 period and 1 closing Invariants parenthesis . . period . .. Note that T underbar i s the full transformation algebra 384 R . P o¨ schel et al . : Completeness Citeria and Invariants . . . \noindentover A underbar( a c t i: n = g angbracketleft as $ \widehat A semicolon{m} $ max ax right pointwise angbracket on with $Aˆ s elector{ A open} , parenthesis $ \quad = isee d sub\quad A closing1 . parenthesis 4 . 1 ) period . \quad .. Note that ( acting as m ax pointwise on AA, see 1 . 4 . 1 ) . Note that T i s the full transformation For$ T the{\ purposesunderline ofb this{\}} example$ we i shall s the consider full transformation algebra algebra over A := hA; max i with s elector (= i d ). For the purposes of this example we shall overonly subsemirings $ A {\underline of T underbar{\}} containing: the = identity\ langle mapA periodA ; $ max $ \rangle $ withselector $( =$ i $d { A } consider only subsemirings of T containing the identity map . )In . accordance $ \quad withFor notation the purposes from Theorem of 3 this period example 5 comma putwe S shall : = A toconsider the power of A comma U : = open brace max closing brace and for S := AA,U := { F ⊆ AA Fonly subsetIn accordance subsemirings equal A to the with power of notation of $ A T {\ fromunderline Theorem{\}} 3 . 5 ,$ put containing the identitymax } and map for . let (F ) := F = AA∩ (F ∪ { let Srg clone max } ) (F ) \noindentSrgbe open the parenthesiscorrespondingIn accordance F closing closure with parenthesis operator notation : = ( overbar s from ee 3F Theorem . =1 .A 1 to ) the . 3 Itpower . follows 5 , of put A from cap $ clone2 S . 3open : that parenthesis = Srg A ˆ{ FisA cup the} open, brace U max : closing = brace\{ $ maxclosingleast\} parenthesisand f o r $ F \subseteq A ˆ{ A }$ , ◦)− F ∪ { d }. F l ebe t( the max correspondingsemiring closure containing operator open parenthesisi A We s ee shall 3 period say 1 that period 1is closings emiring parenthesis - complete period ....if It Srg follows from 2 period 3 that Srg (F ) = T ,U− T open parenthesis. Therefore F closing parenthesissets is inthe the least t erminology of Theorem 3 . 5 are j ust subsemirings of \ centerlineopencontaining parenthesis{ Srg the max $ identitycomma ( F circ ) closing map : parenthesis ( = or\ overline 1 - hyphen dimensional{\}{ semiringF containing} operation= A F ˆcup{ algebrasA open}\ bracecap i as d$ sub cl defined A o ne closing $ brace ( Fperiod\cup \{ $ maxWein\} shall) 1} say . 5 that ) . F is S s ince emiring max hyphen is complete a if Srg open parenthesis F closing parenthesis = T underbar sub period A Thereforeso - called comma S lupecki U hyphen operation sets in the ( it erminology. e . surj of Theorem ective and 3 period essentially 5 are j ust binary subsemirings ) , clone of T(A underbar∪ { max containing \noindentthe} ) .. is identity thebe clone ..the map corresponding of .. allopen operations parenthesis closure on or ..A, 1 hyphenso operator by Theorem dimensional ( s 3 ee .. . operation 53 we . 1 now . .. 1 algebras know ) . that\ ..h fas i inl .. l defined orderIt follows to .. in find .. 1 from period 2 5 .closing 3 that parenthesis Srg period$(maximal .. F S ince )$ .. max istheleast is .. a soU hyphen− sets calledit suffices S l-suppress to find upecki those operation Rosenberg open relations parenthesis that i period are invariant e period .. under surj ective max and . essentially In the next binary closing parenthesis comma clone\noindentlemma open parenthesis we( max s ingle A$ toout , the possible\ powercirc of candidates A) cup− open$ forbrace semiring these max relationsclosing containing brace . closing $ parenthesisF \cup is the\{ $ i $ d { A }\} . $ Weclone4 shall . of 1 all 0 say .operations that Lemma on $ A F comma . $ isIf so s a by emiring Rosenberg Theorem− 3 re periodcompleteifSrg lati 5 on we now on knowA that= $( in{1 order, ...,F n} to ) findis = invariant maximal T {\ underunderline {\}} { . }$ ThereforeUmax hyphen, sets then$ it , suffices i t U belongs to− find$ those tosets one Rosenberg in of the the relations fol t erminology lowing that c are lass invariant of es Theorem of re under lati 3 maxons . 5:period are .. j In ust the next subsemirings of $ T {\underline {\}}$ c o nlemma t a i n i• nwe g sbounded ingle out partial possible orders candidates on forA thesewhere relationsn is period either the least or the greatest element ; the4 period\quad• 1 0iequivalence period d e n t i ..t y Lemma\quad relations periodmap ..whose\quad If a Rosenberg b( locks or \ arequad re lati intervals1 on− .. ondimensional in A =the open usual brace\ linquad 1 ear commaoperation order periodhA period;\≤iquad; perioda l g ecomma b r a s n\quad closingas brace\quad .. is d e f i n e d \quad in \quad 1 . 5 ) . \quad S i n c e \quad max i s \quad a invariant under max comma .. then i t • proper subsets of A; \noindentbelongs toso one− ofc the• a l l fol ebinary d lowing S \ l central cupecki lass es relations of operationre lati ons where : (1 i .is e not . \quad a centralsurj e lement ective . and essentially binary ) , clone $ ( A ˆbullet{ProofA }\ .. .boundedcupWe partial\{ examine$ orders max the on\} A s ix)where iclasses sn the is either of Rosenberg the least or relations the greatest ( assuming element semicolon the reader is familiar clonebulletwith of.. them equivalence all , operations otherwise relations see onwhose e .$ b gA locks . , are [$ 1 intervals 0so ] byor in Theorem the [ 9 usual , 4 3. lin 3 . .ear 5 2 we order 1 now] ) angbracketleft and know eliminate that A in semicolon those order not less to orfind equal maximal right angbracket semicoloninvariant under max . \noindentbulletBounded .. proper$ partial U subsets− orders$ of A sets semicolon. itLet suffices4 be ato bounded find those partial Rosenberg order on relationsA with the thatleast element are invarianta under max . \quad In the next lemmabulletand we .. the binary s greatest ingle central out element relations possible whereb and candidates1 suppose .. is not a central4∈ forInv e these{ lementmax relations period} . Then . clearly a 4 n 4 b. If Proofa < b periodthen .. from We examinea 4 n theand s ixb classes4 b ofwe Rosenberg infer b = relationsmax (a, open b) 4 parenthesismax (n, assuming b) = n so theb reader= n, i is .familiar e .n is with \noindentthemthe commagreatest4 . otherwise 1 element 0 . \ seequad of e period4Lemma. Analogously g period . \quad .. openIf , if square aa Rosenberg > b bracketthen n 1i 0re s closing the lati least square on element\quad bracketon of or4 .. $. open A square= \{ bracket1 9 comma , . 4 period . 3 . period,Nontrivial n 2 1 closing\} $ equivalence square\quad bracketis re invariant closing lati ons parenthesis . underConsider and max eliminate , \ anquad equivalence thosethen not i invariant t relation under% ∈ maxInv period{ max } and let 0 0 0 belongsBoundedB be one to partial of one it orders s of blo the period cks .fol .. Let Takelowing preccurlyeq any cb, lass b ..∈ beB esand a bounded of let reb lati≤ partialx ≤ onsb order. :From on A with(b, b ) the∈ % leastand element(x, x) ∈ a% andwe 0 0 theget greatest ( max element(b, x b), andmax suppose(b , x)) preccurlyeq = (x, b ) in∈ Inv% whence open bracex ∈ maxB. closingConsequently brace periodB ..is Then an interval clearly a w preccurlyeq . r n preccurlyeq b period ..\ centerline If a. less t . b the then{ order$ \ bullet≤ . $ \quad bounded partial orders on $ A $ where $ n $ is either the least or the greatest element ; } fromPermutational a preccurlyeq n and relations b preccurlyeq . bLet we infer% b == max{( openx, α(x parenthesis)) | x a∈ commaA} b closing ∈ Inv parenthesis{ max } preccurlyeqbe .. max open parenthesis n\ centerline commaa b permutational closing{ $ parenthesis\ bullet relation =$ n\ soquad b given = nequivalence comma by some i period permutation relations e period n isα whose theof greatestA bwhere locks all are cycles intervals of α have in the the usual lin ear order $ \elementlanglesame prime of preccurlyeqA; length period\ leq .. Analogously\rangle comma; $ if} a greater b then n i s the least element of preccurlyeq period −1 Nontrivialp. Thus equivalence , in particular re lati ons period, n 6= α( ..n) Consider. On an the equivalence other hand relation from rho in(α Inv(n open), n) ∈ brace% and max(n, closing α(n)) brace∈ % and let B −1 \ centerlinebeit one foof it llows{ s blo$ \ that cksbullet period ($ max .. Take\quad(α any(propern b), comma n), max subsets b to(n, the α(n power of))) $Aof = prime (n, ; n in) $ B∈} and%, let bi less . e or. equal α(n) x less = orn. equal b to the power of prime periodContradiction .. From open parenthesis . Therefore b comma , no permutationalb to the power of relation prime closing is invariant parenthesis under in rho max and . open parenthesis x comma x closing parenthesis in rho\ centerline weAffine re{ lati$ \ onsbullet . $Let\quadhA; +binary, −, 0i centralbe an elementary relations abelian wherep 1− \groupquad onis notA and a suppose central e lement . } getthat .. openλ+ := parenthesis{(x, y, u, v max) | openx + y parenthesis= u + v} ∈ bInv comma{ max x closing} . parenthesis Let a ∈ commaA \{n} max. Then open parenthesis(n, a, n, a) b∈ toλ+ the power of prime comma x closing\noindentand parenthesisProof closing . \quad parenthesisWe examine = open parenthesisthe s ix x classes comma b of to Rosenberg the power ofrelations prime closing ( parenthesis assuming in the rho whence reader x is in B familiar period .. with Consequentlythem , otherwise B is an interval see we period . g . r\ periodquad t[ period 1 0 ] or \quad [ 9 , 4 . 3 . 2 1 ] ) and eliminate those not invariant under max . the order less or equal period Consequently (n, a, a, n) ∈ λ . \noindentPermutationalBounded .. relations partial period orders .. Let .. . \ rhoquad = openLet brace $ \ openpreccurlyeq parenthesis $ x comma\quad alphabe a open bounded parenthesis partial+ x closing order parenthesis on $ Aclosing $ parenthesiswith the bar least x in Aelement closing brace $ a in $ .. Inv and open brace max closing brace .. be .. a .. permutational ( max (n, n), max (a, a), max (n, a), max (a, n)) = (n, a, n, n) ∈ λ therelation greatest given by element some permutation $ b $ alpha and of suppose A where all $ cycles\ preccurlyeq of alpha have the\ in same$ prime Inv +\{ lengthmax \} . \quad Then clearly $ a \ preccurlyeqp period .... Thus n comma\ preccurlyeq in particular comma b n . equal-negationslash $ \quad I f $ alpha a open< parenthesisb $ then n closing parenthesis period .... On the other hand fromfrom open $ parenthesis a \ preccurlyeq alpha to the power n $ of minus and 1 $ open b parenthesis\ preccurlyeq n closing parenthesis b$ weinfer comma n closing $b parenthesis =$ max in rho $( and open a parenthesis , b n) comma\ preccurlyeq alpha open parenthesis $ \quad n closingmax $( parenthesis n closing , b parenthesis ) = in rho n$ so $b = n ,$ i.e $. n$ isthegreatest elementit .. fo llows o f that $ \ ..preccurlyeq open parenthesis max . $ open\quad parenthesisAnalogously alpha to the , power if $ of a minus> 1 openb $ parenthesis then $ n closing n $ parenthesisi s the least comma element n closing of parenthesis$ \ preccurlyeq comma max . open $ parenthesis n comma alpha open parenthesis n closing parenthesis closing parenthesis closing parenthesis = open parenthesis n comma n closing parenthesis in rho comma .. i period e period alpha open parenthesis n closing parenthesis = n period .. Contradiction period\noindent Nontrivial equivalence re lati ons . \quad Consider an equivalence relation $ \varrho \ in $ Inv \{ max \} and l e t $ BTherefore $ comma no permutational relation is invariant under max period beAffine one re of lati it ons s period blo cks .. Let . ..\ angbracketleftquad Takeany A semicolon $b plus , comma bˆ{\ minusprime comma}\ 0 rightin angbracketB $ and .. bel e t an elementary $ b \ leq abelianx p hyphen\ leq groupb ˆ{\ onprime A and} suppose. $ that\quad From $ ( b , b ˆ{\prime } ) \ in \varrho $ and $ ( x , x ) \ inlambda\varrho sub plus$ : = we open brace open parenthesis x comma y comma u comma v closing parenthesis bar x plus y = u plus v closing brace in Inv open brace max closing brace period .. Let a in A backslash open brace n closing brace period .. Then open parenthesis n comma a comma n comma a\noindent closing parenthesisget \quad in lambda(max$( sub plus and b , x ) ,$max$( bˆ{\prime } , x ) ) = ( x ,Equation: b ˆ{\prime open parenthesis} ) n\ in comma\varrho a comma$ a comma whence n closing $ x parenthesis\ in B in lambda . $ sub\quad plus periodConsequently .. Consequently $ B $ is an interval w . r . t . theopen order parenthesis $ \ leq max open. parenthesis $ n comma n closing parenthesis comma max open parenthesis a comma a closing parenthesis comma max open parenthesis n comma a closing parenthesis comma max open parenthesis a comma n closing parenthesis closing parenthesis = open parenthesis n\noindent comma a commaPermutational n comma n closing\quad parenthesisr e l a t i o n in s lambda . \quad subLet plus \quad $ \varrho = \{ ( x , \alpha ( x )) \mid x \ in A \}\ in $ \quad Inv \{ max \}\quad be \quad a \quad permutational relation given by some permutation $ \alpha $ of $A$ where all cycles of $ \alpha $ have the same prime length

\noindent $ p . $ \ h f i l l Thus , in particular $ , n \ne \alpha ( n ) . $ \ h f i l l On the other hand from $ ( \alpha ˆ{ − 1 } ( n ) , n ) \ in \varrho $ and $ ( n , \alpha ( n ) ) \ in \varrho $

\noindent i t \quad fo llows that \quad ( max $ ( \alpha ˆ{ − 1 } ( n ) , n ) ,$max$( n , \alpha (n)))=(n,n) \ in \varrho , $ \quad i . e $ . \alpha ( n ) = n . $ \quad Contradiction . Therefore , no permutational relation is invariant under max .

\noindent Affine re lati ons . \quad Let \quad $ \ langle A ; + , − , 0 \rangle $ \quad be an elementary abelian $ p − $ group on $ A $ and suppose that $ \lambda { + } : = \{ ( x , y , u , v ) \mid x + y = u + v \} \ in $ Inv \{ max \} . \quad Let $ a \ in A \setminus \{ n \} . $ \quad Then $ ( n , a , n , a ) \ in \lambda { + }$ and

\ begin { a l i g n ∗} \ tag ∗{$ ( n , a , a , n ) \ in \lambda { + } . $} Consequently \end{ a l i g n ∗}

\ centerline {(max$( n , n ) ,$max$( a , a ) ,$max$( n , a ) ,$ max$(a ,n))=(n ,a ,n ,n) \ in \lambda { + }$ } R period P o-dieresis schel et al period : .. Completeness Citeria and Invariants period period period .. 385 \ hspacewhence∗{\ n =f i a l periodl }R . .. P Contradiction $ \ddot{o period} $ .. schel Therefore et comma al . : ..\ noquad affineCompleteness Rosenberg relation Citeria lambda andsub plus Invariants .. i s invariant . . un . hyphen\quad 385 der max period R.P o¨ schel et al . : Completeness Citeria and Invariants . . . 385 \noindentRegular relationswhence .. and $n .. central = .. a re lati .$ ons\ ..quad of arityContradiction .. at .. least .. three . \ periodquad ..Therefore These .. relations , \quad .. areno .. totallyaffine Rosenberg relation whence n = a. Contradiction . Therefore , no affine Rosenberg relation λ i s invariant $ \reflexivelambda and{ of+ arity}$ at\quad least threei s period invariant .. Suppose un that− rho in Inv open brace max closing brace+ i s a nontrivial h hyphen ary open parenthesis derun max - der . max . h greaterRegular equal relations 3 closing parenthesis and central re lati ons of arity at least three . These totally reflexive relation period .... Take any open parenthesis a sub 1 comma period period period comma a sub h closing parenthesis element- relations are totally reflexive and of arity at least three . Suppose that % ∈ Inv { max } i slash\noindent rho periodRegular .... We have relations open parenthesis\quad and a sub\ 1quad commac e 1 n commat r a l \quad periodre period l a t period i ons comma\quad 1 closingo f a r iparenthesis t y \quad commaat \quad open parenthesisl e a s t \quad thre e . \quad These \quad r e l a t i o n s \quad are \quad t o t a l l y s a nontrivial h− ary (h ≥ 3) 1reflexive comma a sub and 2 comma of arity 1 comma at period least period three period . \ commaquad Suppose 1 closing parenthesis that $ comma\varrho \ in $ Inv \{ max \} i s a nontrivial totally reflexive relation . Take any (a , ... , a )element − slash%. We have $ hperiod− period$ ary period $ ( comma h open\geq parenthesis3 ) 1 $comma period1 periodh period comma 1 comma a sub h closing parenthesis in rho period .... (a , 1, ... , 1), (1, a , 1, ... , 1), Applying1 max to these tuples2 component hyphen wise and recalling the fact ... , (1, ... , 1, a ) ∈ %. Applying max to these tuples component - wise and recalling the fact \noindentthat .. rhototally i s .. invarianth reflexive .. under relation .. max yields . ..\ h open f i l l parenthesisTake any a sub $ 1 ( comma a period{ 1 } period, period . . comma . a sub , h a closing{ h parenthesis} ) element in −s l a s h that % i s invariant under max yields (a , ... , a ) ∈ %. Contradiction . Thus rho\varrho period .. Contradiction. $ \ h f i l l periodWe have .. Thus $ .. (no .. asuch{ ..1 rho} .. is,1,...,1),(1,a1 h { 2 } ,1,...,1),$no such % is invariant under max . invariantUnary underand b max inary period re lati ons . Note that unary central relations are j ust proper subsets of Unary and b inary re lati ons period .. Note that unary central relations are j ust proper subsets of A A and that every subset of A is invariant under max . Finally , let % ∈ Inv { max } be a \noindentand that every$...,(1,...,1,a subset of A is invariant under max period .. Finally comma .. let .. rho in{ Invh open} ) brace\ maxin closing\varrho brace be. a $binary\ h f i l l Applying max to these tuples component − wise and recalling the fact binary central relation . Suppose that 1 i s a central element of %. Then for every x, y ∈ A centralwe have relation period .. Suppose that .. 1 i s a central element of rho period .. Then for every x comma y in A we have \noindentopen parenthesisthat \ xquad comma$ 1 closing\varrho parenthesis$ i s comma\quad openi n vparenthesis a r i a n t \ 1quad commaunder y closing\quad parenthesismax y in i e rhol d s whence\quad open$ parenthesis ( a { x1 comma} , (x, 1), (1, y) ∈ % whence (x, y) ∈ % by applying max component - wise . Thus % = A2 i s trivial y. closing . parenthesis . , in a rho{ byh applying} ) max\ in component\varrho hyphen. wise $ \ periodquad ..Contradiction Thus rho = A to the . \ powerquad ofThus 2 i s\ trivialquad no \quad such \quad which contradicts the requirement that central relations are nontrivial . line − line $ \whichvarrho contradicts$ \quad the requirementi s that central relations are nontrivial period line-line invariantSome of theunder relations max . % li sted in Lemma 4 . 1 0 need not produce maximal semirings End {%} Someand of a the careful relations examination rho li sted in would Lemma improve 4 period ( 1i . 0 e need . not shorten produce ) themaximal li st semirings. However End open , all brace maximal rho closing brace and asemirings careful examination would improve open parenthesis i period e period .. shorten closing parenthesis the li st period .. However comma all maximal\noindent semiringsUnary and b inary re lati ons . \quad Note that unary central relations are j ust proper subsets of $ Aare $ among the s emirings End {%} produced from the above relations . This already provides areus among the fo the llowing s emirings : End open brace rho closing brace produced from the above relations period .. This already provides andus the that fo llowing every : subset of $ A $ is invariant under max . \quad F i n a l l y , \quad l e t \quad $ \varrho \ in $ 4 . 1 1 . Proposition ( Completeness criterion ) . A s et F ⊆ AA is s emiring Inv4 period\{ max 1 1\} periodbe a.. Proposition binary .. open parenthesis Completeness criterion closing parenthesis period .. A s et F subset equal A to the power - complete if and only if for every re lati on % lis ted in Lemma 4 . 1 0 there is an f ∈ F such ofcentral A is s emiring relation hyphen . complete\quad ifSuppose and that \quad 1 i s a central element of $ \varrho . $ \quad Then for every that f 6∈ End {%}. line − line $ xonly if , for yevery\ rein lati onA rho $ lis we ted have in Lemma 4 period 1 0 there is an f in F such that f negationslash-element End open brace rho closing brace We specialise this in two ways . First , we shall describe 1 - element generating s ets of T period line-line . We shall say that an equivalence relation % on A i s ≤ − regular if it i s nontrivial and all it s \noindentWe specialise$(x,1),(1,y) this in two ways period .. First comma we shall describe 1 hyphen\ in element\varrho generating$ whence s ets of T $( underbar x sub , period y .. ) We \ in blocks are intervals in hA; ≤i of the same length . \varrhoshall say$ that by an applying equivalence max relation component rho on A− i swise less or . equal\quad hyphenThus regular $ \ ifvarrho it i s nontrivial= and A ˆ all{ it2 s} blocks$ i s t r i v i a l which4 . 1 contradicts 2 . Proposition the requirement ( 1 - Generators that central of relationsT ). Let aref nontrivial∈ AA. Then Srg $ .(f) line = AA−lif i n e $ areand intervals only if in angbracketleft A semicolon less or equal right angbracket of the same length period 4 period 1 2 period .... Proposition .... open parenthesis 1 hyphen Generators of hline to the power of T closing parenthesis period .... Let f in A \noindentf Someis a of cyclic the permutation relations of $ \varrhoA such that$ lifelement sted in− slash LemmaEnd 4 .{% 1} 0for need al l not≤ − produceregular maximal semirings End to theequivalence power of A period re lati .... ons Then Srg open parenthesis f closing parenthesis = A to the power of A .... if and only if $ \{\f .... is avarrho cyclic permutation\} $ and of A such that f element-slash End open brace rho closing brace for al l less or equal hyphen regular equivalence re on A. In particular , if | A | is prime then Srg (f) = AA if and only if f is a cyclic latia ons careful examination would improve ( i . e . \quad shorten ) the li st . \quad However , all maximal semirings permutation of A. on A period .. In particular comma if bar A bar .. is prime then Srg open parenthesis f closing parenthesis = A to the power of A if and only if f Proof . “ ⇒: ” Suppose Srg (f) = AA. Then f preserves no relation l ist ed in Lemma 4 . is\noindent a cyclic permutationare among the s emirings End $ \{\varrho \} $ produced from the above relations . \quad This already provides 1 0 , in particular no ≤ − regular equivalence relation and no proper subset of A. The latter usof the A period fo llowing : implies that f has to be a cyclic permutation . Proof period .. quotedblleft double stroke right arrow : quotedblright .. Suppose Srg open parenthesis f closing parenthesis = A to the power of “ ⇐: ” Suppose f is a cyclic permutation of A with felement−slash End {%} for all ≤ − regular A\noindent period .. Then4 . f preserves1 1 . \quad no relationProposition l ist ed in Lemma\quad 4( period Completeness 1 0 comma in criterion ) . \quad A s et $ F \subseteq A ˆ{ A }$ equivalence relations on A. By 4 . 1 1 it suffices to show that f preserves no relation mentioned i sparticular s emiring no ..− lesscomplete or equal hyphen if and regular equivalence relation and no proper subset of A period .. The latter implies onlyin Lemma if for every re lati on $ \varrho $ lis ted in Lemma4 . 1 0 there is an $ f \ in F $ such that that4 . f 1 has 0 . to be a cyclic permutation period $ fquotedblleft\not\ in double$ End stroke $left\{\ arrow :varrho quotedblright\} .. Suppose. l i n f e is− al i cyclic n e $ permutation of A with f element-slash End open brace rho closing A cyclic permutation preserves no bounded partial order and no proper subset of A. If brace for all less or equal hyphen regular equivalence f preserves an equivalence relation %, then it acts on the set of blocks \noindentrelations onWe A specialiseperiod .. By 4 periodthis in 1 1two it suffices ways to . show\quad thatFirst f preserves , we no relation shall mentioned describe in 1 Lemma− element generating s ets of of % as a cyclic permutation as well , whence fo llows that all the blo cks of % have the $ T4 period{\underline 1 0 period {\}} { . }$ \quad We same length . Therefore , % i s ≤ − regular , and f preserves no ≤ − regular equivalence shallA .... cyclic say that .... permutation an equivalence .... preserves relation .... no .... bounded $ \varrho .... partial$ on .... $order A $ .... andi s .... $ no\ leq .... proper− $ .... subset regular .... of if A periodit i s .... nontrivial If and all it s blocks relation by the assumption . Finally , suppose f ∈ End {%} for some binary central relation aref .. preservesintervals .. an in .. equivalence $ \ langle .. relationA; .. rho\ commaleq ..\ thenrangle .. it ..$ acts of .. the on .. same the .. setlength .. of blocks . .. of .. rho .. as .. a .. cyclic . Take any (x, y) element − slash % and let s be an permutation as well comma whence fo llows that all the blo cks of rho have the same length period .. Therefore comma integer such that f −s(x) is a central element of %. Then (f −s(x), f −s(y)) ∈ % and from the \noindentrho i s less4 or .equal 1 2 hyphen . \ h regular f i l l Proposition comma and f preserves\ h f i l l no( less 1 − or equalGenerators hyphen regular of $ equivalence\ r u l e {3em relation}{0.4 by pt the} ˆ assumption{ T } ) period . $.. Finally\ h f i l l Let $ f fact\ in that Af preserves ˆ{ A } %.we $ get\ h f i l(x, l yThenSrg) ∈ %. Contradiction $( f . ) Thus = Aˆf preserves{ A }$ no\ h binary f i l l if central and only if commarelation . suppose f in End open brace rho closing brace .. for some binary central relation period .. Take any .. open parenthesis x comma y closing parenthesis\noindent element-slash$ f $ \ rhoh f i and l l letis s a be cyclic an permutation of $ A $ such that $ f element−s l a s h $ End $ \{\varrho \} integer$ f o suchr a l that l f $ to\ theleq power− of$ minus regular s open equivalence parenthesis x closing re lati parenthesis ons .. is a central element of rho period .. Then open parenthesis f to the power of minus s open parenthesis x closing parenthesis comma f to the power of minus s open parenthesis y closing parenthesis closing parenthesis\noindent inon rho and $ A from . the $ \quad In particular , if $ \mid A \mid $ \quad is primethen Srg $( f ) =fact A that ˆ{ A f preserves}$ if rho and we only get .. if open $ parenthesis f $ is x a comma cyclic y closing permutation parenthesis in rho period .. Contradiction period .. Thus f preserves no binaryo f $central A . $ relation period \noindent Proof . \quad ‘ ‘ $ \Rightarrow : $ ’ ’ \quad SupposeSrg $( f ) = Aˆ{ A } . $ \quad Then $ f $ preserves no relation l ist ed in Lemma 4 . 1 0 , in particular no \quad $ \ leq − $ regular equivalence relation and no proper subset of $ A . $ \quad The latter implies that $ f $ has to be a cyclic permutation .

\noindent ‘ ‘ $ \Leftarrow : $ ’ ’ \quad Suppose $ f $ is a cyclic permutation of $A$ with $ f element−s l a s h $ End $ \{\varrho \} $ f o r a l l $ \ leq − $ regular equivalence relations on $A . $ \quad By 4 . 1 1 it suffices to show that $ f $ preserves no relation mentioned in Lemma

\noindent 4 . 1 0 .

\noindent A \ h f i l l c y c l i c \ h f i l l permutation \ h f i l l p r e s e r v e s \ h f i l l no \ h f i l l bounded \ h f i l l p a r t i a l \ h f i l l order \ h f i l l and \ h f i l l no \ h f i l l proper \ h f i l l subset \ h f i l l o f $ A . $ \ h f i l l I f

\noindent $ f $ \quad p r e s e r v e s \quad an \quad equivalence \quad r e l a t i o n \quad $ \varrho , $ \quad then \quad i t \quad a c t s \quad on \quad the \quad s e t \quad o f blocks \quad o f \quad $ \varrho $ \quad as \quad a \quad c y c l i c permutation as well , whence fo llows that all the blo cks of $ \varrho $ have the same length . \quad Therefore , $ \varrho $ i s $ \ leq − $ regular , and $ f $ preserves no $ \ leq − $ regular equivalence relation by the assumption . \quad F i n a l l y , suppose $ f \ in $ End $ \{\varrho \} $ \quad for some binary central relation . \quad Take any \quad $( x , y ) element−s l a s h \varrho $ and let $s$ bean

\noindent integer such that $ f ˆ{ − s } ( x ) $ \quad is a central element of $ \varrho . $ \quad Then $ ( f ˆ{ − s } ( x ) , f ˆ{ − s } ( y ) ) \ in \varrho $ and from the fact that $ f $ preserves $ \varrho $ we get \quad $ ( x , y ) \ in \varrho . $ \quad Contradiction . \quad Thus $ f $ preserves no binary central r e l a t i o n . 386 .. R period P o-dieresis schel et al period : .. Completeness Citeria and Invariants period period period \noindentFor the second386 part\quad of theR statement. P $ \ itddot suffices{o} to$ observe schel that et if albar .A bar : \quad .... is primeCompleteness comma no nontrivial Citeria and Invariants . . . equivalence relation on A is less or equal hyphen regular period line-line 386 R . P o¨ schel et al . : Completeness Citeria and Invariants . . . \noindent4 period 13For period the .... second As a further part i llustration of the statement of 4 period 1 it 1 we suffices shall describe to all observe maximal that semirings if on $ A\mid = openA brace\ 1mid comma$ 2\ h comma f i l l 3is prime , no nontrivial For the second part of the statement it suffices to observe that if | A | is prime , no nontrivial closing brace \noindentequivalenceequivalence relation on relationA is ≤ − onregular $ A $. line is− $line\ leq − $ regular $ . line −l i n e $ in4 order . 13 to . get a completenessAs a further criterion i for llustration the full transformation of 4 . 1 1 we .... shall open describe parenthesis all max maximal comma circsemirings closing parenthesison hyphen s emiring on A period A = {1, 2, 3} \noindentThe relations4 . l ist 13 ed . in\ Lemmah f i l l As 4 period a further 1 0 are the i followingllustration : of 4 . 1 1 we shall describe all maximal semirings on $ Ain order= \{ to get1 a completeness , 2 , criterion 3 \} for$ the full transformation ( max , ◦)− s emiring on A. BoundedThe relations partial orders l ist ed : tau in sub Lemma 1 23 : 4 = . 1 1 prec 0 are 2 prec the 3 following and tau sub : 2 1 3 : = 2 prec 1 prec 3 open parenthesis as well as their duals which we do not have to consider s ince End open brace tau closing brace = End open brace tau to the power of minus 1 closing brace closing parenthesis Bounded partial orders : τ := 1 ≺ 2 ≺ 3 and τ := 2 ≺ 1 ≺ 3( as well as their duals which semicolon\noindent in order to get a completeness123 criterion213 for the full transformation \ h f i l l ( max $ , \ circ ) we do not have to consider s ince End {τ} = End {τ −1}); − $Equivalence s emiringon relations : $A epsilon . sub $ 1 2 bar 3 : = Capital Delta sub A cup open brace open parenthesis 1 comma 2 closing parenthesis comma Equivalence relations : ε := ∆ ∪ {(1, 2), (2, 1)} and ε := ∆ ∪ {(2, 3), (3, 2)}; Unary re lati ons open parenthesis 2 comma 1 closing parenthesis12|3 A closing brace and epsilon1|23 sub 1A bar 23 : = Capital Delta sub A cup open brace open parenthesis 2 \noindent: σ1 :=The{1}, σ relations2 := {2}, σ3 := l{ ist3}, σ12 ed:= in{1, Lemma2}, σ13 := 4{ .1, 13}, 0 σ23 are:= { the2, 3}; following : commaand 3 closing parenthesis comma open parenthesis 3 comma 2 closing parenthesis closing brace semicolon Unary re lati ons : sigma sub 1 : = open brace 1 closing brace comma sigma sub 2 : = open brace 2 closing brace comma sigma sub 3 : = open Binary central re lati ons : ζ2 := ∆ ∪ {(1, 2), (2, 1), (2, 3), (3, 2)} whose only central element is brace\noindent 3 closingBounded brace comma partial sigma orderssub 1 2 : = $ openA : brace\tau 1 comma{ 1 2 23closing} brace: = comma 1 sigma\prec sub 1 32 : = open\prec brace 13 comma $ and 3 closing $ \tau brace{ 2 2 , and ζ3 := ∆ ∪ {(1, 3), (3, 1), (2, 3), (3, 2)} whose only central element is 3 . comma1 3 sigma} : sub = 23 :A= 2 open\prec brace 2 comma1 \prec 3 closing3 brace ( semicolon $ as well as their duals which We know ( see 3 . 5 ( 1 ) ) that the maximal subsemirings of the full ( max , ◦)− semiring and AA are \noindentBinary centralwe do re lati not ons have : zeta to 2 : = consider Capital Delta s ince sub A End cup open $ \{\ brace opentau parenthesis\} = 1 comma $ End 2 closing $ \{\ parenthesistau ˆ comma{ − open1 }\} parenthesis) among End {%} where % i s one of the above relations . It i s straightforward to determine the 2; comma $ 1 closing parenthesis comma open parenthesis 2 comma 3 closing parenthesis comma open parenthesis 3 comma 2 closing parenthesis closing partially ordered set N ( ordered by inclusion ) of all these End {%}, see Figure 5 . brace whose only central element is Figure 5 . The partially ordered s et N \noindent2 comma andEquivalence zeta 3 : = Capital relations Delta sub $ A : cup\ openvarepsilon brace open parenthesis{ 1 2 1 comma\mid 3 closing3 } parenthesis: = \ commaDelta open{ A parenthesis}\cup 3 comma\{ Thus , by 3 . 5 ( 1 ) , there are precisely 8 maximal subsemirings of T containing the identity 1(1,2),(2,1) closing parenthesis comma open parenthesis 2 comma 3 closing\} $ parenthesis and $ comma\ varepsilon open parenthesis{ 1 3\mid comma 223 closing} : parenthesis = \Delta closing brace{ A } \cupmap\{ , (2,3),(3,2) \} ; $ whosenamely only central element is 3 period UnaryWe know re .... lati open ons parenthesis $ : see\sigma 3 period{ 5 open1 } parenthesis: = 1\{ closing1 parenthesis\} , closing\sigma parenthesis{ 2 ....} that: the = .... maximal\{ 2 subsemirings\} , .... of\sigma the .... full{ 3 ....} open: parenthesis = \{ max3 comma\} circ, closing\sigma parenthesis{ 1 hyphen 2 } semiring: = A to\{ the power1 of , A .... 2 are\} , \sigma { 1 End{%}for% ∈ {ε , ε , τ , τ , σ , σ , ζ2, ζ3}. 3 }among: End = open\{ brace1 rho , closing 3 brace\} where, rho12\|sigma3 i s1 one|23 of123{ the23213 above} 1 : relations13 = period\{ ....2 It i , s straightforward 3 \} ; to $ determine the partiallyTherefore ordered , we set can N open now parenthesis infer the following ordered by completeness inclusion closing criterion parenthesis : of all these End open brace rho closing brace comma see Figure 5\noindent period4 . 14 .and Proposition . Let A = {1, 2, 3}. A s e t F ⊆ AA is s emiring complete if and only Figureif for 5 period The partially ordered s et N \noindent Binary central re lati ons $ : \zeta 2 : = \Delta { A }\cup \{ ( 1 , 2 ) Thusevery comma% ∈ by {ε 312 period|3, ε1|23 5, τopen123, τ parenthesis213, σ1, σ13, ζ 12, closing ζ3} there parenthesis is an commaf ∈ F theresuch are that preciselyfelement 8 maximal− slash subsemiringsEnd of T underbar containing the,(2,1),(2,3),(3,2) identity{%}. line map− commaline \} $ whose only central element is namelyReferences \noindent 2 , and $ \zeta 3 : = \Delta { A }\cup \{ ( 1 , 3 ) , ( 3 , 1 End open[ 1 ] brace Aichinger rho closing , brace E . for ; rho Ma in opensˇ ulovi bracec´, epsilonD.;P sub 1 2o¨ barschel 3 comma , R epsilon . ; subWilson 1 bar , 23 comma J . S tau sub 1 23 comma tau sub 2 1),(2,3),(3,2) 3 comma.: sigmaCompleteness sub 1 comma for sigma concrete sub 1 3 comma zeta 2 comma zeta\} 3$ closing whose brace only period central element is 3 . Therefore commanear we - ringscan now. infer Technical the followingcompleteness Report MATH criterion - : AL - 1 4 - 2002 , TU Dresden , \noindent4September period 14We period know 2002 ....\ Propositionh f i (l l 25(see3.5(1)) period .... Let A = open\ h brace f i l l 1that comma the 2 comma\ h f i l 3 l closingmaximal brace subsemirings period .... A s e\ th F f subset i l l o equalf the A\ toh thef i l l f u l l \ h f i l l ( max power$ , of\ Acirc is s emiring) complete− $ semiring if and only if$Aˆ for { Apages}$ )\ h . f i l l are [ 2 ] Hebischevery , rho U . in ; Weinertopen brace , epsilonH . J . sub : 1 2Halbringe bar 3 comma. epsilon B . Gsub . 1 Teubner bar 23 comma , Stuttgart tau sub 1 1 993 23 comma . Z tau bl sub 0 8 2 2 1 9 3 . comma 1 6 0 sigma 3 5 sub 1 comma sigma\noindent sub 1 3among comma Endzeta 2 comma $ \{\ zetavarrho 3 closing brace\} $ there where is an f in $ F\ suchvarrho that$ f element-slash i s oneof End the open above brace rho relations closing brace . period\ h f i l line-linel It i s straightforward to determine the References \noindentopen squarepartially bracket 1 closing ordered square set bracket $ N .. Aichinger ( $ ordered comma by.. E inclusion period semicolon ) of .. all Ma caron-sthese Endulovi c-acute $ \{\ subvarrho comma .. D\} period, $ semicolonsee Figure .. P o-dieresis5 . schel comma .. R period semicolon .. Wilson comma .. J period .. S period : .. Completeness for .. concrete near hyphen rings period .. Technical .. Report .. MATH hyphen AL hyphen 1 4 hyphen 2002 comma .. TU .. Dresden comma .. September .. 2002\ centerline .. open parenthesis{ Figure 25 5 . The partially ordered s et $ N $ } pages closing parenthesis period \noindentopen squareThus bracket , by 2 closing 3 . 5 square ( 1 bracket ) , there .. Hebisch are comma precisely U period 8 maximalsemicolon Weinert subsemirings comma H of period $ JT period{\underline : .. Halbringe{\}} period$ .. B containing the identity map , period G period Teubner comma Stuttgart 1 993 period .. Z bl 0 8 2 9 period 1 6 0 3 5 \noindent namely

\ [ End \{\varrho \} f o r \varrho \ in \{\ varepsilon { 1 2 \mid 3 } , \ varepsilon { 1 \mid 23 } , \tau { 1 23 } , \tau { 2 1 3 } , \sigma { 1 } , \sigma { 1 3 } , \zeta 2 , \zeta 3 \} . \ ]

\noindent Therefore , we can now infer the following completeness criterion :

\noindent 4 . 14 . \ h f i l l Proposition . \ h f i l l Let $ A = \{ 1 , 2 , 3 \} . $ \ h f i l l A s e t $ F \subseteq A ˆ{ A }$ is s emiring complete if and only if for

\noindent every $ \varrho \ in \{\ varepsilon { 1 2 \mid 3 } , \ varepsilon { 1 \mid 23 } , \tau { 1 23 } , \tau { 2 1 3 } , \sigma { 1 } , \sigma { 1 3 } , \zeta 2 , \zeta 3 \} $ there is an $ f \ in F$ such that $ f element−s l a s h $ End $ \{\varrho \} . l i n e −l i n e $

\noindent References

\ hspace ∗{\ f i l l } [ 1 ] \quad Aichinger , \quad E.; \quad Ma $ \check{ s } $ u l o v i $ \acute{c} { , }$ \quad D.; \quad P $ \ddot{o} $ s c h e l , \quad R.; \quad Wilson , \quad J. \quad S.: \quad Completeness for \quad c o n c r e t e

\ hspace ∗{\ f i l l } near − r i n g s . \quad Technical \quad Report \quad MATH − AL − 1 4 − 2002 , \quad TU \quad Dresden , \quad September \quad 2002 \quad ( 25

\ centerline { pages ) . }

\ centerline { [ 2 ] \quad Hebisch , U . ; Weinert , H . J . : \quad Halbringe . \quad B . G . Teubner , Stuttgart 1 993 . \quad Zbl0829.16035 } R period P o-dieresis schel et al period : .. Completeness Citeria and Invariants period period period .. 387 \ hspaceopen square∗{\ f i bracket l l }R . 3 P closing $ \ddot square{o bracket} $ schel .. Jakub et acute-dotlessi al . : \quad kov acute-aCompleteness hyphen Studenovsk Citeria acute-a and Invariants comma D period . . semicolon . \quad ..387 P dieresis-o s chel comma R period : .. Cayley repres entation of algebras with addi hyphen R.P o¨ schel et al . : Completeness Citeria and Invariants . . . 387 [ 3tional ] \quad s emigroupJakub s tructure $ \acute and{\ theirimath invariant} $ re kov lati ons $ \ periodacute ..{ Technicala} − $ Report Studenovsk MATH hyphen $ \ ALacute hyphen{a} , $ D . ; \quad P [ 3 ] Jakub ´ı kov a´− Studenovsk a,´ D.;P o¨ s chel , R . : Cayley repres entation of $ \7ddot hyphen{o} 200$ 1 commas c h e l TU , RDresden . : \ commaquad OctoberCayley 200 repres 1 .. open entation parenthesis of 1 algebras 9 pages closing with parenthesis addi − period tialgebras onal swith emigroup addi - s titructure onal s emigroup and their s tructure invariant and retheir lati invariant ons . re\quad lati onsTechnical. Technical Report MATH − AL − openReport square MATH bracket - 4 AL closing - 7 square - 200 bracket 1 , TU .. Dresden Lausch comma , October H period 200 semicolon 1 ( 1 N 9 o-dieresis pages ) bauer . comma W period : .. Algebra of polynomials period7 − 200 .. North 1 , hyphen TU Dresden Holland Publishing , October Company 200 1 comma\quad ( 1 9 pages ) . [ 4 ] Lausch , H . ; N o¨ bauer , W . : Algebra of polynomials . North - Holland 1Publishing 973 period .. Company Z .. b l 0 2 , 8 31 period 973 . 1 2 Z 1 0 .. b 1 l 0 2 8 3 . 1 2 1 0 1 [ 4open ] \ squarequad Lausch bracket 5 ,H. closing ;N square $ bracket\ddot ..{o} Pilz$ comma bauer G , period W . : ..\quad Near hyphenAlgebra Rings of periodpolynomials Volume 23 . of\quad NorthNorth hyphen− HollandHolland Publishing Company , 1 973[ .5 ]\quad PilzZ ,\quad G . : bl0283.1210Near - Rings . Volume\quad 23 of1 North - Holland Mathematics Studies . MathematicsNorth - Studies Holland period Publishing North hyphen Company Holland , revised edition , 1 983 . Z b l 0 5 2 1 . 1 6 2 8 Publishing Company comma revised edition comma .. 1 983 period .. Z b l .. 0 5 2 1 period 1 6 .. 2 8 [ 6 ] P o¨ schel , R . : Concrete repres entation of algebraic s tructures and a general [ 5open ] \ squarequad bracketP i l z , 6 G closing . : square\quad bracketNear − .. PRings o-dieresis . Volume schel comma 23 of R period North :− ..Holland Concrete repres Mathematics entation of Studies algebraic s. tructures North − andHolland a PublishingGaloi− s theory Company. In, revised : H . editionKautschitsch , \quad , W .1 B 983 . M . u¨\quadl ler ,Z and b l W\quad . N o¨0bauer 5 2 1, editors . 1 6 :\quad 2 8 generalContributions Galoi to the power to General of minus Algebra s theory period , 249 – 272 . Verlag J . Heyn , Klagenfurt 1 979 In : .. H period Kautschitsch comma W period B period M u-dieresis l ler comma and W period N dieresis-o bauer comma editors : .. Contributions [ 6. ] \quad ( ProcP . $ \ Klagenfurtddot{o} $ sConf c h e l . , R 1 . 978 : \ )quad . Concrete repres entation of algebraic s tructures and a general to General Zbl 0396.08003 $ GaloiAlgebra ˆ comma{ − }$ .. 249 s theory endash 272 . period .. Verlag J period .. Heyn comma .. Klagenfurt .. 1 979 .. period .. open parenthesis Proc period .. In : [\quad 7 ]H P .o¨ Kautschitschschel , R : ,W.A general B . Galois M $ \ theoryddot{u for} $ operations l ler ,andW.N and relations $ and\ddot concrete{o} $ bauer , editors : \quad Contributions to General Klagenfurtcharacter .. Conf - period ization .. 1 of 978 related closing algebraic parenthesis s period tructures . Report R - 0 1 / 80 , Zentralinstitut f AlgebraZ b l .. 0 , 3 9\quad 6 period249 0 8−− 0 0272 3 . \quad Verlag J . \quad Heyn , \quad Klagenfurt \quad 1 979 \quad . \quad ( Proc . \quad Klagenfurt \quad Conf . \quad 1 978 ) . u¨ r Mathematik und Mechanik , Akademie der Wissenschaften der DDR , Berlin 1 980 ( with open square bracket 7 closing square bracket .. P o-dieresis schel comma R : .. A general Galois theory for operations and relations and concrete \ centerlineGerman and{Z b l \quad 0 3 9 6 . 0 8 0 0 3 } character hyphen Russian summaries ) , 1 0 1 pp . Z bl 0 4 3 5 . 0 8 0 0 1 ization of related algebraic s tructures period .. Report R hyphen 0 1 slash 80 comma Zentralinstitut f u-dieresis r Mathematik [ 8 ] P o¨ schel , R . : Galois connections for operations and relations . In : K . [ 7und ] Mechanik\quad P comma $ \ddot Akademie{o} $ der s Wissenschaften c h e l , R : \ derquad DDRA comma general Berlin Galois 1 980 .. theory open parenthesis for operations with German and and relations and concrete character − Denecke , M . Ern −e´ and S . Wismath ( eds . ) : Galois connections and applications . izationRussian summaries of related closing algebraic parenthesis, s comma tructures .. 1 0 1 . pp\quad periodReport .. Z bl 0 R4 3− 5 period0 1 / 0 80 8 0 ,0 .. Zentralinstitut 1 f $ \ddot{u} $ r Mathematik undKluwer Mechanik , Dordrecht , Akademie 2003 der , Wissenschaften der DDR , Berlin 1 980 \quad ( with German and open square bracket 8 closing square bracket .. P( o-dieresis to appear schel ) . comma R period : .. Galois .. connections for operations and relations period In : .. K period Denecke comma M period Ern minus acute-e sub comma \ centerline[ 9 ]{ PRussiano¨ schel , summaries R . ; Kalu ) , zˇ\quadnin ,1 L 0 . 1 pp A . .: \quadFunktionenZbl0435.0800 - und Relationenalgebren\quad 1 } and. S Deutscherperiod Wismath Verlag .. open der parenthesis Wissenschaften eds period , Berlin closing 1parenthesis 979 . Z : b .. l Galois 0 4 connections 2 1 . 0 30 and 4 9 applications period .. Kluwer comma .. Dordrecht 2003 comma Birkh a¨ user Verlag Basel , Math . Reihe Bd .67, 1979. Z−− b − 0−−4−1 − 0−3−0−−4− [ 8open ] \ parenthesisquad P $to\ appearddot{ closingo} $ parenthesis s c h e l , Rperiod . : \quad Galois \quad connectionsl for8. operations4 and relations . In : \quad K . Denecke , M . Ern [ 1 0 ] Quackenbush , R . W . : A new proof of Rosenberg ’ s primal algebra characteri−zati− $−{open \acute square{e bracket}} { 9, closing}$ square bracket .. P dieresis-o schel comma .. R period semicolon .. Kalu caron-z nin comma .. L period .. A period andon S the . Wismath - orem \.quad In( : eds Finite . ): Algebra\quad andGalois Multiple connections - valued Logic and applications , volume 28 . of\quad CollKluwer . , \quad Dordrecht 2003 , : .. FunktionenMath . Soc hyphen . ..J und. Bolyai Relationenalgebren , 603 – 634 . period North ..- Deutscher Holland Verlag , Amsterdam 1 98 1 . ( Proc . Szeged der Wissenschaften comma Berlin 1 979 period .. Z b l .. 0 4 2 1 period 0 30 4 9 \ centerlineConf .{ 1( 979 to appear ) . ) . } Birkh dieresis-a user Verlag Basel comma MathZ bl period 0 4 7 Reihe 9 . 0 Bd 8 0 period 0 3 67 comma 1 979 period Z to the power of minus minus b minus sub l 0 to the power of minus minus 4 to the power of minus 1 minus sub 8 period 0 to the power of minus 3 to the power of minus 0 to the power of minus [ 1 1 ] R e´ dei , L . : Algebra I . Akademische Verlagsgesellschaft Geest & Portig minus[ 9 ] 4\ minusquad subP 4 $ \ddot{o} $ s c h e l , \quad R.; \quad Kalu $ \check{z} $ nin , \quad L. \quad A.: \quad Funktionen − \quad und Relationenalgebren . \quad Deutscher Verlag K.-G ., Lei− − pzig 1 959 . Z b l 0 0 9 2 . 0 2 9 0 1 deropen Wissenschaften square bracket 1 0 , closing Berlin square 1 979 bracket . \ ..quad QuackenbushZ b l \ commaquad R0 period 4 2 1 W . period 0 30 : 4 .. 9 A .. new proof of Rosenberg quoteright s primal [ 1 2 ] Rosenberg , I . G . : U¨ ber die funktionale Vollst a¨ ndigkeit in den algebra characteri to the power of minus zati to the power of minus on the hyphen mehrwertigen L ogiken . Rozpr .Cˇ SAV Rˇ ada Mat . P rˇ ir . V eˇ d . , Praha , 80 ( 4 ) ( 1 \ hspaceorem period∗{\ f i .. l l In} Birkh : .. Finite $ Algebra\ddot{ anda} Multiple$ user hyphen Verlag valued Basel Logic ,Math. comma .. volume ReiheBd 28 of .. $ Coll . period 67 Math , period 1 Soc979 period . .. JZˆ period{ − − }$970 b ) , $ 3− – 93{ .l } Z0 bl ˆ 0{ 1 − 9 9 − . 3 } 0 24 0 ˆ{ 1 − } 1 − { 8 . } 0 ˆ{ − } 3 ˆ{ − } 0 ˆ{ − − } 4 − { 4 }$ BolyaiReceived comma October 603 endash 23 634 , 2002 period North hyphen Holland comma Amsterdam 1 98 1 period .. open parenthesis Proc period Szeged Conf period .. 1 979 closing parenthesis period \noindentZ bl 0 4 7 9[ period 1 0 ] 0\ 8quad 0 0 3 Quackenbush , R . W . : \quad A \quad new proof of Rosenberg ’ s primal algebra $ characteri ˆ{ − } z a topen i ˆ{ square − }$ bracket on the 1 1− closing square bracket .. R acute-e dei comma .. L period : .. Algebra I period .. Akademische Verlagsgesellschaft .. Geestorem .. .ampersand\quad In Portig : \ Kquad periodFinite hyphen Algebra G period comma and Multiple Lei to the− powervalued of minus Logic minus , pzig\quad volume 28 o f \quad Coll . Math . Soc . \quad J. Bolyai1 959 period , 603 ..−− Z b l634 .. 0 .. . 0 North 9 2 period− Holland 0 2 9 .. 0 1 , Amsterdam 1 98 1 . \quad ( Proc . Szeged Conf . \quad 1 979 ) . open square bracket 1 2 closing square bracket .. Rosenberg comma .. I period .. G period : U-dieresis ber die funktionale .. Vollst a-dieresis ndigkeit\ centerline in .. den{Zbl0479.08003 .. mehrwertigen L ogiken period } Rozpr period C-caron SAV caron-R ada Mat period P caron-r ir period V e-caron d period comma Praha comma 80 open parenthesis 4 closing parenthesis\noindent ..[ open 1 1 parenthesis ] \quad 1R 970 $ closing\acute parenthesis{e} $ comma d e i , 3\ endashquad L.: 93 period\quad .. Z blAlgebra 0 1 9 9 period I . 3\quad 0 2 0 1Akademische Verlagsgesellschaft \quad Geest \quad $ \Received& $ Portig October K 23 . comma− G 2002 $ . , Lei ˆ{ − } −{ pzig }$ 1 959 . \quad Z b l \quad 0 \quad 0 9 2 . 0 2 9 \quad 0 1

\noindent [ 1 2 ] \quad Rosenberg , \quad I. \quad G $ . : \ddot{U} $ ber die funktionale \quad V o l l s t $ \ddot{a} $ ndigkeit in \quad den \quad mehrwertigen L ogiken . Rozpr $ . \check{C} $ SAV $ \check{R} $ ada Mat . P $ \check{ r } $ i r . V $ \check{e} $ d. ,Praha ,80(4) \quad ( 1 970 ) , 3 −− 93 . \quad Zbl0199.30201

\noindent Received October 23 , 2002