T1p chi Todn hoc Journal of Mathematics t9(1)(1991), 1-21
INFORMATION SEMIMODULESA}ID ABSORBINGSUBStrMIMODULtrS
JONATHANS" GOLAN
Dedicatedto ProfessorNguyen Dinh Tri on his strtiethbirthday
Abstract. The purpse ol this short surueyis to intrduce the notion ol an informatil:n semimdrfu, and indieate some ol the apphcatioru such structutasin theorcticd comprder science. Abo, we uill indicale some ol the matlumatical thnry availdle lor inlormailon seminahiles by bingtng Poyatos' cotutruction ol a uersion ol the Jordan - Hrjlder theorcrnfor inlormdion semi- mdules. To this end, we ttuolcerurie ol the notion ol an absorbingsubsemimdule ol a semimdule, studid indep.ndently bg Poyotos ond T&dresfui.
1. THERE ARE PLENTY OF INFORMATION ALGEBRAS OUT THERE
A. semiring is a nonempty set .R on which operationsof addition and multi- plicationhave been de5nedsuch that the followingconditions are satisfied: l) (8, *) is a corrmutativemonoid with identity element0p; 2) (R,.) is a monoid with identity elementlB I Op; 3) Multiplication distributes over addition from either side; 4) Opr - 0R : rQp for all r € R. We will usually write 0 insteadof 0p and l instead of 1p if there is no room for confusion. Also, multiplication will normally be written as concatenation. We will always denote E \ {0} by ft.. We will follow the standard conventions: if a is an element of a semiring R and fr is a positive integerthen the sum o * ... * a (t summands) will be denoted by /co and the product a . . . a (k factors) will be denotedby oft. We alsoset oo:lR for all a €.R'. For a detailedintroduction to the theory of semirings,as well as many examplesand applicationsof this theory, referto IGolan,1991]. A semiring R is zerosumfreeif it satisfiesthe condition that r-l-s € E- for all r € R' and all ,s€ R. Countably complete semiringsare always zerosumfree,as Ionathan S. Golan are those partially+rdered semirings in which 0 is the unique minimal element. See [Golan, 1991]for details. Thus, for example, the semiring of linear relational oper- ators introduced in [Ioannidis & Wong, 1990] to describe the opertors performed on the relations in a database systems is zerosumfiee. Note that a zerosumfree semiring is as far from being a ring as possible: for every nonzero element r of a ring R there exists an element s such that r*s:0. A semiring R is entireif it satisfies the condition that rs € fi* for all r € R* and all s € ft*. It is clear that a sufficient condition for a semiring ,R to be entire is that iI be left multiplicatiuely cancellatitse,namely if sr: ro'implies that s: s'for all r €,R-. Similarly,a suf- ficient condition for R to be entire is that it be right multiplicatively cancellative. Thus, in particular, a diuision semiring, i.e. a semiring in which every nonzero element has a (two-sided) multiplicative inverse, is surely entire. Entire zerbsumfree semirings arise naturally in graph theory, especially in the consideration of path problems, and prove considerableinformation about the structure of graphs. See, for example, [Gondran & Minoux, f984]. With this in mind, Kuntzmann [1OZZ]called such structtres int'ormation algebras(algibres d,erenseignement). The semiring N of nonnegativeintegers, the semiring Q* of nonnegative rational numbers, and the semiring R+ of nonnegative real numbers are all natural examplesof information algebtas. With these examplesobviously in mind, in [Eilenberg,1974], information algebrasare called as "positive semirings" and many other authors follow this terminology. However, "positive" is also used in a different sense in semiring.theory, and so it is best to avoid its use in this context. Iwano and Steiglitz [fooO] have defined the structure of an information algebra on the set of all convex polygons, with the sum of two polygons being taken to be the convex hull of their vector summation. Anothei important class of information algebrasconsists of thosesemirings of the form (R, max, min), where R is some bounded totally-ordered set having minimal element 0 and maximal element 1. In particular, we can take ,R to be B = {0,1} or I : [0,1], obtaining l}neboolean semiring and the fvzzy semiring respectively. If R is an integral domain then the semiring of all ideals of .R is an information algebra. A division semiring is either an information algebraor a division ring [Mitchell & Sinutoke, 1982]. One of the most widely studied and applicablesemirings is an information algebra: if ,4 is a nonempty set then the free monoid F A is the set of all finite stringsataz...o,. of elementsof A. (Note: this set is usuallydenoted A*; herewe have deliberately chosena nonstandard notation in order not to causesonfusion * with the use of given above.) This can be turned into a monoid by taking rnul- tiplication to be concatenationof strings" The identity of this monoid is just the empty string, which we denoteby fl. The elementsof A are called symbolsor let- ters and the elementsof F,4 are called words. Subsetsof F,4 are called (formal) languoges on A. The set sub(FA) of all formal languageson A has the struc- ture of an information algebrain which addition and multiplication are definedby h,lorm at ion semim od ules an d absorbing su bsemimodules
L- L' : LUL'and LL' : {r*'lw € L andwt€ L'}. The additiveidentity of this semiring is @ while its multiplicative identity is {D}. This information algebrawas first consicleredas part of Kleene'salgebraic formulation of the theory of machines [Kleene, 1956] and lies at the heart of modern algebraic automata theory. For further refer to & Reutenauer, .Lallement, details [Berstel 1988],fEilenberg,l974), f979]or[Salomaa&Soittola, 1978]. If S: {LCFAIL:$or n e I} then S is an information subalgebraof sub(F,A). Another important example of an information algebra is the following. Let R : RU{-} and defineoperations @ and I on ^Rby settingoOb: max{a,b} arrd a 8 6 : a 16. Then (R, O,8) is a zerosumfreedivision semiring and so is an in- formation algebra,called the schedulealgebra, having very important applications in optimization theory, graph theory, and the modeling of industrial processes. Refer;for example,to [Cuninghame-Greene,1979]. Similarly,(R u {*},rnin,f) is an information algebra with importanb applications in graph theory and opti- mization,such as the solutionof the shortest-pathproblem lGondran & Minoux, 19841.This semiringhas a subsemiring(N u {*}, min, +) rvhichis an information algebra as well and has important applicationsin the theory of formal languages and automata theory, including the study of the nondeterministiccomplexity of a finite automaton;it is alsoused for gost minimizationin operationsresearch. See \fascle, 1986]and ISimon,1988]. For a generalzationof this constructionin which R is replaced by-an arbitrary linearly-orderedabelian group, refer to [Butkovic, le85l. If R is a semiring which is not a ring then 1 * r * 0 for all r € R. In this case,we set P(fi) : {0} U {1 + rlr e R} and note the followingresulr.
1.1 PROPosrrroN: If R is a semiringwhich is not a ring rfien p(ft) is a -.ubsemiring of R which is an information algebru.
Let B be a semiringand let (X,- ) be a finite faetorizationmonoid, namely a monoid satisfying the condition that for each c € X there set {(r/, r,,) € X x x x'-2" : r) is finite. on the set R << x )) of all functionsfrom X to i? define operationsof addition and multiplication as follows: t) (l + g)("): f(r) + s(r); z) (f :,I{/(r')g(r't)lrt-xtt: s)(') "}; for all z€X and/,g€R<
1.2 PROPOSITION [Hebisch & weinert, 1990b]: If .R.is an infot- mation algebra and X is a finite factorization monoid then any subsemiring of R << X >> containing R < X ) is an information algebta. The conversr of Proposition 1.2 is not necessarilytrue. Conditions under which it is have been investigated in [ilebisch & Weinert, 1990b1. In particular, if X is a right - absorbing monoid and R is a semiring then n <,X > is an information algebra if and only if R is. Similarly, if X is the free abelian group generated by an arbitrary set then R < X ) is an information algebra if and only if ,R is. (Note that, in this case,X doesnot have the finite factorization property so R << X >> cannot be defined') The conclitions of being zerosumfreeand entire are independent and there are plenty of sernirings satisfying one of these conditions but not the other. For example, if l? is a commutative integral domain then R is entire but not zerosum- free. Entire senniringswhich are not zerosumfreehave been studied in [Hebisch & Weilert, 1990]. On the other hand, if ,R is a commutative ring which is not an integral dornain then the semiring of all ideals of R is zerosumfreebut not entire. Countably complete semirings, which have important applications in theoretical computer science, are zerosumfreebut not necessarilyentire and hence not nec- essarily information algebras [Krob, 1987]. A linkage between the two conditions does exist, however, for finite semirings. 'Weinert, 1.S PROIIOSITION [Hebisch & 1990alz Every finite entire semiring wtrrichis not a fing is an information algebra. Any semiring R can be embeddedin an information algebraR0 constructed in the following manner: let u be an elementnot in R and set Rd : RU{u}. Extend the operationsof addition and multiplication on R to Re by setting r lu, : ll*r : r forall r e Re andru-:r!,r: uforall r € R0.It isstraightforwardtoverifythat R0 is an information algebrasatisfying having additive identity equal to u. Actually, in order for r?d to be an information algebra we can even relax the conditions on a semiring and insist oniy that (n, +) be a commutative semigroup rather than a monoid or than it is a monoid but that condition (a) in the definition of a semiring is not completely satisfied. Such a situation occurs in the follbwing exarnplescoming from theoretical computer science. To begin with, let us considerthe following model of nondeterministiccom- puter programs defined in [Main & Black, 1989]. We are given a nonempty set D of "states" in which the c;-rmputercan be, one of which is the distinguishedstate Irtorm ation semimoduleE end zbaorbing su bsemimodules
I of being in an unending loop. Let -B be the family of all relations r on D ( i.e. nonempty subsets of D x D) satisfying the condition that (J-, d) e r if and only if d,:I. The operations of addition and multiplication on .E are defined as follows : 1)r*s:rUs; 2) rs: {(d,d.") € Rlthere exist an elementdt e D such that (d,,d,,)€ r and (dt,dt')e s\. Then (n, +) is a commutative monoid with identity element 0 : (f ,I) and (R,.) is a monoid with .identity element I : {(d,,d.)ld e \. Moreover, multiplication distributes over addition from either side while 0r : 0 for al r € R. It is not true, however, that r0 : 0 for all r € R. Therefore R is not a semiring. Nonetheless,it can be embeddedin an information algebra Rd by the above construction. We also note that R' : {r € RlrO : 0} is a subsemiringof R. The elementsof .B correspondto nondeterministicprograrns on D. Addi- tion correspondsto a nondeterministicchoice ("either r or s") and multiplication correspondsto sequentialcomposition ("first r then s"). Related to this example is the notion of a command,olgehra defined in [Hes- selink, f990], which is in turn a special case of the processalgebras studied in [Baeten, Bergstra & Klop, 1987]. Such an algebra consistsof a set .R on which operations of addition ( nondeterministicchoice) and multiplication ( sequential composition) are defined such that (R, +) is an idempotent commutative semi- group, (.R,.) is a semigroup, and multiplication distributes over addition from either side. Such algebras lack both an additive and a multiplicative identity so th.t .Rd, defined as above, still lacks a multiplicative identity. This can be remedied by the process of Dorroh ertension; embed .R0 in S : Rd x N via the map o - (a,0) and define operations of addition and multiplication on S by - (o,n)*(o',n'): (a* at,n*nt) and (a,.n)(o',n,) (no,*nta* aat,nn,).This turns 5 into an information algebrahaving multiplicative identity (u,l), where u is the additive identity of Ra An interpretation of the element 0 in a general process algebra is given in [Baeten & Bergstra, lg90], where it is understood to mean the processof "pred- icatable failure", which the system under considerationwill try' to avoid if at all possible (as opposed to "deadlock", which the system may enter but cannot try to avoid). If .R and S are semirings then a function ,y i R --+ ,S is a rnorphism of semirings if and only if : 1)r(on) : o*i 2)t(tn): l"i - 3) r(t * ,y(r) + 1(r') for all r,rt € R; "') ) 1(rrt) :1(r\1(r') for all r,r' € R. A morphism of semirings which is both injective and surjective is called an isomorphism. If there is an isomorphism between semirings R and ,S we write Jonathan S, Golan n=s. We remark that the embedding of R into Rd is not a morphism of semirings since it does not preserve additive identities. Also, note that a morphic image of an information algebra needs not be an information algebra, which suffices to show that the class of information algebra is not a variety . We also note that this class is, clearly, not closed under taking direct products as well. If. B : {0, f } is the boolean semiring (with idempotent addition and multiplication) and R is an --+ a,rbitrary semiring then the function 1: R B definedby f(0) :0 and i(") : f for r / 0 is a morphism. of semirings if and only if .R is an information algebra. Finally, for the record, we mention some rather obvious algebraic properties of inforrnation algebras. Clearly an information algebra can have no nonzero nilpotent elementsnor any nonzeronilpotent ideals. An elementa of a semiring R is cornplementedif and only if there exists an element 6 of R satisfying ab : ba : O and a*6 : 1. Complementedelements are important in the study of semirings,and central complemented elements are used to determine direct-sum decompositions. See [Golan, 1991]for details. If R is an information algebra, however, R cannot have any complemented elements other than 0 and 1.
2. OVER INFORMATION ALGEBRA WE CONSTRUCT INFORMATION SEMIMOD ULES
If l? is a semiring then a commutative monoid (M, +) with additive identity 01a is a left R-semimod.uleif there exists a function R x M -, M, denoted by (r,*) e rrn and called scalar multiplicatron, which satisfies the following condi- tions for all elements r and r' of E and all elements rn and mt of M ; I) (rr')m: r(r'm); 2) r(m + *') : rm + rm'l 3) (r + rt\m - rm 1- r'rn; 4) Lpnr,- nni 5) rDy - Oya: ORm. Right R - semimodulesare definedin an analogousmanner^ We will usually work with left semimodules,with the correspondingresults for right semimodules taken as proven without explicit mention. Any semiring is clearly both a left and right semimodulesover itself. When we speakof a semiring as a semimoduleover itself, we will mean a left semimoduleunless the contrary is specificallyindicaterl, Again, we will denoteM \ {O-} by M*. We begin with an example: A left seminearringis a structure (p, +, ") satis- fying all of the axioms of a semiringexept that distributivity of mutiplication over addition from the right may not hold. That is to say,if a,b, and c are elementsof : P then a(b + c) ab * oc but (b + c)a is not necessarilyequal to ba * ca. Such Information semimodules and absorbing subsemimodules
structures arise, inter alia, in the study [Golan, 1991]. They also arise naturall senseof [Baeten, Bergstra & Klop, 1g€ .processes" situation we have a set of { * ("alternative composition;) and . (.. left seminearring p6 then : {frf 1ft e I to verify that in p6 both distributive chain of subsemirings of p then so is U. every seminearring has a maximal subs r?-semimodule. A similar situation occurs for the quemirings introduced in [Elgot, 1gT6]to study computation using abstract automata. A quemiring is a strricture of the form -R X M, where R is a semiring and,M is a tert R-semimodule, on which addition is defined componentwiseand multiprication is given by (o, *i.@,,*,) : (y.a',arn-'1rn). This is not a semiringsince (a, m) (0,0) : (0,0) onry when rnO: 0. Also, while right distributivity of multiplication ovei addition always holds, left distributivity holds only sometimes. However, the quemiring .R x M certainly R' : e .R} as a subsemiring :o:t1i"r {(a,O)la and can profitably be studied as left -R'-semimodule. a A left R-semimodule having more than one element is nontriuial. An ^R- semimoduleM is zerosumfreeif and only if it satisfiesthe condition that m + m, € M* for all e M. and all m, e M. It is enfrre if and only if it satisfiesthe condition that rm € M'for all r € R" and all m € M-. A semimodulewhich is both zerosumfree and entire is an fnfar mation $emimod.ure.
2'r PROPOSITION: A semiring R is an information algebraif and only if there exrsts a nontriviar left information R-semimodure.
lf M is a left -E-semimodule then a nonempty subset N of M is a subsemi_ module of M if and only if it is closed under trr.ing sums and scalar products. clearly{o}isalwaysasubsernimotluleofaleftR-."rr"imodtleM.AleftR-semi- module which has no subsemimodules other,than {0} and itself is sintple:.lr N' are ry subsemimodulesof a left E-semimodule lti "na fh"n .nr* N, : {n + n,ln e N and n' € N') is a subsemimodureof M containing both N and N,. If .d is a nonempty family of subsemimodules of a left - B Jemimodule M then o.{ is a subsemimoduleof M, as : is I A {mr + ... + mtlm; € UA f.. *.fr"i}." Let R bL a semiring and ret M be a reft r?-semimodule, Furthermore, let ideal(-R)be the sdt of all ideals of ft and t* s;i"1u4 be semimodules the set of all sub- of M.' Then ideal(^B)has a for details) l.9ntl and (ssrn(M),+; i, , tion in ssm(M) being defined * uborr" IN: {otrr+.".+ anrnla;€I; x;€N}" 8 lonathan S. Golan
If A is a nonempty subset of a left R-semimodule M then the set .R.4 of all elementsof M of the form alrnr * . . . * anrnn with a; € R and m; € A,is a subsemimoduleof. M and, indeed, is the intersectionof all subsemimodulesof M containing .4, called the subsemimoduleof M generatedby A" By convention, we set R6: {0}. If RA: M then A is a setof generatorsfot M. A nonempty subset A of a left -R-semimoduleM is weaklylinearly independentif and only if for any finite subset{rt,...,rn} of A and finitesubset {or,...,an} of .B we have I o;*; oniy when ai : O for all t; it is linearly iniependenf if and only if for any finite subset{"r,...,rn} and ,4,and finitesubsets {or,...,ar,} and -- {6t,... ,6r,} of .R,we haveLotrt Db;rt only when ai: bi for all l <. i 1n. If M is an information semimodulethen any nonempty subset of M- is weakly linearly independent. A [weakly]linearly independentset of generatonsfor a left R- semimoduleM is a fweak]6csfs for M. Thesenotions are discussedin I Takahashi, 1e84bl.
2.2 FROPOSITION: .Let R be an information algebra. A left ft-semi- module Iv{ is an information semimoduleif and only if it fras aweak basis.
A completeclassification of all cyclic informationsemimodules over the in- tbrntation alget'ra N of natural numbers is given in fTakahashi,1g85]. If &f is a left E-semimodulethen V(M): {m€ Mlm*m, :0 for some elementm' af ,rt{} is surely a subsemimoduleof M. The semimoduleM is zerosum- free preciselywhen v (M) : {0}. At the opposite extreme, a left fi-semimodule .&f is an R-rnoduleif and only if V (M) - Jy[. A subsemimoduleof a zerosumfree[resp. entire] left E-semimodule is again zerosurnfree[resp. entire]. Thus the classof information semimodulesis closed under taking subsemimodules. If N is an information subsemimoduleof a left R-semimoduleM then N $ N': {0}U {n*n,ln € N- and n, e N,} is a subsemimoduleof l{ * N' which contains N but doesnot necessarilycontain ly'r. nfm€Mwe setT(rn):RmEM. An elernentu of a left .R-semimoduleM \s infinitein M if and only if utffr, - w for all m € M. If u,'and u)' are infinite elementsof M then u : w :1r,. 'Ihus *,u, any left ft-semimoduleM can haveat most one infinite element. The element u is strongly infi,nite\n M if and only if rw : w for all r € .8" . If the semiring _Ris antisimple then every infinite elementof a left ,R-semimoduleis strongly infinite. indeed, in this situation any element r of R- is of the form 1 * r, foi r, € .R- and so ru: (1+ rt)w ==w{r'w: u'. If M hasastrongly-infiniteelem'ent ru then the set C(M) : {0,tr,'} is the crur of M; if M has no strongly infinite elementswe set C(h[) : {0}. Note that C(M) os alwaysan information subsemimoduleof M. We say that M is crucial if and only if M : C (M). Otherwiseit is noncrucia/.If w is a strongly-infinite elemenLof a left r?-semimoduleM and if l/ is a subsemimodule of lt[ then l[ u {r} is also a subsemimoduleof M. Also, we note that if u is a Infor m ation semimodules and abcorbing s u bsemim od ules
strongly-infinite element of a left R-semimodule M lhen T(w) : C(M). If u is a . strongly-infinite element of a left N-semimod:uleM then for each element m of M the set w(*): {mt e Mlrn*m' : u} u {o} is a subsemimoduleof M. clearly C(M) c W (rn) for each m € M. Let R be a semiring and let M be a left R-semimodule. If (X,- ) is a finite factorization monoid then, as we have seenin Section l, we can define the semiring .R << X >> and its subsemiringR < X >. Moreover, if M << X >> is the set of all functions from X to M then M << X )) has the structure of a left R << X >>-semimodule if we define addition and scalar multiplication as follows: if u,u € M <
2.3 .PROPOSITION: If N is a subsemimoduleof a left R- semimodule M then: IJ N C P(N, M) if and only if N is zerosumfreel 2) P(I,I,P(N, M)) : P(N, M) for any subsemimodule1{ of a left R-semi- module M; 3) If M' r'sa subsemimoduleof M tlen P(N, M'): P(N, M) nMt; 4) If {M,li € O} is a family of R-semimodules and N is a subsemr'moduleof M : n{M;lu € n} then P(N,M) : n{P(N, Mi)li € O}; 5) If {N,la e n} is afamily of subsemr'modulesof M satisfying t-lN; : N then nP(N, ,M) I P(N,M); 6) If Mt is a subsemimodule of P(N, M) then N + Mt : N U Mt and N $ &/': N. If M and N are left R-semimodules then a map a : M - N is an R-ho- momorphisrnif and only it (m I m')a: ma * mta and (rrn)a -: ,(ma) for all m,nl' € M and r € R. Note that we follow the standard convention of writ- ing homomorphism as acting on the side opposite scalar multiplication so that homomorphism of right -R-semimoduleswill be written as acting on the left. A bijective E-homomorphism is an R-rsomorphism. The kernel of an .R-homomor- phism a ; M -+ N of left .R-semimodulesis /cer(a) : {m € Mlma: 0.,v}. This is surely a subsemimodule of M. Indeed, this subsemimodule of M which is su6- troctiue,i.e.itsatisfiestheconditionthatif m'e ker(a)andm'*m€lcer(a) for some m e M then rn € ker(a). Subtractive subsemimodulesplay an important part in the developrnent of semimodule theory. See [Golan, 1991] for details. If o and,p are -R-homomorphisrns from a left E-semimodule M to a left .R-semimod- ule N then the map d * 0 fram M ta N defined by o * 0, * H mat * rn| is also an.R- homomorphism. Indeed, it is easy to see that the set Ilorr-rp(M,N) of all R-homomorphisms from M to N is a monoid (:N-semimodule) under this operation, the identity element of which is the'constant map rn r+ 0. Information eemimodulee and absorbing subsernimodules 1l
Let M be a left R-semimodule. An R-homomorphism from M to itself is called an R-endomorophismof M. The set End,p(M) of all R-endomorphisms of M is a semiring under the operations of addition and multiplication defined by setting m(a -t fi) : ma I mp and *("F) : (ma)p for all rn e fu[ and all u,0 € Endp(M). The additive identity of Endp(M) is the map given by m *-, A for all m € M and its multiplicative identity is the map given by rn + nl for all meM. lf M is a left ,R-semimoduleand S : EndR(M) Lhen M is a right S- semimodule,with scalar multiplication definedby m.a: rrld.for all m {: M and ak s. For example, let -4 be a nonempty set and as before,let S : sz6(F,4) be the semiring of all formal languageson A. Followingthe terminology of [Abranisky & Vickers, 1990]we say that a transition system (P, r) over A consitsof a nonenrpty setP togetherwithasubset -- of Px Ax P,wherewewritep\ q insteadof (p,o,q) €-. The elementsof A are the atomic actionsof the transition system, while the elementsof P are the processesof the system. Each element a of A defines a function 0o : sub(P) --* sublP) given by rn|o: {q e Plp 3 q for somep e *}. we can expandthis notion by definingd,,, for each w €E A recursivelyas follows: 1) If ro - n then du, is the identity map; 2) If w : uafor u € FA and o €,4 then m0* : (.m0,,)0o. Furthermore, if L € S then we can define a function 01 : sub(P) -- sub(p) bv - setting m0n u{rn|-lw e L}. Moreover,(sub(P),u) is a left B- semimodule. If , € ,9 then 07 is a B-endomorphism of sub(P) for each L e s and, indeed, {|LIL e S} is a subsemiring of the semiring of B-endomorphisms of sub(P). Moreover, the map L ,- 0t, is a morphism of semiringsfrom 5 to the semiring of B-endomorphisms of sub(P). Thus sub(P) is canonicallya right ^g-semimodule, where, plp for each m e sub(P) and eachL € 5, we havemL: {q e 3 qfor some p € M and u € I). This semimoduleis zerosumfreebut not necessarily entire. If ,9' is the subsemiringof ,9 satisfiedby ,S,: {L € .Sl, : 0 or tr € ,} then sub(P) is an information ,5t-semimodule. Let,R be an information algebraand let {(Mn,+;)li e o} be a family of information left .B-semimoduleseach of which has a strongly-infinite element u;,.. Without loss of generality,we can assumethat the M; are hi.loint as sets. Set : Iv[ o{Mi \ c(t\{)lf e ni u {0,?r}, where 0 and u do not belongto any of the M;. Define addition and scalar multiplication of elementsaf IuI as follows: 1) 0+ nt : rn,*0 : m andrQ:0 for all m € M and all r € Rz 2) w*m:m*w: u,and ru):n) forall m€ M andall r € R: 3) If m,Tn'€ M \ {0,u'} then rn*rn, is definedto be m*;mtif both m and m' belong to IVi \ C(M) for somef e O; otherwise,mI m, : w: ) It m € M \ C(Mi) then rrn is the sameas the correspondingvalue in 14. t2 Jonathan S. Golan
It is straightforward to check that M, as definde above, is an information semi- module having a strongly-infinite element ur. Moreover, for each l € O we have an injective E- homomorphism \; : M; '-+ M which sends the zero-element 0; of M; to 0, sends ur; to ur, and sends each element of M; \ C(Mr) to itself. We will denote the semimodule M constructed in this way by lJ;ErM;. An important method of constructing surjective .R-homomorphismsof semi- modules is via .R-congruencerelations. An'equivalence relation : on a left -B- semimodule M is an r?-congruencerelation if and only if m - n1,t and n = nl imply that neI n: rn' + nt and rrn : rn' for all r € R. If : is an .R- congruence relation on M and if rn € M then we denote the equivalenceclass of rn with respect to this relation by rnl= and set M f =: {*l= lrn e M}. We can defineoperations of addition and scalarmultiplication on M l= by settingrnl= +nl=: (rn + n) l: and r(mf =) : (rm)l: for all m,n € M and all r e R. These operations turn M l= into a left R-semimodule,called the factor semimoduleof h[ by =. Moreover, we have a surjective,-R-homomorphismM ---+Ml= defined by ** ml= for all m € M. For further details and examplesof congruencerelations on semimodules, refer to IGolan, 1991]. If R is an information algebrathen any nontrivial information subsemimod- *contracted" ule of a left R-semimoduleM can be to a strongly-infinite element of a factor semimoduleof a subsemimoduleof M. Indeed,let R be an information algebraand let N be a nontrivial information subsemimoduleof a left B-semimod- ule M. Let Mtbe a subsemimoduleof P(N,M) properly containing N. Define a relation -N on M' bysetting nt- Nnttif and only if rn : m' or {m,m,} eN'. It is straightforward to checkthat this is an .r?-congruencerelation on M,and so we can definethefactor semimoduleMtf -N.Indeed, -n: if m € M,\N thenmf {m} so M'l_-!! isjust[M'\ff']u{u.'},whereuisastrongly-infiniteelementtu,j-1r.lt a: -* Mt M'l-x is the canonicalsurjection deficned by a: rn,-- mf -tt then a i,s not injective unlessN has preciselytwo elements.On the other hand, we always have ker(a) : {O}. For notational convenience, we will denote the semirnodule M,f - x bv M'llN and, for each element m of Mt, we will write ,nl lN instead of mf -7,1. Thusnrl lN:{rn}forall rn€M' \Nand rllN ={u}iorall n€N-. Follow- ing [Takahashi,lg84], we will call MtllN the .Reesfactor semimodule of M,by N.
3. INFORMATIONSEMIMODULT]S HAVE ABSORBINGSUBSEMIMODULES
We now introduce poyatos a construtionfirst givenby [tO7Z,1g73a, lg7gbl in his construction - of a versionof the Jordan Holderih"o."- which *""ri i"ij lnfo r m at ion semimod ules an d a bsorbing s u bsemim od ules 13 for semimodules,and later studied independentlyin [Takahashi,1984a]. A sub- semimodule N of a left -B-semimodileM is ahsorbingif and only if it is entire and M : P(N,M). In this case we write N e M. That is to say, N a M if and only if trf is entire and M +N. - N-. This surely implies that N is zero- sumfree. Indeed, an entire left .R-semimoduleM is an information semimodule if and only if M a M. Thus we seethat absorbingsubsemimodules are information subsemimodules.The converseneed not be true. By Proposition 2.1 we seethat it is meaningful to talk about the existenceof nontrivial absorbingsubsemimodules only forsemimodulesoverinformationalgebras. If n € N then {0}u{i € Nlz > n} is an absorbing subsemimoduleof N. Trivially, {0,1a} a M for every left ft-semimodule M. If r.uis an infinite element of a semimoduleM then ur is strongly infinite if and only if {0,r'rl} C M . Let M be an entire subsemimodule. If N is a subtractive subsemimoduleof M satisfying the condition that N : (M \ M') U {0} is a subsemimoduleof M then surely N a M. Thus, for example, if M is a left N-semimodule which is not an abeliangroup then V(M) satisfiesthe givencondition and so N : (M\y(lZ))u{O} is an absorbingsubsemimodule of M.
3.r PROPosrrloN: For subsemimodu,fesN, N/. lvf' of a ]eft .R-semi- module M we have: l,) If N E M and N q M' tlen N I L4'; 2) If N = M' then M' e P(N,,Lf); 3) If N is zerosumfreethen lf I P(,n/,I4); 4) If A is a nonempty family of absorbing su bsemimodulesof M having intersectionN then N C M; 5) lf A is a nonemptyfamily of absorbingsub.semim odules of h[ having gnion N then N E M; 6) If N,N'a M thenNaN'+ {0} andN+N'C M,wherein thissituation we in fact have N + l/' : ly' U N'; 7)IfNCMthenN.'rhttIM'; 8) If M is entire and N, Nt a Lt tien N $N'C N n l/,; 9) If N C M then N u N/ is a subsemirnoduleof M. Thus, in particular,from (2) and (:) of Proposition3.1 we seethat if N is an information subsemimoduleof a left rt- semimoduleM then P(N, M) is the largestsubsemimodule of M containingN as an absorbingsubsemimodule. lVe also note that, bV (a) and (5) of Proposition3.1, the family assm(,tf) of all absorbingsubsemimodules of M is a sublatticeof the lattice ssm(M) of all subsemimodulesof M.Indeed, this lattice is distributiveby Proposition3.t (6). The following example is in fTakahashi,1984a1: Let R be an antisimple serniring.If M is a left ,R-semimodulewhich is not a group then N : {0} U [M \, V (M)l is an absorbingsubsemimodule of M. Indeed,it is clearly closedunder 14 Jonathan S. Golan addition. If n € N* and r € R* then r : 1 *s for some s € R. Thus rn]- n' : O .implies that z f (sz * n') - 0, which is impossible, proving that rn € N. Finally, let rn € M and rz€ N* : M\ V(M). If m*n f N- then there existsan element m' of M satisfying0: rn+n*m' -- n-t (mI*'), contradictingthe assumption thatn€N*. Thus NCM. Wealsonotetheconverse:if Nf MandNsatisfies the conditionthat M\V(M) g N* thenM\V(M): -l[* for if reV(M) nN* then there exists an element y of V (M). satisfying y + r : 0, contradicting the factthatM+N.:N*. Let M be a left R-semimodule containing an entire simple submodule No, let ssm(M) be the set of all subsemimodulesof M, which we have already noted is a left semimodule over the semiring ideal(R) if all ideals of R. LetU : {N € ssm(M)lN I N"). If {0} *I e ideal(,R)andNe UthenlN f INs:1rgo. Thus U' : U u {{0}} is an absorbingsubsemimodule of ssm(M). Let R be an information algebra and let {M,li € 0} be a family of infor- mation left R-semimodules,each of which has a strongly-infinite element. Let M : lj;e n ,U{; and, for each i € CI,let ),; : M; --+M be the injective -B-homomor- phism. Then M;\; I M for each i € f,l. I{ote that if R is an information algebra then it is zerosumfreeand so R a R but it does not follow from this that if ft is an information algebra then every nontrivial left R-semimodule has a nontrivial entire subsemimodule. Thus, for example, N is an information algebrabut any nontrivial additive abelian group is a left N-semimodule having no nontrivial absorbingsubsemimodules. If M and N are disjoint left R-semimodulesthen a Takahashiertension of M by N is a left .R-semimodule? the underlying set of which is M u N. and the operations of addition and multiplication on which are so defined that N f 7. These extensionswere first consideredin [Takahashi,1984a]. We will denote the family of all Takahashiextensions of M by N by Tak(M,N). By what we have already seen,a necessarycondition for Tak(M, N) to be nonempty is that N be an information semimodule. If N is an information ft-semimodulethen a function rpfromN" to N'is atranslationif andonlyif ,h("+n'):,h(")+rL': n+rb("') for all n,n' e N-. Denote the set of all translationsof N- by tr(N-); this set is always nonempty since it surely contains the identity map. It is also closed under composition of functions and, indeed, is easily seento be a monoid under cornposition. An elementm of a left R-semimoduleM \s cancellableif and only if rn,lmt -- lzlrmt'implies mt : m,'tfor all rnt,mtt € M. The semimoduleM is cancellatiueif and only if everyelement of M is cancellable.See lGolan, 1991] for details. If N is an informationsemimodule having a nonzerocancellable element ns then tr(N.) is an abelian monoid. Indeed, if ns is a cancellableelement of lV- then for p and - rl' in tr(N-) and n € N we havepth(n1 ro) : p,h(n) * no rh(") +9:(n6) and ,hp(" 1ro) : rztr/tp("o): rh(")+\D(no)and so erb(n)tn6: rhe(")*no. Since gemirnodules Informatiorl and absorbing subsemimodules 15
z6 is cancellable,this implies that erh(n) : thp(n) for all n € N* . rf € Tak(M,N) then each element rn of M induces a translation p* € tr(N-) given" by p.,(x) : rn * r. Thus we have a function pr : M -- tr(N=) given by pr ; n1 + tprn, and this is'in fact a morphism of monoids since prn*rn, : emgn,, for all ntrnl,' e M. This morphism satisfiesthe additional condition: (.) If r € R, m € M,, and z € N* then r[pr(^)(")]: er(rm)(rn). A morphism of monoids from M to tr(N.) with this property will be called cd- missible, Thus, for example, if M is any left R-semimodule and N is an infor- mation semimoduledisjoint from M then the morphism e : M -* tr(N-) defined : by e(nz)(n) n is admissible. The set of all admissible morphisms irom the monoid (r\a,+) to tr(N') will be denotedby Adm(M,N). If O i. the operation onAdm(M,N) definedby setting(oO 0)(*): a(*)p(rn) for altm, e M then (Adm(M,N),e) is a monoid with identity ui"^"rrt r. If N has a nonzerocan- cellable elementthen, by what we havenoted above,this monoid is ahelian (i.e., it is a N-semimodule). Let
3.2 PROPOSITION: Let R be a semiring and let a: M , Mt be an R_ homomorphismof left R-semimodulessatisfying the condition that ker(cr) ; M. If Nt a M' thenN: N'a-r = M.
If N is an absorbingsubsemimodule of a left fi-semimodule M then the left R-semimodule Ntlltrl is defined for any R -semimodulely'' of M containing 1y'. It is straightforward to verify that the map N, r- NtllN inducesa bijectiveorder - preserving correspondencebetween the family of all subsemimodulesof ,4,fcon- taining N and the family of all subsemimodulesof M I lN, which in turn restricbs to a bijectivecorrespondence between the family of all absorbingsubscmirnorlules of M containing N and the family of all absorbing subsemimodulesof M I lN. The following 'converse' result givesa of this constructionfor semirn6dulesover information algebras. 16 Jonathan S. Golan
--+ B.g pROPOSITION z Let R be an information algebraand let a: M N be an R-homomorphism of left R-semimodules.If w' is a strongly-infinite element of N containedin Ma then M' : u)'d.-r U {0} is an absorbingsubsemimodule of M.
3.4. PROPOSITION [Poyatos, 1973a]: Let N and Nt be absotbing subsemimodules of a left R-semimoduleM ' Then: r)NqNuN'; z)NnN'fN'; 3) (N u Nt),tlN is R-isomorphicto N' I l(N n N'). 3.5 PROPOSITION [Poyatos, 1973b]: Let R be an information alge- bra and 1etM be a left R-semimodule.If N, N', W andW' aresubsemimodules of (J,(Jt,V M satisfyingN' Q N andW' a W, and if and Vt arethe subsemimodules of M defined by: i)U: N'u(N)W); ii)ut : N'u (N n W'); iii)V:W' u(NnW): iv)Vt - W'u (N'n W); Then: 1)UtiU andVtCV; 2)u I lu' = vI lv'. !'rom these results we can then concludethe following.
3.6. PROPOSITION [Poyatos, 1973a]: If N c N' are proper ab- sorbing subsemimodulesof a left R-semimoduleM then (M I lN) I lw I lN') is B-isomorphicto MllN'. Let M be a left rE-semimodulehaving a nonempty family I of absorbing subsemimodulesand let N : Ul. By Proposition3.1(5) we seethat N is again an absorbing subsemimoduleof. M and, indeed,is the unique maximal absorbing srrbsemimoduleaf M. We will denote this subsemimoduleby ,a(M).
4. CON{POSITIONSERIES AND THE JORDAN- HOLDER THEOREM
If 14 is a semimodulewe seethat C(M) a NL An absorbingsubsemimoduk: N of M is quasiminimalif and only if it properlycontains C(M) and there is no absorbingsubsemimodule of M properlycontaining C(M) and properlycontained in N. That is to say,if Af is a quasiminirnalleft R-semimodulethen either A1 has no proper absorbingsubsemimodules or it has preciselyone suchsubsemimodule, narnelyits crux. A nontrivial left -R-semimoduleM is quasisimplelf and only if it is quasiminimaland has no primitive elements.That is to say,a quasiminimal Iaformation semimodules and absorbing subsemim odules fi
left ,R-module is qua^sisimpleif it either has no strongly - infinite elements or has one which is not primitive. If ur is a strongly - infinite element of an information ft- semimodule M which has no proper absorbing subsemimodulesother than C(M) then we know by Proposition 3.1(6) that M* + M* equalseither M* or {tr}. In the first case, M is quasisimple.In the secondcase, u, is'a primitive elementof M. consider the following example I Poyatos,1gz3a]: Let T be a nonempty set and let z,w be distinct elementsnot in ?. Definethe operation * on X : TU{z,w} by setting: 1) t + t' : n) for all t,tt € T; 2) x + z : z * x :r for all r € X; 3) c* rD: u) t x :tr., for all c € X; Moreover, for each /c e N and c € X define the element /cc of X as follows: 4) Or: z; 5) lc: r; 6) kr: ur if k > 1 and r I z; 7)kz:zforallfteN. Then X becomes a left N-semimodule with strongly- infinite element nr. Indeed, if ?' is any subset of ? then ?' u {",u} is an absorbing subsemimoduleof x. Moreover, tr.,is primitive in X. From the definitions we seethat if N is an absorbing subsernimoduleof a left R-module of M and N' is an absorbing subsemimoduleof M properly containing N then the following conditions are equivalent: l) N' I lN is quasiminimal; 2) There is no absorbing subsemimodule of M properly containing N and properly cpntained in N'. we alsonote that'tf A(M) f M then,by proposition 3.4, the ft- semimodule M I I A(M) is quasisimple.
4.r PRoPosrrroN [poyatos, lgz2,l973a]: Let R be an informatian algebra. If M is a left R-semimodule and if m is ^n ui"^"nt of M* then: 1) m € A(M) if and only if f(*) . M; 2) It m e A(M) then Rm $a(*) is t.heunique smal/estabsorbrng subsem,'- module of M containing m. Moreover,Rm $,4(M) : T(m) In particular; if rn e, A(M). tl=. q rlml for all m, e T(m)" . set T'(*): {0}u {*, e rQe-lr@)+ r@}.'".(:"') 4.2. PRoPosrrroN [poyatos, rg73b]: Let R be an information algebra and left M be a \eft R-semimodule having an absorbing subsemimodule" If m A(M)* € thenT'(*) is a maximal proper absorbingsubsemimodule of T(m). we note that u.'€ A(M) and if me A(M) therrm*tu: u for eachr €.8 andso w €T(m). Thuswe conclude that ciui e iq*'1for each*a-itaq. 18 Jonathzn S. Golan
4.3 PROPOSITION [Poyatos, L972]z Let R be an information atrgebra and let M be a left .R-semimodule.havin g a strongly-infrnite element w. Then:' 1) L(M) : {0} u {* € A(M)lf @) : C(M)} t's an absorbing subsemri- module of M w.hiclr is t.he unique maximal subsenilmoduleN of A(M) satisfying n I rn : w for aII n € N* and nl e A(M)-. 2) If m € A(M) \ C(M) then T(m) : C(M) if and onlly it T(m) is quasi- minimal.
4.4 PII,OPOSITION [Poyatos, 1973a]: tret R be an infarmation al- gebra and let M he a noncrucial information R- semimodule having a strcngly - infinite element 'w. Then M is quasisimp/e if and only if T(*) - lvf for al] m€M\c(M). 4.5 COROLLARYI Let R be an information semiring and let M be a noncrucial qua^sisimpleinformation R-semimodule having a strongly - infinite elementw. Then T'(*') : C(M) for all m' € M \ C(,Vf).
4.6 PROPOSITION [Poyatos, 1973a]: Let R be an information alge- bra and let M be a left R-semimodulehaving a strongly - infi.niteelement w. lf N is a quasiminirnal absorbingsubsemimodule of M then either N.has a primitive element or it is quasisimple. If R is an information algebra and 14. is a left R- sernimodule having an absorbing subsemimodulethen, for every rn € A(M), the left ft-semimodule f @) : f (*)llT'(*) is the principalfactor of M at m. By what we have noted above, f @) has a strongly - infinite element. 4.7 PROPOSITION: Let R be an information algebra and let M be a Ieft R-semimodule havingan absorbingsubsemimodule. If m € A(M)' thenT(^) either ha^sa primitive element or rs quasisimple.
4.8 PROPOSITION [Poyatos, 1973b]: Let R be an information alge- bra and let M be a left R-senimodule. If N' is a rnaximal proper absorbing sub- module of a subsemimoduleN of A(M) then N I lN' = T(m) for anyrn e i/ \ N'.
If fu[ is a left R-semimodule then an absorbingseries for M is a descending chain M: No -l Nr f ...1 Nt:C(M) of subsemimodulesof M. An ahsorbing quasiseriesfotM is an absorbing seriesfot A(M). Any chain obtained from a given absorbing series by inserting further terms is a refinement of that series" If new subsemimodulesare actually inserted, such a refi.nementis proper. Two absorbing seriesM : NaI Nr I ... f Nt: C(M) andM : Lo) Lra... I tr" : C(M) fot M ate isomophicifand only if t: s and there is a permutationo of {1,".. ,r} such that N;-tl lM = L,(il-rl I L"(;) for each 1 < i < t. Finally, we come to Poyatos' extension of the Jordan - Hdlder theorem. Infiotm ation semimodules and absorbing su beemimodules 19
4.9 PROPOSITION [Poyatos,1973b]z Let R be an information alge- bra and let M be a left R-semr'module.Then any two absorbing quasiseries of M fi ave isomorphic rellnements.
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Department of Mathematics & Computer Science Uniuersity of Hada 31905 Haifa