Distributive Lattices

CEU eTD Collection nprilflumn fterqieet o h ereof degree the for requirements the of fulfulment partial In eateto ahmtc n t Applications its and Mathematics of Department itiuieLattices Distributive uevsr rf P´al Heged˝us Prof. Supervisor: eta uoenUniversity European Central uaet Hungary Budapest, atro Science of Master ase Khan Taqseer umte to Submitted 2011 by CEU eTD Collection uaet 9My2011 May 19 Budapest, for of education higher part of no institution the that other of declare any part to also no form I and this degree. in academic others, copyright. declare submitted an of institution’s I been work or and has bibliography. of person’s thesis research and made the any my was notes on on use in infrignes based illegitimate credited thesis work, and properly own unidentified the as no my at declare informaion that Science exclusively Applications, of external is its Master such thesis and present only Mathematics of the of degree that Department the herewith for candidate University Khan], European Taqseer Central [ undersigned the I,

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Applications special to my encouragement its utter and and I generosity Mathematics his all of for department Morosanu Gheorghe the Professor to Mathematics. thank of to Institute books Renyi the wish the use I of to library possible the me from made Gy˝ori articles who Ervin collect Professor and to due also are thanks to My me enabled that valuable level thesis. final his the this P´al Heged˝us for to write initial Professor the from supervisor encouragement my and support to guidance, gratitude profound my owe I Acknowledgments iii CEU eTD Collection w esoso h ento hnw aemnindtecneto dasi lattices in ideals of concept the them. to mentioned have related we results the some Then of and equivalence . the definition showing the theorem of a versions stated have two and lattice lattice-as-an-algebraic- the a given of have tried We definition have included. structure we lattices texts, the than of most better of speak diagrams pictures give to As here. given been have example counter their found have lattices which in life. areas main practical some current in described applications the have to we logics chapter formalising this lattice In for of attempt research. development Boole’s chronological Georege almost from mentioned starting have theory We subject. the give of I duction preface this In lot. a learnt have thesis. I the feel of I account writing thesis brief of this Writing a process of the sources. research in these the to but references on hard provide based been to has is best however, my text, done the have of I most and me; others, by P´al Heged˝us. The Professor solely been made has been thesis has the thesis for supervisor My Degree Master’s me. a for for requirements Mathematics the in of fulfilment partial in submitted being is thesis This Preface hpe he icse pca lmnsi atcswt examples. with lattices in elements special discusses three Chapter and examples Several theory. lattice of concepts basic the of consists two Chapter intro- short a given have we chapter first the In chapters. six of consists thesis This iv CEU eTD Collection hpe atc n h te novsteji-reuil lmnso h distributive the of elements join-irreducible the involves a lattice. of other congruences the involves and one lattice lattices: result chopped distributive a finite proved of illustrated and theorems and stated stated representation have we two have Finally, we lattices. chopped Then lattice of importance chopped the on example. The light throwing with illustration. introduced with We been lattice has lattices. non-atomistic in concept a atomisticity of and example atoms an like given notions have more some defined have We lattices. lattices; important three lattice. of isoform definitions and the lattice uniform with lattice, chapter regular this close We have kernels between and connection introduced. lattice the factor been of discussed concepts have The we Then lattices. distributive concept. and lattices this congruence of feel a have to given been lattices. have in We morphisms introduced. of discussion been the has with lattices chapter of Boolean this modularrity of ended Then equivalence the rings. shown Boolean have and and Boolean algebras rings of concepts Boolean the and introduced matters algebras have lattice Boolean We any lattices, lattice. in that lattices of two character these distributive of the theorem presence for a the how and shows diagrams which their stated prototypical with Two been given has been sets. have of lattices lattices non-distributive of of terms examples in lattices distributive of characterisation the ncatrsxw aedsusdrpeettoso itiuieltie scongruence as lattices distributive of representations discussed have we six chapter In have examples Some lattices. in congruences of concept the introduces five Chapter mentioned have we Here detail. in lattices distributive discussed have we four chapter In v CEU eTD Collection ilorpy47 27 17 34 1 13 4 iii Bibliography Lattices Distributive of Representation 6 Lattices in Congruences 5 Lattices Distributive 4 Lattices in Notions Special 3 Definitions Background 2 Introduction 1 Preface Acknowledgments Copyright Contents of Table vi iv ii CEU eTD Collection regeDualgruppe erzeugte Teiler gr¨oßten gemeinsamen ihre durch Zahlen which von lattices, of theory basic the develop called to he led was Dedekind questions, related and this e.g., sums, and intersections taking subgroups Dirich- of edition enlarged and revised let’s a on working was Dedekind Richard his 1890’s, Schr¨oder early in Ernst the by earlier out worked been had book lattices of some theory thus elementary and logic, the in of naturally Lattices, arise algebras, century. Boolean nineteenth and late and lattices distributive the Pierce especially in S. lattice Charles of algebras, concept the Boolean Boolean Schr¨oder discovered introduced of Ernst Boole axiomatics George for- the century, to investigating nineteenth While attempts the nineteenth-century of algebras. half the first to the back In dates logic. concept malise lattice the of origin The Introduction 1 Chapter olsne ¨uberVorlesungen Zahlentheorie i ler e Logik der Algebra Die Dualgruppen ,B C B, A, fa bla group abelian an of 10) ntesbeto .Ddkn ruh h hoyt iewell life to theory the brought Dedekind R. of subject the on (1900), h ulcto ftetofnaetlpapers fundamental two the of publication The . ihr eeidas neednl icvrdltie.In lattices. discovered independantly also Dedekind Richard . A n se isl h olwn usin ie three Given question: following the himself asked and , + B G ( , o aydffrn ugop a o e by get you can subgroups different many how , A 1 + B ) ∩ C 19)and (1897) t.Tease s2.I okn at looking In 28. is answer The etc. , u e i o riModuln drei von ¨uber die ¨uber Zerlegungen CEU eTD Collection edr n h eiiaercies urneiga h aetm,ta ounauthorized no that time, same the at legitimate guaranteeing the receivers, between new information legitimate of the of main exchange and safe The definition senders the ensure the cryptanalysis. to in is in cryptosystem and role a primitives of significant goal cryptographic a of played study has the in theory cryptosystems, lattice area, latter as Cryptography. such the Public-Key areas and In theoretical programming various integer in polynomials, applications integer lattice of to factorization computer led of century evolution The 20th numbers. the of in exclusively geometry and science we motivated theory Minkowski the hand, to in in theory due lattice results thesis of important use century, the to the 19th In has the In other space. the case. first while in the sets points consider ordered of partially arrangements with regular do with to has do one first The lattices. called algebras. especially Mathematics, is of distributivity areas various of in condition arising the lattices applications on are many imposed lattices In on conditions distributivity. Many of theory. provided forms lattice have weakened general lattices in These results many role. of for development vital motivation the a the In played have field. lattices this distributive developed theory, others lattice and Ore Oystein Menger, Neumann, Karl Von Glivenko, Valere John that After disciplines. unrelated for mathematical the framework many demonstrated unifying in a he developments provides papers it that of showed series and theory a lattice In of that importance thirties subject. mid the the of in development work general Birkhoff’s the Garett started was It theory. J´onsson, lattice Later of development prominently contributed the subject. Birkhoff to a Garrett and as Tarski, Neumann, theory von lattice Ore, which Malcev, of Kurosh, theory development lattice the and for algebra impetus modern the between provided connection the recognised He lattices. later many inspired have and classical are pappers mathematicians. two These ago. years hundred one over nbbigah,teeaetoqiedffrn ahmtclsrcue htaeusually are that structures mathematical different quite two are there bibliography, In distributive of form weakend are which lattices modular defined Dedekind Richard 2 CEU eTD Collection frslswihlea h er ftesubject. the of heart the at development lie systematic which the results with of concerned primarily are Gr¨atzer. They G. and Dilworth information. the of part any recover to able is party h motn urn eerho atc hoyhsbe ntae yG iko,R P. R. Birkhoff, G. by initiated been has theory lattice on research current important The 3 CEU eTD Collection aifig o all for satisfying, that set empty non Sub a of subsets all of collection system a is set, (poset) Set Ordered Partially Definitions Background 2 Chapter vr o-imlelement non-minmal every of element minimal if (iii) if (ii) (i) oeigrelations Covering is set ordered an of example natural most The abound. posets of Examples Atoms ( G b x ,tecleto falsbruso group a of subgroups all of collection the ), covers ≤ x x ≤ ≤ x e ( Let : (reflexivity) , y y a and and if P ,y z y, x, aunion atc.Telattice The lattice. relation a lattice. a forms it elements, two the of enesup Definie sup defining By Q poset A : e ( Let : scle t raetlwrbud(l)o nmm(n)of (inf) infimum or (glb) bound lower greatest its called is A : ≤ , { B ,b a, as P, P n inf and Q ≤ } { h ( L, ,b a, Q S eapstand poset a be ) if slmof lcm as of ) ≤i a L scle t es pe on lb rspeu (sup). supremum or (lub) bound upper least its called is } 6 ≤ { stebge ftetoeeet n inf and elements two the of bigger the as n S stefollowing the is saltiei sup if lattice a is ,B A, let , x bv sapstudricuin e sdfiesup define us Let inclusion. under poset a is above o all for , Q } iue21 Lattice 2.1: Figure L a if , sitreto of intersection as n N b y 2 etesto l oiiedvsr of divisors positive all of set the be n inf and ≤ x = a Q ∈ b { ≤ o all for , 1 Q easbe of subset a be , 6 5 1 b 2 h raetmme ftesto l lower all of set the of member greatest The . { , ⇐⇒ { ,b a, 3 ,b a, ... , } y } } a 3 n inf and ∈ ihteuua re of order ususal the with sgdof gcd as | A L b Q , 6 B h mletmme fteset the of member smallest The . P Then . { nelement An . ,b a, a } , xs o all for exist P b Then . ( S { eoe lattice. a becomes ) ,b a, Q n } a Dually, . e sdefine us Let . L ∈ stesmaller the as n ≤ ,b a, P eoe a becomes saposet. a is { ssi to said is ∈ ,B A, L b . ∈ } as P CEU eTD Collection 2.3. e-hoei nescinof intersection set-theoretic a is there If and them. group plane containing a the those in to subgroups ones smaller smaller the the joining above line occure subgroups larger the that such h symbols (123) osdrtesmercgroup symmetric the Consider If Examples group the of subgroups the all lists which diagram a is group a of diagram lattice A G i h atc iga of diagram lattice The . G = { 1 , V , H 2 4 h li-orgop hnteltiedarmof diagram lattice the then group, Klein-four the , , ∨ : 3 } K hntenntiilsbrusof subgroups non-trivial the Then . = h ,K H, i ,K H, iue22 atc fSubgroups of Lattice 2.2: Figure h ugopgnrtdby generated subgroup the , S iue23 atc of Lattice 2.3: Figure hnisltiedarmi ie yfiue2.2. figure by given is diagram lattice its then , 3 sgvnb h gr 2.4. figure the by given is S H 3 H H = ∨ ∧ { K K { identity, G 6 e } K (12) V S 4 , (13) 3 ,K H, are , (23) ,K H, h and (12) G , (123) H sgvnb h figure the by given is i r w ugop of subgroups two are , ∧ h (13) , K (213) = i , H } h (23) ∩ nthree on K i the , and CEU eTD Collection sadvsrof divisor a is G H (ii) (i) = = atc igasfrcci ruso nt resaees oda.W nwta if that know We draw. to easy are orders fnite of groups cyclic for diagrams Lattice groups cyclic of diagrams Lattice h h G G x d i i = = = sacci ru forder of group cyclic a is Z Z d 6 p Z m a h lattice the has n n a h lattice the has npriua,when particular, In . stecci ugopgnrtdby generated subgroup cyclic the as (12) iue25 atc of Lattice 2.5: Figure iue24 atc of Lattice 2.4: Figure (13) n hnaysbru so h form the of is subgroup any then , G = ( : ( p Z { p S 2 e n ( ) ) 7 Z 3 } p stecci ru eeae y1 ewrite we 1, by generated group cyclic the is p . . . m m ) (23) d. Z 1. S p 3 m (123) H = h x d i where d CEU eTD Collection atc sa leri structure algebraic an as to Lattice belong not does then 2,3 above, of 3 gcd example the in as inf and sup and order partial leri tutr sasto hc w iayoeain r end called defined, are operations binary two which on meet set a is structure algebraic (iii) ihr eeiddsoee h leri hrceiaino atcs atc san as lattice A lattices. of characterisation algebraic the discovered Dedekind Richard Consider lattice a is poset every Not eoe by denoted , G = Z 12 S a h lattice the has = ∨ { ,3 ,... . . 4, 3, 2, and ∧ aifigtefloigaxioms: following the satisfying , } h e fntrlnmesdltd1 e sdfiethe define us Let 1. deleted numbers natural of set the , iue27 atc of Lattice 2.7: Figure (4) iue26 atc of Lattice 2.6: Figure S : (2) . : (2) { 0 } { Z 0 8 6 } Z 12 (6) (3) Z (3) Z 12 6 S sntaltiea,frexample, for as, lattice a not is join and CEU eTD Collection aifigteaoeaxioms above the satisfying 1. Theorem that proved Gr¨atzer (1971) G. mathematician Hungarian all for Asrto a : law Absorption iii) Ieptn a : law Idempotent iv) Ascaielw: law Associative ii) Cmuaielw: law Commutative i) ,b c b, a, ∈ o-mt set non-empty A L . 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( , scle nieliff ideal an called is and + = ( or S L stesto oiieitgr and integers positive of set the is ∨ S A ). n b I . G Then . , ∩ ∈ ≤ L 6= 1 G a sasbatc of sublattice a is ) B ∨ I L ( ntefloiglattice, following the In . = . S G rprideal proper A . sntasbatc of sublattice a not is ) G { 2 10 x P ( h ugopgnrtdby generated subgroup the = S ∈ ( S lofrsaltieudrteoperations the under lattice a forms also ) L saltiewt h prtosdefined operations the with lattice a is ) L | ∗ S i a ⊆ ∈ h oe set power the - ≤ S L I x then , ⊆ } and I I + L . of scle ultieif sublattice a called is a L L ∈ ∗ spieiff prime is edntb sublattice a be not need P L I a ( = ml that imply S P ≤ si sntclosed not is it as ) ( { S b 0 of ) } ⇐⇒ G sa da and ideal an is 1 S ,b a, ∪ a and G a | ∈ b 2 ∧ Then . Then . L G i ∈ S ( and S I is ), . CEU eTD Collection t.Then etc. intersection containing ideals L of sublattice hoe 2. Theorem lattice. a of subset a by P containing = hrceiaintermo nieli sfollows: as is ideal an of theorem ideals. characterisation also A are ideals two of join and Meet ideals of Join and Meet Let generated ideal the and sublattice the define also can we structures, algebraic other Like { 0,a H } easbe of subset a be sapieiel while ideal, prime a is I L H ∩ Let iial nielgnrtdb subset a by generated ideal an Similarly . eeae by generated J H The . L eoe y(] If (H]. by denoted , ealtieadlet and lattice a be join U U L 2 1 ( ( hntesals ultieof sublattice smallest the Then . ,J I, J I, sbttik enda olw:Define follows: as defined tricky bit is H eoe y[] ti h nescino l ultie of sublattices all of intersection the is It [H]. by denoted , The : = ) = ) I I ∨ { { sntprime. not is iue28 figure 2.8: Figure J U x x a 0 | | meet = x x ( H H ,J I, ≤ ≤ ∪ = and ( u u U 11 = ) 0 i ftoideals two of a ∨ ∨ i ( ,J I, hn( then I v v I ; ; ennodsbesof subsets nonvoid be ,v u, v u, ∪ ) b | J i<ω H ∈ ∈ ( = ] U U ) 1 0 H ( ( I ,J I, J I, a , scle rnia ideal. principal a called is ] of J L ) ) L } } of containg steitreto fall of intersection the is L sdfie ob the be to defined is L . H scle the called is CEU eTD Collection ,d a, For (iii) (ii) (i) sa xml,i h olwn atc h rnia da eeae yteelements the by generated ideal principal the lattice following the in example, an As a ewitnas written be can I hr exist there x I ( = ∈ sa da iff ideal an is I a H . ∈ ] L iff , h ( I a 0 h , = ] sa ideal, an is 1 ,b a, h , . . . , { x ∈ ∈ L I n − : mle that implies 1 H x ∈ ≤ ⊆ H ( a iue29 figure 2.9: Figure a I } uhthat such b = ] n o all for and ( = d = ] { { x 0 d a a 12 ,b ,d c, b, a, , 1 0 ∧ { ∨ 0 a d , b i : ≤ ∈ } x i c h I ∈ ∈ } 0 and , L ∨ I } . . . hr xssa integer an exists there a ∨ ∈ h n ,x I, − 1 . ∈ ,x L, ≤ a n ml that imply ≥ 1 and CEU eTD Collection o every for every lattices. in elements of types special some define we Here Lattices in Notions Special 3 Chapter Telattice The iii) Tentrlnme stezr lmn nteltieo xml .Ti lattice This 2. example of lattice the in element zero the is 1 number natural The ii) I hpe ,teltieo xml a h mt set empty the has 1 example of lattice the 2, chapter In i) Examples lattice Bounded Element Unit Element Zero lmn.Hence, element. bounded. not is lattice This element. unit the possess not does lmn n h nteeetrsetvl.Therefore respectively. element unit the and element a ∈ a L ∈ Dually . L : . L neeet1i aldteui lmn rtealeeetof element all the or element unit the called is 1 element An : nalattice a In : 6 feape3 hpe a siszr lmn n sisunit its as 6 and element zero its as 1 has 2 chapter 3, example of lattice A : L 6 sbounded. is L L neeet0i aldtezr lmn of element zero the called is 0 element an ih0 scle one lattice. bounded a called is 1 0, with 13 P ( S φ sabuddlattice. bounded a is ) n h set the and L S f0 if L siszero its as if ≤ a a ≤ for , 1, CEU eTD Collection lmn n whenever and element h iderwo h imn ffiue42aytoo h he lmnsaecomplements are elements three the of two third. any the 4.2 of figure of diamond the of row middle the of complement while a then irreducible, whenever and element zero a sme reuil u o onirdcbe sji reuil u o etirreducible, meet not but irreducible join is d irreducible, join not but irreducible meet is Remark element an of Complement irreducible Meet irreducible Join o xml,i h bv atc,tecmlmn of complement the lattice, above the in example, For Example ,c b, r irreducibles. are facmlmn fa lmn xss tmyntb nqe o xml,in example, For unique. be not may it exists, element an of complement a If : nteltiedarmbelow diagram lattice the In : a a sdfie ob nelement an be to defined is ssi ob irreducible. be to said is nelement An : nelement An : a = b ∨ a = c hneither then , b a a ∨ Let : iue31 figure 3.1: Figure a ∧ nalattice a in c b hneither then , nalattice a in b 0 = L d a 14 a , 1 0 b eabuddltieand lattice bounded a be a ∈ ∨ = L L b fsc neeeteit,sc that such exists, element an such if , L 1 = b ssi obe to said is c a or is = etirreducible meet a b b or = is c a c If . = n vice-versa. and onirreducible join c a Dually . sbt onadmeet and join both is iff a a ∈ sntaunit a not is L iff hna Then . a snot is CEU eTD Collection lmn in element a y unique. is it . frany for (ii) ∈ (i) Example Lattice Pseudocomplemented of pseudocomplement the that noteworthy is It as defined is complement a to alternative Pseudocomplement an issue, non-uniquness the around get To Example words, other In L b if ∧ a L 0 = h ezn stepedcmlmne lattice. pseudocomplemented the is Benzene The : nteltieo gr . cle ezn)tepedcmlmn of pseudocomplement the Benzene) (called 4.3 figure of lattice the In : a pseudocomplemnt. a has c uhthat such b stemxmleeeti h set the in element maximal the is c nelement An : ∧ a 0, = iue32 Lattice 3.2: Figure c iue33 figure 3.3: Figure x a ≤ a : b L b . ∈ scle suoopeetdltiei every if lattice pseudocomplemented called is 15 0 1 0 i L b ih0i aldtepedcmlmn of pseudocomplement the called is 0 with y c b y M snot is { 3 c ∈ a L i si fact in is (it : c ∧ a 0 = } n,i texists, it if and, x ) a is CEU eTD Collection a n vr lmn of element every and a any for of plement. Let (ii) 6 6 Telattice The (i) U opc Elements Compact Lattice Complemented Sectionally set open an of pseudocomplement The Examples lattice Complemented opeeLattices Complete Example ∨ . ∨ suoopeetdlattice. pseudocomplemented a S S 1 a . o nabtaysubset arbitrary an for ≤ X b eatplgclsae hntecollection the Then space. topological a be h atc ffiue44i etoal complemented. sectionally is 4.4 figure of lattice The : in L M hr sa element an is there , 3 bv scmlmne lattice. complemented is above L a ewitna onof join a as written be can : nelement An : L : scmlt if complete is L S scle opeetdi vr lmn in element every if complemented called is of iue34 figure 3.4: Figure L a c hnteeeit nt subset finite a exists there then in a U : sup of L 16 0 L b of uhthat such L scle etoal opeetdltieif lattice complemented sectionally called is and X scle opc if compact called is s( is opc elements compact c inf U c xs o n rirr ustof subset arbitrary any for exist a ) L o ∧ h neiro h complement the of interior the , ( X c faloe ust of subsets open all of ) ,and 0, = . a a S ∈ ∨ 1 c L ⊆ = L ssc that such is S a com- a has b . uhthat such X L is CEU eTD Collection h atc ftuhvle.LtrteHnainmteaiinAda unintroduced Huhn Andras mathematician of Hungarian concept the the Later values. truth of lattice the (2 all for lattice a is lattice distributive A Lattices Distributive 4 Chapter Remark , ∧ L , ∨ scalled is ,b c b, a, .I steol w-lmn atc.Ti atc etrspoietyi oi as logic in prominently features lattice This lattice. two-element only the is It ). nteeuiy(3 equlity the In : ∈ L n n itiuieltieo udmna motnei h w-lmn chain two-element the is importance fundamental of lattice distributive A . − − distributive distributivity . ) ti rva that trivial is it 1), x ( c ∧ c fin if ∧ ( : ∧ L i _ ( =0 n a a aifigtelw(aldtedsrbtv identity) distributive the (called law the satisfying ) L y ∨ ∨ i = ) h olwn dniyholds: identity following the b ( ( = ) c ∧ i _ =0 n 17 b c ( ) x ∧ ≤ ∧ a c ) ( ∧ j ∨ ( 6= _ ( ( n i a c )=0 ∧ ∨ y b b j (4.1) ) ) )) CEU eTD Collection ot rv atc ob itiuie eol edt rv that prove to need only we distributive, be to lattice a prove to so Teltie( lattice The ii) Dsrbtv atcsaeuiutu.Tepooyia xmlso uhstructures such of examples prototypical The ubiquitous. are lattices Distributive i) Examples hct igo sets. of ring a to phic 3. Theorem stated as completely theorem scenery and following the union the describe set in by sets given of be lattices can these operations Indeed, lattice intersection. the which for sets of collections are : I + G iko n .H tn)Altiei itiuiei ti isomor- is it iff distributive is lattice A Stone) H. M. and Birkhoff (G. , ≤ etoe bv sdistributive. is above mentioned ) ( c ∧ a ) ∨ ( c ∧ 18 b ) ∧ ≥ ( a ∨ b ) CEU eTD Collection and smrhcto isomorphic em 5. Lemma 4. Theorem M j where t 1 ≤ 3 ∈ esalcl subset a call shall We lattices Non-distributive o esaeatermwtotpofwihrvasta h ultie smrhcto isomorphic sublattices the that reveals which proof without theorem a state we Now Proof i and N J ∨ hsipisthat implies This . 5 i j . 1 with , N = Suppose : 5 t lya motn role. important an play lattice A ∧ lattice A M i i ∈ ∈ 3 I or I , , j L j N L 1 ∈ 5 sdsrbtv n e stake us let and distributive is L = sdsrbtv fffraytoideals two any for iff distributive is J respectively. 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( then so and 13) osdrteltieo l ugop fagroup a of subgroups all group of lattice the Consider (1938). unique. is complement onirdcbeeeet of elements join-irreducible lattices: distributive x x ≤ ∧ ∧ Proof not is exists, if lattice, a in element an of comlement that 3 chapter in seen have We that suppose converse, For eew eto iecaatrsto fdsrbtv atc u oOsenOre Oystein to due lattice distributive of characterisation nice a mention we Here Definition finite of structure the investigate which theorem(s) and complement. definitions unique some have a we has Next element every lattice Boolean a in Thus Lattice Boolean a y y i 1 ∧ 2 ) G ≤ ) a ∨ c ∨ slclycci ffteltieo ugop of subgroups of lattice the iff cyclic locally is ∈ ( 0 = a Let : y ( , 2 I y nabudddsrbtv atc neeetcnhv nyoecomplement. one only have can element an lattice distributive bounded a In j ∧ 1 ∨ b < y ∧ ≤ Let : J 1 L 0 = ) y u ecamthat claim we But . a ∈ 2 eadsrbtv atc n ups,i osbe nelement an possible, if suppose, and lattice distributive a be hnas Then . y 0 = ) 1 ( D b opeetddsrbtv atc scle ola lattice. Boolean a called is lattice distributive complemented A : and = ] ∨ eadsrbtv atc and lattice distributive a be ( y ∨ I 2 y Thus . ∧ 2 ( D hnuigdistributivity, using Then . y y I 1 j Then . 1 ( = I = ) ∧ ∈ ∨ y J a b 2 J y ] = ) J , = 2 ( = = ∧ J a i ( ( = y { ∨ D a o ewitnas written be not can c 1 y i ], 1 j = sapstudrteprilodrn inherited ordering partial the under poset a is ) ∨ c ∧ j ∈ .(epn nmn h figure the mind in (Keeping ]. 20 j y y ≤ 1 I : 2 ∧ i hs w ieus give two These . 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( t ) This Θ). = ¯ t 0 ⊆ u 1 i ∨ ( 1 I ∧ i<ω } [ J t ∨ So . u 1 ]Θ. 1 T ≡ )) ). . 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Theorem lattice chopped finite a lattice distributive finite a represent 6 ( . ,c b, Illustration esalilsrt h bv hoe ntefr ftefloigtheorem: following the of form the in theorem above the illustrate shall We [7] Schmidt T. E. Gr¨atzer and G. to due theorem representation a state we Now ffiue51t osrc h nt hpe lattice chopped finite the construct to 5.1 figure of J M ftegadget. the of ) sioopi to isomorphic is Let vr nt itiuielattice distributive finite Every eilsrt tb aigachain a taking by it illustrate We : P M eafiieodr hnteeeit hpe lattice chopped a exists there Then order. finite a be 0 M fji reuil lmt of elemets irreducible join of iue63 Lattices 6.3: Figure P a since , 1 where , b a 2 ( Con 0 b ) Con D ( iue6.2: Figure M sacnrec atc,i ssffiin oconstruct to sufficient is it lattice, congruence a as J b b M ) 1 ∼ = 41 b steltieo onirdcbecnrecsof congruences irreducible join of lattice the is c Con a N ( b ,b a, D 1 ( P L IdM and ) a erpeetda h congruence the as represented be can . D D c b 2 M = ) ( 0 oehrwt .W s h lattice the use We 0. with together ffu lmnsadcondsidering and elements four of c etk w copies two take We . ) N Con ( ,c b, c c ( 1 ) L ,where ), L = M N IdM ( uhthat such ,b a, . and ) CEU eTD Collection aedn ncami hpe ) W hl eemn nyoeo hs ogune,say, congruences, these of one only determine shall (We Con 5): chapter in claim in done have (0 Con pairs the by generated congruences lattice chopped the of congruences of atom elements an maximal two exactly with lattice chopped complemented is conditions problem. following the answered have [7] Roody M. and Lakser Gr¨atzer H. G. this, proving lattice hudas eogt .Nwsince Now Ψ. to belong also should ncniuto fteilsrto o hoe 0 ene ofidteji irreducible join the find to need we 20, theorem for illustration the of continuation In Solution Problem that fact the of proof The ideal common a have They ydfiiin h lmn (0 element the definition, By congruence the determine We (0 (0 b , c , 1 1 )adsallist shall and )) a edtrie pliglma1 eetdya olw smlrt swe as to (similar follows as repeatedly 12 lemma applying determined be can ) M .G¨te .Lke n .Roypoe htif that proved Roody M. and Lakser Gr¨atzer H. G. : Let : IdM then , etoal complemented? sectionally M IdM eafiiescinlycmlmne hpe atc.Udrwhat Under lattice. chopped complemented sectionally finite a be Con sscinlycomplemented. sectionally is b c iue64 h egdlattice merged The 6.4: Figure a L (0 M = I a , b , = IdM = Con 1 1 M and ) eog oΨTe h lmns( elements the Ψ.Then to belongs ) a Merge b { 1 2 0 nfiue(6 figure in (0 ∧ b , sscinlycmlmne snts ay Towards easy. so not is complemented sectionally is a , b , b 1 b Con 1 1 } a 2 1 ,(0 ), ( 0 = ( yapyn em 2a follows: as 12 lemma applying by Ψ = ) ow a eg hmadfr h chopped the form and them merge can we So N 42 b ) (0 ( b ,b a, b , ≡ c , b . 0 1 ) easr htteeaeteprinciple the are these that assert We 4). 1 1 b ) ,(0 ), directly). ) 1 N , Ψ,s ylma12(3), lemma by so (Ψ), b ( ( c , ,c b, c c ) 2 1 ,where ), )) c M c 1 m Con M 1 , b sfiiesectionally finite is m (0 1 , 2 a , ) (0 0), and 1 b ), 2 ≡ , Con m ) ( 0), b 1 1 ∧ (0 ∨ b m 1 b , b b , 2 2 1 1 = is ), ) CEU eTD Collection (0 ( (0 si ow hl ae( have shall we so Ψ in is ( element ( the elements have should we lemma, the of (3) ( b n ( and ht( that ( transitivity by ( so elements Ψ, the in that are gives lemma the of (2) part of application ( as ( pairs the ( have we Lastly, Ψ. ( ( (0 ( ( ( b c ,c c, c c ,a b, ,c c, a Ψ ..( i.e. (Ψ) 2 1 1 1 c , b , c , 1 b , c , b , c , Con b , Con and Con congruences the determine we Similarly, 2 2 ( 1 1 ( 2 1 1 ) 1 ) c hc unott be to out turn which ) ) b 1 b , , nΨto etpc h elements the pick Next too. 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