PUBLICATIONES MATHEMATICAE

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Sl ERKESlTOrUlOTTSA G:

ACZEL JANOS, GYIRES BELA, RENYI ALFRED,SZElE TfBOR t ES VARGA OTTO

A SZERKESzrOBIZOTTSAG TITKARA:

KERTESZ ANDOR

A DEBRECENI TUDOMANYEGYETEM MATEMATIKAI INTEZETE

275

Characterizations of relativel:r complementeddistributive lattices.

By G. GRATZER and E. T. SCH;\lIDT in Budapest

The beginning of laitice theoretical researches goes back to the investigation of Boolean algebras, which are - in the terminology of latticetheory - relatively complemented distributive lattices with zero and unitelements. In the course of further research it was found necessary to generalize in different ways the concept of Boolean algebra. In this way weregot the relatively complemented distributive lattices, one of the most important classes of lattices. So it is desirable to be able to characterize this classof lattices in different ways. There are many well-known theorems to thiseffect. These theorems mostly show the equivalence of some propeliy to therelatively complementedness, supposing distributivity. If we want to prove atheorem of such kind, th,:n the greatest difficulties lie in proving the necessity of the relative complementedness. This is the reason why in our paperwe introduce a theorem which makes us possible to prove the necessity ofthe relative complementedness in a very simple way. Applying this theorem,we get simple proofs of several theorems. We treat G. BIRKHOFf'S problem73 in a new way, we give an affirmative answer to a conjecture of K. IsEKI.After some generalizations of the above mentioned problems we deal withsome new questions. - We should like to express thanks to Prof. L. FUCHS

for helpful suggestions in the preparation of the paper.

§. 1. The Main Theorem.

Let A be a class of distributive lattices, in other words A is a propertyof distributive lattices. A is said to be homomorphically invariable (hereafterHI property) if and only if A contains all homomorphic images of its elements. We prove the

Main Theorem. Let A be an HI property of distributive lattices. If theelwin of three elements does !lot have the property A thi>!l all the latticeshaving property A are relatively complemented.

276 G. Gratzer and E. T. Schmidt

PROOF. The following assertion implies the Main Theorem:

Any distributive lattice which is not relatively complemented has a homomorphic {mage isomorphic to the elwin of three elements. For, if L is a distributive lattice with the property A and L is not relatively complemented.~

then by this assertion L has a homomorphic image isomorphic to the chainof three elements. Hence A being an HI property of L, in this case the chainof three elements also has the property A, contradiction.

Now we prove this assertion. If the lattice L is (distributive, but) notrelatively complemented, then there exist in L three elements b < c < a suchthat c has no relative complement in the interval [b, a]. Let us consider thedual ideaP) D = {d; a ::::; due) and the dual ideal E = [c) uD. Those elements of E wich are included in c are of the form end. We see, b is notan element of E, because otherwise b = end for a suitable dE D, and sob = an b= c n (a nd) would hold. It is evident that and ED, hence a::::; c UU(a nd), thus in consequence of c < a we get a = c U(a nd). In summary weget. that if bEE, then and is a relative complement of c in the interval[b, a], contradicting the hypothesis.

Therefore by STONE'S theorem") there exists a prime ideal P, whichcontains b and is disjoint to E. Now consider the ideal 1= (c] uP. Thoseelements of I which are greater than or equal to c are of the form CUp{p E P). The element a is not in I, because in this case a = cup for asuitable pEP, hence p ED by the definition of D, which contradicts thedisjointness of P and D. Again by the Theorem of STONE there exists aprime ideal Q which does not contain a and Q~ I~ P.

Let us define the following congruence relation 0: x y «(-) if andonly if x and y are elements at the same time of P or Q-P or L-Q.')Evidently, 0 is an equivalence relation. Taking into account that P and Qare prime ideals, we can easily establish the substitution property of 0. The

1) In what follows [c) denotes the dual ideal and (c] the ideal generated by c; if Dand E are dual ideals then DUE is the dual ideal generated by D and E, and D n Ethe dual ideal, which is the set-theoretical intersection of D and E. Furthel more, it is wellknown, tha't in a distributive lattice the elements of DUE are of ll:e form d ne (see [2),p. 140.). Thus if 11~· v and 11 E [I") U D, then 11 - V nd, for suitable dE D, because, as wementioned above, 11 =c VI nd (1"1 E[1"», hence 11 = 11 n1; ~ V nd.

At last, let a (x) denote a property of elements of a set H; then {x; « (x)} willdenote the set of those x, for which a.(x) holds.

2) The theorem of M. H. STONE [10] asserts the following: Let L be a distributivelattice and ! and D a disjoint ideal and a dual ideal resp. of L. There exists a primeideal P with P;2! such that P and D are disjoint.

3) A-B denotes the set-theoretical difference of A and B.

Relatively complemented distributive lattices. 277

image of L under the homomorphism induced by 0 is isomorphic to thechain consisiting of three elements, q. e. d.4

)

In what follows the elements of the chain of three elements will bedenoted by 0, a, 1, where °denotes the least and 1 the greatest element ofthe chain.

§ 2. G. Birkhoff's. problem 73.

This problem of G. BIRKHOFF is the following:

Find necessary and sufficient conditions, ill order that the correspondencebetween congruence relations and ideals of a lattice be one-one.

More precisely,

Find necessary and slIfficient conditions, in order that every ideal be akernel of one alld only one homomorphism, and every homomorphism hasa kernel.

Firstly]. HASHIMOTO has given an answer to this problem in his paper[4J. In [3] we have solved this problem independently of ]. HASHIMOTO. Thefollowing solution is essentially simpler than those in [3] and in [4], moreover, it is also suitable to generalizations.

Theorem 1. En L there is a one-one correspondence between congruence relations and ideals if and only if L is a relatively complemented distributive lattice with zero element.

Proof of necessity. L must contain the element 0, since otherwise theidentical mapping of L, which is obviously a homomorphism, has no kernel.

It· is known that if L is non-distributive, there exist in L threeelements b < c < a such that in the interval [b, a] the element c has at leasttwo relative complements d and e. It may be supposed that d ~ e. We assertthat the principal ideal (e] is no kernel of any homomorphism. For, if wesuppose that (e] is a congruence class under a congruence relation 0, thene c b (0), consequently, d = d n (e Uc)==d n (b uc) = b, but dEE(e], a contradiction.

The necessity of relative complementedness in the case of distributivitymay be proved by the aid of the Main Theorem. Let A] denote the followingproperty: Any ideal of the distributive lattice L is the kernel of precisely onehomomorphism. Let L be a homomorphic image of L, and 1 an ideal in L.

4) The most substantial part of the proof is that in a distributive lattice which isnot relatively complemented there exist two different prime ideals one of which containslhe other. For this assertion see MONTEIRO [8], and in case of Boolean algebras NACHBIN [9].

D 19


If I is the kernel of more than one homomorphism, then its complete inverseimage has the same property, i. e. A is an HI property. The chain of threeelements does not have the property AI' because the ideal (OJ is a congruence class under the identical congruence relation, and under the congruencerelation in which 1~ (C and (C 0. Hence the necessity of relative complementedness follows from the Main Theorem.

Proof of sufficiency. In consequence of the existence of 0, every homomorphic image of L has a least element, i. e. every homomorphism of L hasa kernel. In a distributive lattice every ideal is neutral (see [2J, p. 142, Ex 3)and every neutral ideal is a kernel of a suitable homomorphism (see [2J, p.80, Ex. 3 (c». The last part of the theorem, i. e. the assertion that everyideal is the kernel of at most one homomorphism follows from the fact:every interval [0, a] as a lattice is complemented (see [2], p.23, Theorem 3).

In distributive lattices Theorem 1 may be sharpened:

Theorem 2. If any prime ideal of the distributive lattice L is the kernelof precisely one homomorphism, then L is relatively complemented.

For the proof let us consider the property A2 : Every prime ideal of thedistributive lattice L is the kernel of precisely one homomorphism. That A2 isan HI property may be shown in a similar way as in Theorem 1, if we takeinto consideration that the complete inverse image of a prime ideal is againa prime ideal. The chain of three elements does not have the property A2

since its ideal (0] is prime and, as we have seen above, it is the kernel oftwo different congruence relations. This completes the proof of Theorem 2.

§ 3. Characterizations by residue classes.

In a distributive lattice L every ideal is a kernel of a suitable homomorphism, in other words, every ideal is a congruence class under a suitablecongruence relation. We consider only the case when L has a zero and a unitelement. By a theorem of O. BIRKHOFF ([2J, p.23, Theorem 4), to each ideal1 of L there exists a least congruence relation under which / is a congruenceclass. We shall denote by jl the dual ideal consisting of all x such thatx 1 under this congruence relation. We say j1 is the last residue class ofI. In a similar way, under the minimal congruence relation of II we mayconstruct the ideal consisting of all x with x . 0; it is the last residue classof the last residue class of I, and we denote it by jl'l. It is obvious thatP' I.

The following theorem is due to V. S. KRISHNAN [6J and T. MICHIURA [7J.


Theorem 3. The distributive lattice L with zero and unit elements isa Boolean algebra if and only if /JO / for all principal ideals I of L.

Before the proof we formulate Theorems 4 and 5 too.In his paper [5] K. ISEKI proposed the following problem:

Let L be a distributive lattice with zero and unit elements. Is the latticeL necessarily a Boolean algebra if every dual ideal of L is the last residueclass of precisely one ideal?

The answer is affirmative; moreover, there holds the following more general

Theorem 4. Let L be a distributive lattice with zero and unit elements.L is a Boolean algebra if and only if to every dual ideal D of L there existsan ideal J with D fl.

A generalized form of Theorem 4 is the following:

Theorem 4'. A lattice L with unit element is Boolean algebra if andonly if to every dual ideal D of L there exists an ideal I with D P.

Another characterization of Boolean algebras may be got in the following way.

Theorem 5. A distributive lattice L with zero and unit elements is aBoolean algebra if and only if for any fixed ideal I of L the same elements(t. e. the elements of P) are congruent to 1 under every congruence relationunder which I is a congruence class.

For the proof of Theorems 3, 4 and 5 let us consider the followingproperties concerning distributive lattices with 0 and 1:

An. For any principal ideal I of the lattice, I = po is valid.

A4 • For any dual ideal D, there exists an ideal I such that D fl.A". Any ideal I satisfies the condition that under every congruence rela-

tion, under which I is a congruence class, the same elements are congruent to 1.

First of all we show that A,; is equivalent to

A;. For any ideal I of L, 1= 110 is valid.

An is a part of A;, so it is enough to prove that A;; implies A~. If As isvalid and I is an ideal with I='FPo then") 1=:;/1°. Let us choose aE/-Po.We know, that (a] (aJI0 (by A'l) and from (a] I it follows that (apoc::po,in contradiction to a EE Jlo•

Next we prove that A::, A4 , At, are HI properties. Let [ be an arbitaryhomomorphic image of L.

';) We shall use the following well known results (see e. g. [6]): 1. I? /10, 2. IfIeJ then 110 c/o; the proofs are trivial.


A;;. If I* flO is valid for an ideal J in L, then let / be the completeinverse image of J and K that of i". Then I ~ K, but clearly K~ /1", thuswe obtain /:::) po, i. e. A; holds neither in L.

A•. Let 15 be a dual ideal of I, and D the complete inverse image of15. By A. there exists in L an ideal I with D = fl. It is clear that 15 = P.

AG• Since the complete inverse image of a compatible classification isagain a compatible classification, it is clear that AG is an HI property.

Finally we show that the chain of three elements has none of the propertiesAi (i = 3, 4,5). To prove this we only consider at the property Ai the ideal (a],at Ai the dual ideal [a), at A, the ideal (0] in the chain of three elements.

Thus by the Main Theorem, the necessity of the condition of relativecomplementedness in Theorems 3, 4, 5 is proved.

", J ~ In relatively complemented distributive lattices, as it is well known,every ideal and dual ideal is a congruence class under precisely one congruence relation, hence the sufficiency of the conditions is obvious. Thus theproof of Theorems 3,4 and 5 is completed.

Theorem 4' is an immediate consequence of Theorem 4, for the necessity of distributivity and of relative complementedness follows in a similarway as in Theorems 1 and 4 while the existence of 0 follows from the factthat there exists an ideal I with fl [1) and it is easy to check that thisimplies / = (0].

§ 4. Semi-complemented and weakly complemented lattices.

We define the following property:

An. In every homomorphic, image, with more than two elements, of thedistributive lattice L there exists an incomparable pair of elements.

Per definitionem Ar, is an HI property and the chain of three elementsdoes not have this property. Applying the Main Theorem we get

Theorem 6. A distributive lattice L is relatively complemented if alldonly if in every homomorphic image of L with more than two elements, thereexists an incomparable pair of elements.

As a consequence of this theorem we get

Corollary 1. A distributive lattice L with zero element is relatively complemented if and only if every homomorphic image of L is semi-complementedG

).

0) A lattice L with zero element is said to be semi-complemented if to each elementx(tl) of L there exists a y~. 0 satisfying xny=o (for this terminology see (11)).A lattice L with zero element is called weakly complemented if to all y < x there exists az with x nz 0 and y nz =~ O.


Corollary 2. (Theorem of G. J. ARESKIN [1 ].) A distributive lattice Lwith zero element is relatively complemented if and only if every 11Omomorphic image of L is weakly complemented").

Remark. We show by an example that a distributive lattice which isweakly complemented is not necessarily relatively complemented. (In otherwords, weak complementedness is not an HI property.) Let us consider arelatively complemented distributive lattice with zero element which has nounit element. We adjoin to this lattice a unit element. This lattice is obviouslyweakly complemented, but it is not relatively complemented.

§ 5. Topological characterizations.

In his paper [4] J. HASHIMOTO deals with certain topologies defined onthe set of dual prime ideals of a lattice L. By the aid of these topologieshe gets a number of topological characterizations of relatively complementeddistributive lattices. The notions used in the sequel are largely identical withthose of J. HASHIMOTO but there are some differences which make it necessaryfor us to state their definitions.

In this section we prove Theorem 7 which contains ten conditions, eachof them being equivalent to the relative complementedness of the given distributive lattice L. The necessity of these conditions is an immediate consequence of the Main Theorem. In proving the sufficiency, it would be simplerto refer to the paper [4]; for completeness' sake we prove here the sufficiencyof the conditions too; these proofs are essentially simpler than the originalones (in [4]).

Let Q denote the set of all dual prime ideals of L (the lattice L itselfis not considered as a dual prime ideal !). Dual prime ideals are denoted bythe capital latters P, Q, R. In what follows A,B, C are subsets of S2, A' isthe complementary set of A in Q and 0 is the void set.

Let us consider the following homomorphism of L onto the set ofcertain subsets of Q: F(x) == {P; x EPl. As a matter of a fact, it is a homomorphism of L onto Lo which consists of all subsets of S2 of the form{F(x); x EL}, for F(x ny)={P; x nyE P}= (P; x EP} n (P; yE P}=F(x) n F(y)and F(x Uy) == {P; xu YEP} = {P; x EP} u {P; YEP} = F(x) u F(y), namelyxu yEP if and only if x or YEP. We get that F is a homomorphism andL" a ring of subsets of S2.

We define the following four operations (5 is a subset of L):

F(S)= /\ F(x),F(0)=Q; 0(5)= V F(x), 0(0)=0;xEN xEN

p-l(A)={x; F(x)=::JA}; 0 l(A)={x; F(x)CA}.

(We mention the fact that F(x) = O(x) for x EL.)


If we define the closure of A by A= FF- 1 (A), then f2 becomes aTo-space7

); and if we define the open kernel of A by A(i = 00- 1 (A), thenQ becomes a 7;)-space too. The set Q with the induced topologies is denotedby f2 p and f2r;, respectively.

Let us show e. g. that Qp is a To-space. Preparatory to this we provethe following assertion:

(*) PEA if and only if P~ 1.1 = /\ P. (1.1 is a dual ideal).I'EA

Since X= FF- 1 (A) = F{x; F (x)::::J A) = /\ F(x), so in the meet occurP(:<'j:::lA

only the elements of the dual ideal h, moreover, if PEA then P E F(x) forall x E 1.1, i. e. 1.1 C P.

By this note, I. A.~. A is obvious; 2. A =~ A because IA = IA ; 3.AuB=cAuB is equivalent to Ij!nln=l"'uB, but obviously l",nlnc;JAuB,and on the other hand if P~IAuli then in case of P~I~IA and P~E)1B wechoose an x E I.1 -P and ayE In-P. It is clear that'x UyEP which is acontradiction for P is a dual prime ideal, i. e, P 21.1 or P ~ In that isIAunSJA n In the assertion is established.

It is easy to see that F(x) n A for a fixed A is also a homomorphismof L. In fact F(x ny) nA = (F(x) nA) n(F(y) nA) and F(x Uy) nA =c(F(x) n A) U(F(y) nA). We denote by f) (A) the congruence relation suchthat the homomorphism induced by f) (A) is just F(x) nA.

With the aid of these notions the following theorem gives ten characterizations of relative complementedness in distributive lattices.

Theorem 7. Each one of the following conditions is necessary andsufficient for a distributive lattice L to be relatively complemented: .

(I) Q o is a T1-space");(2) Qp is a T1-space;

(3) 0 (a)' is a T 2-subspacel') of Qu, for each a EL;(4) F(a) is a T2-subspace of Qr, for each a EL;(5) O(a)nO(b)' is closed in Qu, for each a,bEL;

7) If in the set M there is defined for every subset A a correspondence A -~ A between the subsets of M such that I. A2A; 2.X=Jr; 3. AUB, ==AUi3, 4. 0~l'J, 5. ifP =F Q, (P and Q EM), then P (j, then the set M is a To-space with the closure operationdefined above.

8) A To-space is a Tcspace if j5 == P is valid for all points P of To.0) A Tcspace is a T2-space if and only if for all pairs of points P, Q of T1 some

pair of open sets U containing P and V containing Q is disjoint.


(6) F(a) nF(b)' is closed in gp, for each a, bEL;

(7) if gu has more than one element, then to every P Ego there existsa Q ES2r; such that 15 <:1::: Q and j5:::!~ Q;

(8) if S2 p has more than one element, then to every PE gp there existsa Q ES2 F such that PS!:::Q and P;]"2 Q;

(9) 0- 1 (A') = 0- 1 (B') 0 implies 0 (A) = (-) (B); ]f')

(10) p-I (A) = p-I (B) '* 0 implies 0 (A) = 0 (B).

Remark. With the exception of the conditions (7) and (8), the conditions (1)-(10) are that of ]. HASHIMOTO [4]; see his Theorems 4, 2 and 7, 1.

PROOF. It is enough to prove the assertions (2), (4), (6), (8) and (10),because the others may be proved in a similar wayll).

Necessity. Let us consider the chain of three elements. In this latticeQp={P,Q}, where [a)=P and [l)=Q. It is easy to check thatF=P, Q={P, Q}.

In this lattice the following statements do not hold:

(2), for Q:e,~ Q;

(4), because F(l) is not a T2-space;

(6), for F(l)nF(a),-'--QcQ;

(8), because in S2/.. there are only two dual prime ideals and Pc Q;(10), since F- 1 (Q) == F -I (P, Q) ~~ {1} yet 0 (P) > (-1 (P, Q).

Since homomorphic images and complete inverse images of dual primeideals are again dual prime ideals it is clear that the properties AT,I, AT, ':2 , ... , AT•w ,

which are defined respectively that the conditions (1), ... , (10) are fulfilled,are HI properties. Applying the Main Theorem we get that the fulfilment ofone of the conditions (1)-(10) implies the relative complementedness of thelattice.

Sufficiency.

(2) By (*) it is sufficient to observe that in relatively complementeddistributive lattices every dual prime ideal is maximal (see [10], or [2],p. 160).

1lI) It is known that every e (A) is a congruence relation of L, it is also true thatall congruence relations of L are of the form e (A) for a suitable subset A of Q. We omitthe proof of this fact, since we shall not need it in the sequel.

11) More exactly we refer to Lemma 3,2 of J. HASHIMOTO [4]. Let L* be the dual.lattice of a lattice L, and let Q and Q* be the sets of the dual prime ideals of Land L*respectively. Then between the space Q;, and the space Q,: there exists a homeomor-

phism under which G (a) corresponds to F (a)" and accordingly G (S) corresponds to F (Sr.


(4) Instead of considering the subspace F(a), it is enough to verifythat if a relatively complemented distributive lattice L has a unit element,then gp is a T2-space. We must prove for all P, Q Egp the existence ofopen subsets AI' and AI2 of !;2B' such that P EAI', Q EAQ and AT' nAil 0.Instead of this we prove the existence of closed sets B1' and B II such thatPtf;.Bp, QEfB9 and Bl'UBI2 =SJp. Preparatory to this we show:

Let P and Q be prime ideals of the relatively complemented distributive lattice L with zero element. There exist two ideals X and Y suchthat P, Q and L = X. Y. 12

)

For the proof we consider an element a of P- Q (such an a exists,because, according to (2), Q is impossible). We construct Y from theelements b, satisfying an b = 0. We prove that (a] U Y = L «a] n Y = ° isobvious). Namely if x EL, we suppose that xEf Y i. e. a n x 0, then denotingby ax the relative complement of a nx in the interval [0, x], we havea" n (a n x) = 0, i. e. a, n a = 0, that is, a" E Y. We get x = (x n a) Ua~ whichwas to be proved.

Now we construct B1' and BIJ • We know that there exist dual primeideals XC:;P and Y S Q satisfying X n Y"= (1), Xu Y = L. Let R EBp if andonlyifR Y; REBr.1 if and only if X.I. B1'uBIJ=gp,forifREfBl'uBIJthen there exists an xEX and ayE Y such that x, yEtR, but XU yEXn Y=,"[l),that is, XU Y 1 E R, i. e. R is not a dual prime ideal. 2. B1' is closed (andin a similar way Bli too), for if R EBp , Rtf. B1' , then R EBq thus R-:::) X andR -:::) Y, i. e. R;::2 X u V= L, a contradiction.

(6) Let us suppose that P EF(a) n F(b)", PEf F(a) nF(b)". We choosein L the interval [x, y] so that a, b E [x, y} and [x, y] n P 0. The non-voidintersections of the dual prime ideals of L with the interval [x, y] form allthe dual prime ideals of [x, y] (it is easy to prove this on using STONE'STheorem). So it is clear that F(a) nF(b)' is not closed in the F topology of[x, y]. Hence it remains to prove (6) for lattices with °and 1 elements, i.'e.for Boolean algebras. Under this hypothesis, let b' denote the complementof b. Evidently, F(b') = F(b)', so F(a) nF(b)" = F(a) nF(b') = F(a nb/). ButF(c) is closed for all eEL as it is obvious from (*).

(8) If Q then from (2) Pn Q Pn Q=0.Instead of (10) we prove (9) for the sake of convenience. 0-1(A') is a

congruence class under the congruence relation @(A), for x E 0-1 (A") if andonly if F(x) A" which is equivalent to F(x) nA = 0. We see that (9)asserts: if two congruence relations have the property that the kernel of the

12) X. Y denotes the cardinal product of the lattices X and Y in the sense ofG. BIRKHOfF [2}, p.7.


homomorphisms induced by them are the same, then the two congruencerelations coincide. But this is actually true in relatively complemented lattices(see the proof of Theorem 1).

Thus the proof of Theorem 7 is com pleted.

§ 6. Group operatious on distributive lattices.

If on the set of elements of a distributive lattice L there is defined a newoperation, satisfying the group axioms and having the property that everylattice homomorphism is a group homomorphism too, then we speak of agroup operation defined on the distributive lattice L.

We proceed to the problem which consists in finding a necessary andsufficient condition under which on the distributive lattice L a group operation may be defined.

Theorem 8. On a distributive lattice L a group operation may bedefined if and only if L is relatively complemented. If L is relatively complemented distributive lattice, then all possible group operations on L may bedefined in the following way:

taking any (fixed) element w of L, define x +y as the relative complement of (x n w) n (x u y) n (y u w) in the interval [x n y n w, xu y U wJ.

PROOF. Let As be the following property: In the distributive lattice La group operation may be defined. As is an HI property, as it follows directlyfrom the definition. It is also obvious that the chain of three elements doesnot have the property As, for this chain has a lattice homomorphism ontothe chain of two elements, while the group of three elements has no grouphomomorphism onto the group of two elements. Thus the Main Theoremassures the relative complementedness of L.

Let us suppose that in a relatively complemented distributive lattice Lwe can define two different group operations (+ and 0) with the sameneutral-element w. Then there exists in L a pair of elements x, y such thatx y x 0 y. By STONE'S theorem (see footenote 2) there exists a primeideal P which contains exactly one of the elements x +y and x 0 y. Withoutloss of generality we may suppose that w EP; then P is a normal subgroupand L-P is its coset, for the partition of L into P and L-P is a compatible classification of the lattice L. Hence if x and yEP or x, y EL":"'" P, thenx y and xoyEP, and if xEP,yEL-P, then x+y and x oyEL-P. Inall cases we have got a contradiction.

We have thus proved that there exists at most one group operationwith a fixed zero element. One group operation with the zero element w is

286 G. Gratzer and E. T, Schmidt

defined in the theorem, for it is obvious that the zero of the above mentioned operation is w; the associative law may be proved in a similar way asthe unicity. Obviously x+x = IV is valid for all x, thus we obtain an'elementary (abelian) 2-group on the lattice L. The proof of the theorem iscompleted.

Evidently, the following statement is valid:On the distributive lattice L ring-operations may be defined if and only

if L is relatively complemented.'Indeed, if on L we have defined ring operations, then R+ is a group

and Theorem 8 implies that the lattice L is relatively complemented. On theother hand, if L is a relatively complemented distributive lattice then by Theorem 8 on L there may be defined a group operation with the zero elementw, and we can define the product by the rule xy IV for all x, y EL. (In [3]we have proved that the product may be defined by the following non-trivialrule too: xy = (x Uw) n(x uy) n(y uw).)

§ 7. Some other charaeterizations.

Previously we have seen a lot of applications of the Main Theorem.In some cases it is simpler to prove the necessity of relative complementedness by the aid of the following (obviously) equivalent form of the MainTheOrem:

If the distributive lattice L is not relatively complemented then thereexists in L two different prime ideals one of which contains the other.

Let us see an example.It is well known that the lattice of all ideals of a distributive lattice

with zero element forms a pseudocomplemented lattice (see [10]). Letl* denotethe pseudocomplement of the ideal I.

Theorem 9. If for all ideals I of a distributive lattice L with zero element,/** holds, then L is relatively complemented.

PROOF. Let us suppose that L is not relatively complemented; then in Lt here exist two prime ideals P, Q satisfying Pc Q. Let q EQ-P. If q n x = 0,then xE P, for P is a prime ideal and 0 EP. It follows that Q* i. e.Q* c Q that is Q* = (0] and so Q** (0]* = L Q, q. e. d.

It is easy to see that Theorem 9 may be sharpened:

Theorem 9~. In a distributive lattice L with zero element, 1** holdsjar all ideals I of L if and only if L is relatively complemented and satisfiesthe descending chain condition.


PROOF. If 1= l** identically holds in L, then r is the complement ofI, for otherwise (I n l*)* = (0] i. e. L = (I u l*)** (I u l*), a contradiction. Butthe lattice of all ideals of a relatively complemented distributive lattice L iscomplemented if and only if L satisfies the descending chain condition(see [4], Theorem 4,3).

The sufficiency of the conditions is obvious.Finally we mention the following, almost trivial

Theorem 10. In a distributive lattice L every pair of congruencerelations is permutable if and only if L is relatively complemented.

PROOF. The sufficiency of the condition is well known (see e. g. [2],p. 86, Ex. 3).

Let us define the property A1o : On the distributive lattice L all congruence relations are permutable.

Obviously Ao is an HI property and the chain of three elements doesnot have this property. Thus the necessity of the condition is a consequenceof the original form of the Main Theorem.

Bibliography.

[IJ r. 51. Ape III KHH, 06 OTHOllleHHSlX KOHrpyeHUHH B ItHCTpHoynlBHhlx cTpyKTYpax cHyJIeBblM 3.'leMeHTOM, A0 KJI. A l( a It. Hay l( C C C P 90 (1953), 485-486.

[2] G. BIRKHOFF, Lattice theory (2nd ed.), New York, 1948.{3] G. GRATZER-E. T. SCHMIDT, Hal6k idealjai es kongruencia relaci6i I., Magyar Tud.

Akad. Mat. Fiz. Oszt. K6zl. 7 (1957), 93-109. (Ideals and congruence relations in lattices; the Englisch version appears in Acta Math. Acad. Sci. Hangar.)

[4] J. HASHIMOTO, Ideal theory for lattices, Math. japon. 2 (1952), 149-186.[5] K. ISEKI, Contribution to lattice theory, Pub!. Math. Dcbrecen 2 (1952), 194-203.[6] V. S. KRISHNAN, The problem of the last-residue-dass in distributive lattices, Proc.

Indian Acad. ScL Sect. A. 16 (1942), 176'-194.[7] T. MICHIURA, On Characteristic properties of Boolean algebra, j. Osaka Inst. Sci. Teclm.

Part 1. 1 (1949), 129-133.[8] A. MONTEIRO, Sur l'arithmetique des filtres premiers, C. R. Acad. Sci. Paris 225 (1947),

846-848.[9] L. NACHBIN, Une propriete carasteristique des algebras booleiennes, Portugal. Math. 6

(1947), lI5-1I8.[10] M. H. STONE, The theory of representation of Boolean algebras, Trans. Amer. Math.

Soc. 40 (1936), 37-11I.[11] G. SzAsz, Dense and semi-complemented lattices, Nieuw Arch. Visk. 1 (1953),42-44.

(Received September 3, 1957J

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