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SEPARATUM

ACTA MATHEMATICAACADEMIAE SCIENTIARUM HUNGARICAE

TOMUS IX FASCICULI 1-2

G. GRATZER and E. T. SCHMIDT

IDEALS AND CONGRUENCE RELATIONS IN LATTICES

1958

IDEALS AND CONGRUENCE RELATIONS IN LATTICESBy

G. GRATZER and E. T. SCHMIDT (Budapest)

(Presented by A. RENYI)

Introduction

One of the important tools of the lattice-theoretical researches is theexamination of lattice congruences. In connection with lattice congruencesarises the necessity of the examination of lattice ideals, for fa - the kernelof the homomorphism induced by the congruence relation e - is an ideal(if it is any), and this ideal implicates a lot of properties of e.

In this paper our aim is to examine the properties of lattice congruences and the correspondence @---+ fa. Our main tools in the discussion aretwo special types of congruence relations: the minimal congruence relations,induced by a subset of the lattice L, and the separable congruence relations,respectively.

In this paper we deal also with three problems of G. BJRKHOFF [2).1We prove a result of J. HASHIMOTO [14] (solving problem 73) in a newand more simple way,2 and get a new answer to the question raised inproblem 72; this has more applications to special cases than the originalsolution of this problem given by T. TANAKA [18]. We obtained a moregeneral solution of problem 67 than J. JAKUBIK in [15].

The paper consists of four parts. In Part I we deal with congruencerelations in distributive lattices. First we prove a theorem that describes theminimal congruence relations in distributive lattices. By the help of thistheorem we get a good look at numerous properties of congruence relations indistributive lattices, all of which are able to characterize the distributivity ofa lattice. We prove, finally, a theorem which is a far-reaching generalization

I Numbers in brackets refer to the Bibliography given at the end of the paper.2 In his cited paper J. HASHIMOTO deals with the representations (representation is a

homomorphism of a lattice onto a ring of sets) of a lattice, and with topologies whichare defined by special representations and inverse representations. Among the applicationsof these general discussions one can find the solution of problem 73. This explains thatif we consider a single theorem independently of the others, then the proof seems to berather difficult. It has some interest that while J. HASHIMOTO uses the Axiom of Choiceduring the proof, we succeeded in omitting it.

138 G. GRATZER AND E. T. SCHMIDT

of many known theorems (which are due to J. HASHIMOTO and M. KOLIBIAR),and contains the solution of G. BIRKHOFF'S problem 73 too.

In Part II we discuss with the help of the notion "weak projectivity"(introduced by R. P. DILWORTH [3]) the questions related to congruences ingeneral lattices. After three preliminary lemmas we get an answer to the following question: In which lattices is every congruence relation @ completely determined by the ideal !e consisting of all x with x~ o? Further onwe consider the least congruence relation @[/] under which a given idealis a congruence class. We point out that the correspondence 1-- @[/] is acomplete join-homomorphism and in case of distributive lattices it is moreoveran isomorphism. Finally, we turn our attention to some questions related toweak projectivity.

In Part III we deal with the notion of separable congruence relations.After the definition and typical examples we prove some lemmas of preliminary character, some of which are interesting in their own right. Next weturn to the problem of giving an answer to G. BIRKHOFF'S problem 72, byapplying the results concerning separable congruence relations. Then we usethese results in order to characterize the distributive lattices on which thecongruence relations satisfy the dual infinite distributive law.

In Parts II and HI we get the results of J. JAKUBIK [15] - concerningproblems 72 and 73 of G. BIRKHOFF, in case of discrete lattices - as trivialspecial cases.

We close Part III by analysing a question raised in problem 67 ofG. BlRKHOFF.

After some preliminary theorems we deal in Part IV with the problem:in which distributive lattices maya Boolean ring operation be defined? Wedescribe also in Part IV the types of these lattices and operations.

The kernel of this paper has been published in Hungarian in the papers [6] and [7J. We supplemented the results by several new ones. Forinstance, the results concerning G. BIRKHOFF'S problem 67 are all new.

Preliminaries

Let L be a lattice. The elements of L are denoted by the lettersa, b, C, ••• , x, y, z. If the lattice L has a greatest or a least element, then itwill be indicated by 1 and 0, respectively. Proper inclusion will be denotedby a > b, while the fact that a covers b will be indicated by a >- b. Thelattice operations are denoted, as usual, by u and n, while Vaa and 1\ aa

a a

will mean the complete join and meet of the elements aa, respectively, ifthey exist. If a has a complement, it will be denoted by a'.

IDEALS AND CONGRUENCE RELATIONS IN LATTICES 13-J

{x, a(x)} designates the set of all elements x in L for which the proposition a(x), defined on the elements of L, is true.

The principal ideal generated by a is (a] = {x; x a}, the principaldual ideal generated by a is [a) {x; x a} and the closed interval [a, b]is {x; a :::: x S b}.

The congruence relations on the lattice L are denoted bye, fP, g, 'YJ.The set of all congruence relations on the lattice L is indicated by eel).The universal and the identical congruence relations are designated by t andw, i. e. x=y (t) for all x,yEL; x=y (w) if and only if x=y.

Ideals of the lattice L are denoted by the symb.ols I, j, K. The set ofall ideals of the lattice L is indicated by :to

The sets eel) and :t under suitably defined partial orderings form alattice. This is assured by the following two assertions:

Under the natural partial ordering, @ -< fP (@, fPE @(L)) if and onlyif x y (@) implies x y (fP), @(L) form a complete lattice. Moreover,let A be any set of congruence relations 0 on L. We define two new relations§and 1, by

(i) x= y(g) means x (0) for all 0 EA;

(ii) x-y(1]) means that for some finite sequence x Zo, ZI,".' Zm=ywe have Zi-l =Zi (e,) for some 0 i E A.

Then ~, r, are congruence relations; moreover 1] is the join and ~ isthe meet of all @ E A.

LEMMA 1. Let L be a lattice and :t the set of all ideals of L. Underthe set inclusion, :t is a lattice with complete union. If A is a subset of :t,then we define K as the set of all x for which

x S Yl U Y2 U ••• U Yn, Yi EIi

for some Ii EA. Then K is the complete union of the lEA.

The first assertion is due to O. BIRKHOFF [2]; we were unable to finda proof of the second one in the literature, but the proof is clear from thedefinitions, so we omit it.

Now we define some special congruence relations. Let 5 be a subset of LandA the set of all congruence relations under which 5 lies wholly in one congruenceclass. By BIRKHOFF's theorem (cited above) the meet of the set of congruences A is again a congruence relation, and under this, too, all the elements ofS are in the same congruence class. Hence there exists a least congruencerelation under which the elements of S are in the same class. We shall saythat this is the congruence relation generated by 5, and we shall· denote itby @fS]. A special case of great importance is when 5 contains only two


elements a and b; in this case 0[SJ will be designated by 0a., b. A trivialconnection between the notions 0[SJ and B,l,b is the following

LEMMA 2. Let S be a subset of the lattice L. Then

(1) B[SJ V Ba, b.(l, bE,S

PROOF. Obviously, 0 a, b ::: B[SJ for all a, bE S, hence V Ba,b B[SJ.a, bES

On the other hand, S is in one congruence class under V B a, b, for x, yES anda,bES

X y( V Ba,b) contradict 0r,y <: V Ba,b' Thus B[S] = V Ba,b by thea, bES a, bES a, bES

minimal property of B[SJ, as asserted.If L is a lattice, then I denotes a homomorphic image of L, under the

homomorphism a --+ (i, i. e. (i denotes an element of L as well as the classof those elements x of L for which x --+ a. If a congruence relation B isgiven, then the homomorphic image of L induced by B (i. e. the lattice ofall congruence classes) will be indicated by L(B). If there exists an idealwhich is a congruence class under the congruence relation B, then we denoteit by Ie. Clearly, Ie is the kernel of the homomorphism induced by @.

lf in L all bounded chains are finite, then following J. JAKUBIK andM. KOLIBIAR we speak of a discrete lattice. Further, if in L between all comparable pairs of elements there exists a finite maximal chain, then we callthe lattice semi-discrete. (These notions coincide in modular, moreover insemi-modular lattices, see e. g. [tOJ.)

At last, we shall denote by Sand T the five element lattices generatedby the elements x, y, z such that the following identities hold:

(S) x > y, x Uz = y uz = 1, x nz = y nz = 0 ;

(T) x uy = x U z = y uz = I, x ny = x nz = y nz = O.

I. CONGRUENCE RELATIONS IN DISTRIBUTIVE LATTICES

§ 1. Description of minimal congruence relationsin distributive lattices

It is well known that if 0 is a congruence relation, then a b (B) ifand only if a ub= a nb (B) (see [2]). From this trivial fact it follows thatwe need consider only the problem of determining the comparable pairs ofelements congruent under the minimal congruence relation, which collapsesa comparable pair of elements. (We say that 0 collapses a and b if theyare in one congruence class under @)

IDEALS AND CONGRUENCE RELATIONS IN LATTICES 141

(a U 1') nu =~ ll,

(b uv) nu = v

THEOREM 1. Given two elements a, b of the distributive lattice L witha :>- b, the elements c, dEL with c ~ d satisfy c d (@a, b) if and only if

(2) (a U d) n c = c

and

(3) (b U d) nc= d.

PROOF. We define the relation @ on L by putting

(4) x-y (@)

if and onl y if c = x Uy and d = x ny satisfy (2) and (3).From the identities (a Ux) nx = x, (b Ux) nx = x it follows that @ is

reflexive, and from the symmetry of x, y in the definition of @ it follows thesymmetric property of @. To prove the substitution law for @, let us suppose x _y (@), and let t be arbitrary, then from the distributivity of Landfrom (2), (3), (4) we obtain

{a U [(x Ut)n (y Utm n [(x Ut) U(y Ut)] = {[a U(x ny)] u t} nn [(x uy) Ut] = {fa U(x ny)] n(x uy)} U t = (x Uy) Ut = (x Ut) U(y Ut) ;

and in a similar way

{b U [(x u t) n(y u t)]} n [(x U t) U(y U t)] = (x Ut) n(y u t);

furthermore

{a U [(x n t) n (y n t)]} n [(x n t) U(y n t)] = {[a U(x ny)} n (a Ut)} nn [(x uy) n t] =-= [a U(x ny)] n(x uy) n [(a Ut) n tJ = {[a U(x ny)] n

n(x uy)} n t = (x Uy) n t = (x n t) U(y n t),

and likewise

{b u [(x n t) n (y U t)]} n [(x n t) U (y n t)} = (x n t) n (y n t).

Thus these equations show us that x = y (@) implies x Ut - Y u t (@) andxnt_ynt (@).

We show the transitivity of @ at first in case u ~ v~ w, u _ v (@),v==w (@). By (4) we get

(5)

(6)

and

(7)

(8)

We prove

(9)

(10)

(a U w) n17=1',

(bu w) nv=w.

(a U w) n II = ll,

(b u w) nu=w

142 G. GRATZER AND E.T. SCHMIDT

But

which are by (4) equivalent to a= w (e). Clearly, from (7) we have a uw~ 1',

applying this fact, a >- v, v ~ wand the distributivity of the lattice we get

(a U w) n a = [(a U w) na] U v = (a U wU v) n (a U v) = (a U v) na,

but by (5) a (a Uv) n a, thus (a U w) na a, completing the proof of (9).From v w we get b U v ::: b uw, hence, using (6) and (8),

(b uw) na = (b u w) n (b UI;) na = (b U w) nv w,as asserted.

Now let us suppose a=v (e), tc W (@) for arbitrary a, v, wEL.Applying the substitution law, it follows a Uv (a Uv) U (v nw) a Uv Uw (@),anv=(anv)n(vuw) (anv)n(vnw) anvnw (@), i.e.

au n U w a U v (e),

auv=anv (e),a nv::::= a n I' nw (e).

a U v U W2:: a U v s:. a nv s:. a nv nw,

thus from the previous paragraph it follows that a U v Uw- a nv nw (e).From the substitution law by direct computation we obtain au w= (a Uw) uU(a n w) = [(a Uv U w) n (a Uw)] U (a n w) [(a n v n w) n(a Uw)] U(a nw) == (a n v n w) U(a n w) = a n w (@). This completes the proof of the transiti-vity of e. .

We conclude that @ is a congruence relation. Furthermore, a= b (e),so @ ~ @a,b' @a,b being the least congruence relation with a::::= b. Again,x::::= y (e) implies in view of Theorem 1

xu y [a U(x ny)] n (x uy) and x ny = [b U(x ny)] n (x uy).

a b (ea, b) and so applying the substitution law twice to the elementst= xu y and t x ny, we get [a U(x ny)] n (x uy)= [b u (x ny)] n(x uy) «!-)a, b)which is equivalent to xuy=xny (@a,b) if we take into consideration theabove equations. Hence x _ y (@) implies x -- y (eo, b)' that is, e:;s eo, b,which compared with the inequality proved above gives the desired result,@ = ea., b, completing the proof of Theorem 1.

Some of the most important applications of Theorem 1 will be provedin the following section.


§ 2. Characterizations of distributive lattices

143

Some properties of congruence relations of a lattice are suitable tocharacterize the distributivity of a lattice. We shall deduce such characterizations from Theorem 1.

THEOREM 2. Each one of the folloWing conditions is equivalent to thedistributivity of the lattice L:

(a) if c-:=.d (@a,b)' then a<d (or c<b) is impossible whenever b<a,d < c (a, b, c, dEL);

(b) [b, a] is a congruence class under @a,b for all b < a (a, bEL);

(c) @a,bn@c,(I=wfor all a~b?':.c~d (a,b,c,dEL);

(d) @a,b has a complement in @(L) (for all a ~ b) such that c~ aimplies c a (@~, b) (a, b, eEL);

(e) if C is a chain of the lattice L, then every congruence relation ofC may be extended to L such' that the congruence classes on C remain thesame;

(f) for any ideal 1 and for any x~y, x_y (@[1]) if and only ifx = Y Uv for some v EI;

(g) the condition (f) is valid for all principal ideals 1;(h) every ideal is a congruence class under some homomorphism;

(j) every principal ideal is a congruence class under some homomorphism.

PROOF. First we prove that conditions (a)-(j) hold in a distributivelattice L.

Let us suppose that c - d (@a,b) (a ~ b, c > d) and yet b:>- c, then from (3)d = (b Ud) nc = c contrary to c > d. We get a contradiction in a similar wayfrom a d and (2). Thus the validity of (a) is a simple consequence ofTheorem 1.

From (a) we can easily deduce (b). Indeed, if c-a (@a,b) and cE£[b, a],then either c Ua > a or c nb < b holds (for, in case c Ua = a and c nb = b,we should have b < c < a, i. e. c E[b, aJ). Butfrom a < a Uc and a c (@a,b)we get a-:=.auc (@a,b)' while b>cnb, c-b (@a,b) imply b=::cnb (@a,b),both are in a contradiction to (a).

Now we prove that (c) holds in L. Let a >b?':. c > d, x> y andx-:=.y (@a,bn@e,d). Then x-y «('1c,d)' hence by Theorem 1

(11) (duy)nx=y.

We assert that cn(dux»cn(duy). Indeed, if cn(dux)=cn(duy), thenfrom the equality c U x = c UY (it follows from c Ux - c Uy (@e, d) and

144 O. GRATZER AND E. T. SCHMIDT

from (a» we get c U (d U x) c U (d U y). Thus dux and d Uyare both relativecomplements of c in the interval [c n(d U x), c U xJ, hence dux = d uy. From(11) we infer x = (d ux) nx = (d Uy) nx = y, a contradiction. Obviously,en (du x)=c n (du y) (@,X,y) and @~',y <:@a,b' so we get c n (dU x)=cn(duy) (ea,b) and a>b?;.c"?;.c n(dux»cn(duy), in contradictionto (a). If a = b or c = d, then there is nothing to prove.

Next let us consider condition (d). We define t;P as the join of thecongruence relations @[[a)] and @[(b]]. Then @a,b U t;P i, because for allX-<YEL,[x,y]C[xnb,yua], thus from xnb_yua (@a,bUt;P) we getx=y (@a,bU ([J). Let us suppose that for some x,yEL (X=f=Y) we havex =y (@a, bn t;P). This is equivalent to w < @x,y and @X'1I -< @a, bn (/J.

From the latter 0:1",y= @a,b n @:1",1In ([J and t;P= V @",,,. Thus (using theu>v-=:::aH~t'~b

infinite distributive law in @(L), see in [2J, [4], or in § 1 of Part II) @,r,y= @:1",y nn@a,bn V @""'=@X,ynV(@a.bn@",,,) and from (c) @a,bn0",v=w which

lI>t'~a

u<t'~b

is a contradiction. So, t;P = (i<);, b. Obviously, c::- a implies c - a(t;P).To prove the validity of condition (e), let a chain C be given in L,

and a congruence relation t;P on C. We define the following congruencerelation of L: 0 = V @a, b. We prove that @ has the desired property.

a",=bEOG=sb(<P)Assume x==y (@), x, yE C. Then by BIRKHOFF'S theorem cited in the Preliminaries, there exists a finite number of pairs of elements a" bi such that

n

pairs of elements G;>b,:, ai,biEI (i=I,2, ... ,n) such that x y(V@ni,b).i=1

Let a = Vai and b= 1\ bi , then a, bEl and obviously x y (@a, b)' Then byTheorem 1 x = x n (a U y), thus from the distributivity of L we getx = (x n a) Uy, hence v = x na has the desired property. On the other hand,it is clear that if x = Y uv and v EI, then x y (@[I]), so we have provedthe validity of (f).

n

Gi < bi and ai - bi (t;P), furthermore x=y( V@ai' b). If [x, yJ c U[ai, b;], theni=1

x = Y (t;P) is valid too and there is nothing to prove. If x y( (/J), then thereexists a part [Xl' YI] of [x, yJ with the property that for each i either ai < bi <:

. n

-<XI<YI or x1<YI<ai<bi . Then from (c) @:r,ynV@ai,bi w which con-tradicts @x, 11 -< V@Gi' bi' i=1

To verify condition (f), let the ideal I of L be given and let x >- y,x=Y (@[I]). We prove that x (@a,b)forsome a, bEl. Indeed, from Lemma 2@[IJ= V @O,b' and by BIRKHOFF'S theorem there exists a finite number of

G, bEIa~b

IDEALS AND CONORl'ENCE RELATIONS IN U,TT:CES 145

The conditions (g), (h), (j) are special cases of (f).

Now we prove that each one of the conditions (a)-(j) implies thedistributivity of L.

If L is not distributive, then it contains as a sublattice a lattice, isomorphic to the lattice S or T, defined formerly. Since a lattice has one ofthe properties (a)-(j) only if every sublattice of it has this property, so wemust prove only that the lattices Sand T fail to have this property. Amongthe conditions (f), (g), (h), (j) the last is the weakest one} hence in this stepof the proof we may omit the others. (b) is a consequence of (a), so wemay omit condition (a) too.

First we verify that the interval [0, yJ is a congruence class underno homomorphism in Sand T. Indeed, if y=o (B) for some B, thenx xn(yuz)=xn(Ouz)=O and xE!:[O,yJ, a contradiction. Hence it resultsthat in a non-distributive lattice conditions (a), (b), (f), (g), (h), (j) do nothold. A similar trivial computation shows that conditions (c) (consider in Sthe chain 0, y, x and in T the chain 0, x, I)} (d) (in S the interval [y, xJ, in Tthe interval [0, x] play the role of the interval [b, aD, (e) (see the chainsdescribed at the condition (c», do not hold in the lattices Sand .T. Thusthe proof of Theorem 2 is completed.

We mention that the conditions of Theorem 2 playa fundamental rolein our researches related to all properties of distributive lattices, not onlyin this paper, but in the papers [9J, [lOJ, [11 J, [12J too.

Conditions (h) and (j) are the same as those of Theorem 2. 2 ofJ. HASHIMOTO [14J (conditions (3) and (4».

§ 3. A generalized form of O. Birkhoff's problem 73

In his textbook [2] G. BIRKHOFF proposed the following problem:

Find necessary and sufficient conditions in order that the correspondence between the congruence relations and ideals of a lattice be one-to-one.

More precisely:Find necessary and sufficient conditions in order that the correspond

ence B-Ie be an isomorphism between B(L) and f.Applying Theorems 1 and 2, we get an answer to this question.

LEMMA 3 (J. HASHIMOTO'S theorem). In the lattice L there is a one-toone correspondence (in the natural way) between the ideals and congruencerelations if and only if L is a distributive, relatively complemented lattice withzero element.

10 Acta Mathematica IX/I-2

146 O. GR.l\TZER AND E. T. SCHMIDT

PROOF.

The necessity of the conditions. Obviously, Iw is the zero ideal of L.Every ideal of L is a congruence class under some homomorphism, so thedistributivity of L is assured by condition (h) of Theorem 2.

Now let us suppose that L is a distributive lattice with zero element.We prove that L is relatively complemented. By a theorem of J. VON NEUMANN(see [2J, p. 114), it is sufficient to prove that if b < a, then b has a complementin the interval [0, aJ. Let Va, I> be the ideal which consists of all u withu =°(@a,I». Va, I> is a congruence class under precisely one congruencerelation, hence a=b (@[Va,I>]). From condition (f) it follows that for somevE Va, I> we have(12) bUv=a.

It is clear that v °(@a,b)' hence from Theorem 1 (v and °play the rolesof c and d) ,

(13) b n J! = 0.(12) and (13) show that v is the complement of b in [0, aJ.

The sufficiency of the conditions. From condition (h) of Theorem 2 itfollows that every ideal of L is a congruence class under some homomorphism. Furthermore, every ideal is a congruence class under at most onecongruence relation, as it follows from the complementedness of the intervalsof type [0, aJ" (see [2J, p. 23, or the Corollary of Theorem 4 in this paper).

Now we are ready to prove the general theorem.

THEOREM 3. Let L be a lattice and "a" a fixed element of L. Every convex sublattice of L containing "a" is a congruence class under precisely onecongruence relation if and only if L is distributive, and all the intervals oftype [a, b] (a:> b or a < b) are complemented.

PROOF.The necessity of the conditions. First we show the necessity of the dis

tributivity of the lattice L. Let us suppose that L is not distributive; then itcontains as a sublattice either the lattice S or the lattice T. (x, y, z will indicate the generators of S or T.)

We prove that z = a is impossible. Indeed, since x ny x Uy, that is,@ = @Xf\y, xvy is not equal to 0>, the congruence class which contains aunder @, is different from the congruence class which contains a under w.Thus the congruence class under @ containing a contains a further element x,so we may pick out an element c such that c~a and(14) c=a (@).

Now, according as c > a or c < a, the interval [x ny n0, aJ or [a, x Uy Ua] is

IDEALS AND CONGRU:;NCE RELATIONS IN LATTICES 147

a congruence class under no homomorphism. Let us discuss the case c > a(if c < a, then the proof goes on the same lines). Then a x n y n a (tJ) implies(as in the proof of condition (b) in Theorem 2) x -- Y (tJ), for any tJ), soxuy-xny UP), hence t:P?::cB, c=a (tJ) (see (14)\, but c€!:[xnyna,a];a contradiction.

Thus we have proved that x Uz y Uz and x nz nz areimpossible if z = a. So we may suppose by the Duality Principle thatx ua~ y Ua. We assert that under these hypotheses the interval [y nz na, y Ua}(which contains a) is no congruence class under any congruence relation.Indeed, if y ua__ y n z na, then

z z U (y nz na) _ z U (y U a) (z Uy) Ua = z U x U a,

furthermore x nz - x n(z ux Ua) = x. But x ~= y nz, y U a?: y nz:::.> Ynz naand x€!:[y nz na, y Ua], a contradiction.

Summarizing the above proved assertions, we get that the existence ofthe sublattices S or T contradicts the fact that every convex sublattice of Lcontaining a is a congruence class under precisely one congruence relation.

Our second aim is to prove the complementedness of the intervals oftype [a, b] (a?: b or a < b). Let b1 > b2 > a. Since B"" "2 =1= ill, there exists anelement c, comparable with a, such that c a «9b" b2)' From condition (a) ofTheorem 2 we see c < a is impossible. It follows that the congruence class underB b" b2 which contains a is not empty and it is a part of [a). Hence in [a) thecondition of Lemma 3 holds, that is, [a) is relatively complemented. In asimilar way we get the relative complementedness of (a] too. The necessityof the conditions is therefore proved.

The sufficiency of the conditions. Let L be a distributive lattice such that,.for a fixed a, the lattices (a] and [a) are relatively complemented. First weshow that the distributivity of L implies that every convex sublattice is acongruence class under some homomorphism. Let D be a convex sublattice,I and J the ideal resp. dual ideal generated by D. A trivial computationshows that D is a congruence class under B[I] nBU1.

Secondly we prove that every convex sublattice containing a is acongruence class under precisely one congruence relation. It is enough to'prove in case the convex sublattice consists of a alone, for if D is a congruence class under more than one homomorphism, then let us consideramong these the minimal one, B[D], and let I L(0[D]) be the correspondinghomomorphic image of L. In I there are fulfilled all the conditions as in Lif the fixed element is a, furthermore the one element convex sublattice (j.

is a congruence class under more than one homomorphism of I. SO wesucceeded in reducing the proof to a special case.

10*

148 G. GRATZER AND E. T. SCHMJl!)T

Now let us suppose that x > y. It is enough to prove the existence ofa c with C=Fa and c -a (0,'C.y). From the distributivity of L we obtaina ux > a U y or a n x > any (a U x = au y and a nx = any contradict x =F y).Let c be the relative complement of a uy in the interval [a, a U x] in the first,case, and the relative complement of a nx in the interval [a n y, a] in thesecond case. A trivial calculation shows that x y implies c~ a in both cases,that is, the one element sublattice a is a congruence class only under OJ.

Thus the proof of Theorem 3 is completed.The proof shows us that Theorem 3 may be sharpened by replaciflg

the condition "every convex sublattice containing a..." by the followingweaker one: "every interval containing a.. .".

That the relative complementedness of the whole lattice is not a consequence of the condition, it may be illustrated by the following simplecounterexample: L is the chain of three elements and a, the fixed element,is the only element different from 0 and 1.

An immediate consequence of our Theorem 3 is the

COROLLARY. Every convex sublattice of L is a congruence class underprecisely one homomorphism if and only if L is a relatively complementeddistributive lattice.

Special cases of Theorem 3 were already known. Lemma 3 (the special case a = 0) was first proved by J. HASHIMOTO [14] in 1952; a yearlater O. J. ARESKIN [1] has proved Lemma 3, by supposing that the latticeL is distributive and has a zero element. The Corollary was proved independently of us - by supposing the distributivity of the lattice considered by M. KOLIBIAR [16].

We remark that we may get further theorems, too, as easy consequencesof Theorem 3. For instance, in [9] we have pointed out that the followingassertion of J. HASHIMOTO [14] is also a simple consequence of Theorem 3:

A relatively complemented lattice L is distributive if and only if L hasan element a such that (a] and [a) are prime factorizable.

Using transfinite methods it results [11] that Lemma 3 may be sharpened; in [11] we have published another very simple proof of Lemma 3.Related to these questions we refer to [9] too.

IDEALS AND CONGRUENCE RELATIONS IN L~rT:CES

II. CONGRUENCE RELATIONS IN GENERAL LATTICES

§ 1. Some lemmas on congruence relations

14g>

In this section we prove three lemmas which will simplify the proofsof several theorems in Parts II and III. A part of the merely technical Lemma 4was proved already in Theorem 2.

LEMMA 4. Let ~ be a binary relation defined on the lattice L. ~ is acongruence relation if and only if

(a) xx (~) for all xEL;

(b) x Y (~) is equivalent to xu yx n y mfor all x, YEL;

(c) x~y~z,x==y mand y z (~) imply x z (~);

(d) if x~y and x y (~), then xut yut m, xnt==ynt m forall tEL.

PROOF. Obviously, it is sufficient to prove that a relation ~ satisfyingconditions (a)-(d) is a congruence relation.

By (a) ~ is reflexive, and by (b) it is symmetric too.Let u~ v, uv mand a, bE [v, ul, then we assert a b @. Indeed,.

u:>- au b~ an b~ v and from (d) un (a U b) v n(a U b) mand un (a u b):>~ v n(a U b), thus applying again (d), au b = [u n(a U b)] u (a nb) [v f)

n (a U b)] u (a n b) = an b m, whence from (b) a b (~), and the assertionis established.

Next let x y g) and y =z m. On account of (b) xu Y-x n Y (~),

thus from (d) xuyuz=(xuy)u(yuz)_(xny)u(yuz)=yuz m, similarly, x ny n z y n z (~), that is, x UY uz ~ y U z ~ y n z :>- X ny nz and theconsecutive elements are congruent modulo~, so applying twice (c) we getxu y U z _ x ny n z (~). Considering that x, zE [x ny n z, x u y u z], we concludex· .z (~), i. e. ~ is transitive.

The substitution law may easily be proved too, for if we assume x - y m,then from (b) and (d) xuyxny m and (xuy)ut_(xny)ut m,but xut,yutE[(xny)ut,xuyut], hence we obtain xut yut m andalike x n t = Y n t g), completing the proof of the Lemma.

We note that the conditions of Lemma 4 are independent and may beweakened, e. g. (a) may be replaced by (a /) x - y m for some x, y EL, butwe need only the above described form of Lemma 4.

Now we prove a lemma which sharpens for lattices a similar resultof G. BIRKHOFF for general algebras (see the Preliminaries).

150 G. GRATZER AND E, T. SCHMIDT

LEMMA 5. Let A be a subset of @(L). We define the relation lJ: Xo_y ('J)if and only if there is a finite sequence x uy = Uo::> Ul ::> •••~ u" x nysatisfying Ui Ui-1 (@,) for some @iEA (i=l, ... , n). Then lJ is a congruence relation and l) = V @a.

@a EA

PROOF. It is clear that if 1] is a congruence relation, thenrj [email protected] it remains to prove thatl) is a congruence relation. Obviously, it isreflexive and symmetric. If x y ~ Z, x (1]) and y == Z (1]), then we havetwo chains which connect x and y, resp. y and z, having the desired property.Joining these two chains, we get one from x to z with the desired property.At last if x Zo ::: Zl ~ ... ~ Z" = y, then tux = t UZo ::: t UZ1 ::> ... ::: t UZn =

t Uy, thus x - y (rJ) implies xu t .--: y Ut (l/), and in a similar way we getthat it implies x n t - Y n t (1]) too. We seerJ satisfies the conditions ofLemma 4, that is, ') is a congruence relation.

The importance of Lemma 5 should be revealed by the fact that itdecides in the interval [a, b] whether a- b is valid or not. For instance,applying Lemma 5, it may be proved easilyS the notable theorem of N.FUNAYAMA and T. NAKAYAMA [5], according to which in @(L) unrestrictedlyholds the infinite distributive law

(10) @n V@a=V(@ n @a).

In proving (10) it suffices to show that x ::: y and x==y (V(@ n @..» implyx==y (@nV@..). If x==y (V(@n@a», then by Lemma 5 for some finitesequence we have x = Zo Z1 ~ ..• 2::: Z" = y, Zi-1· Zi (@ n ('9i), henceZ':-1==Z,:(@), further on ZH Z,: (@,:), so Z':-lZ,: (V@a), consequentlyZi-1==Zi (@nV0..), that is, x'-y (@nV@a), thus @nV@.. 2V(@n0a),q.e.d.

According to Theorem 1, in a distributive lattice under @a, b (a b)the elements c, d, (c::> d) are congruent if and only if c = (a Ud) n c,d = (b ud) nc. Now we generalize this theorem to arbitrary lattices.

Obviously, if

(15) r.. ·({[(a Ub) UXl] nx2} Uxg) n ...] Ux" = cUd,

(16) [.··({[(anb)ux1]nX2}UX3)n · ..]ux"=cnd,

then c== d (@a.b)' as it follows from the substitution law.The theory of congruence relations in arbitrary lattices is based upon

the notion of weak projectivity due to R. P. DILWORTH [3].

g The idea of this proof of the theorem of FUNAYAMA and NAKAYAMA is essentially dueto R. P. DILWORTH Pl.

IDEALS AND CONGRUENCE RELATIONS [N LATTlCES 151

DEFINITION 1.4 Let L be a lattice and a, b, c, dEL. The pair of elementsa, b is weakly projective into the pair of elements c, d if for some Xl, ••• , XI! ELthe equations (15) and (16) hold.

In what follows a:tJ - c-:d will denote that a, b is weakly projectiveinto c, d. Obviously, the relation - is transitive.

With the help of this notion we can easily describe the congruencerelation ea,b.

LEMMA 6 (DILWORTH [3]). c=d (ea,b) in the lattice L 1/ and only iffor some finite sequence

(17) cUd = yo ~ Yl ~ ... ::::: Yk. c nd one has a, b - Yi-l, Yi (i = 1, 2, ... , k).

PROOF. It is clear that if c, d satisfy (17), then c d (Ba,b)' On theother hand, let us define the relation ~ such that u v m if and only ifsome sequence {Yi} and c = u, d = v satisfy (17). Repeating word for wordthe trivial calculation of Lemma 5 we get (applying Lemma 4) that ~ is acongruence relation, completing the proof of this lemma.

COROLLARY 1. Let L be a lattice and S a subset of L. S is a congruence class under some congruence relation if and only if a, b, c ESanda,b-c,d imply dES.

PROOF. The assertion "only if" is trivial from the definition, and "if"is obvious from Lemma 2 and Lemma 6.

From Corollary 1 and from Theorem 1 it results the well-known factthat every convex sublattice of a distributive lattice is a congruence class undersome congruence relation. This proof gives perhaps more insight into thecause of the validity of the above statement.

Another trivial consequence of this lemma is

COROLLARY 2. A lattice L is simple if and only if for all a, b, c, dELthere exists a finite sequence cud = 20 ~ 21 2': ••• ~ 2" = C nd such thata:b-2i-l,Z:(i =1,2, ... ,n).

If L is a modular lattice and a covers b, then a, b - c, d implies thatc = d or c covers d. Thus we are led to

COROLLARY 3. If in the simple modular lattice L there exists a pair ofelements a, b such that a covers b, then L is discrete.

4 Definition 1 is that of [6], but it may be shown easily that it is equivalent to thatof R. P. DILWORTH. The notation is the same as in [6].


§ 2. Weakly complemented lattices

The notion of weak complementedness was introduced by H. WALLMAN[I 9] for distributive lattices. Now we define the notion of weak complementedness5 in general such that for distributive lattices this is equivalent tothat of H. WALLMAN.

DEFINITION 2. A lattice L with 0 is weakly complemented if to allpairs of elements a, b (a =1= b; a, bEL) there exists an element c =1= 0 such thata, b ~c,O.

A trivial computation shows6 that in a distributive lattice a> banda, b~ c, 0 (C=1=O) imply a nc > 0, b nc = O. On the other hand, if a nc > 0 andb nc = 0, then putting c' ~~ an c we obviously have a, b~ c', 0 and c' =1=0.This coincides with the original definition of weak complementedness indistributive lattices.

Weak complementedness is not a homomorphic invariable property,that iS,there exists a lattice which is weakly complemented, but a suitablehomomorphic image of it is not relatively complemented. If this lattice isdistributive, then it is necessarily infinite (see the example in [II]), butin the non-distributive case there are finite examples too; e. g. let L be thefollowing lattice:

We can easily verify that this lattice is weakly complemented, yet L(0v,0) which is isomorphic to the chain of three elements - is not weakly complemented.

O. J. ARESKIN [I] proved the following assertion:Let L be a distributive lattice with zero element. Every ideal of L is

the kernel of at most one homomorphism if and only if every homomorphicimage of L is weakly complemented.

5 It seems to be unreasonable to change the definition of weakly complementedlattices, for it is a well-known notion. Our motivation is: the original notion of weakcomplementedness was successfully used only in distributive lattices in discussing theconnection of topological spaces and distributive lattices [19] and in the researches of thecongruence relations of distributive lattices [I]. In general lattices only some theorems wereknown which are based on the original definition. For this reason we propose the notionof "weakly complemented in the stronger sense" for the original one.

r; It follows trivially from Theorem 1 too.


Now we show that this theorem is valid for arbitrary lattices with theabove defined notion of weakly complementedness. This is a solution of themost natural generalization of G. BIRKHOFF'S problem 73.7

THEOREM 4. In the lattice L every congruence relation is the minimalone of a suitable ideal if and only if L has a zero and every homomorphicimage of L is weakly complemented.

For the proof a preliminary lemma is needed.

LEMMA 7. Let L be a lattice with zero element. The zero ideal is acongruence class under precisely one congruence relation if and only if L isweakly complemented.

PROOF. If the lattice L is weakly complemented, then the zero ideal isa congruence class only under w, for if x y, then there exists a z =F 0, withx;y-- z, 0, that is, Z _ °(@x, y), i. e. the zero ideal is not a congruenceclass. On the other hand, let us assume that to the elements x, y there isno element Z with x, Y --Z, 0. Then z--O (@X,y), Z > 0, is impossible, fodf thisheld, then from Lemma 6 it would follow the existence of a Zj >°witha, -- Zj, 0. Thus the zero ideal is a congruence class under wand @x, y too,a contradiction.

Now we prove Theorem 4. Let I be an ideal which is a congruenceclass under at least one congruence relation. Obviously, I is a congruenceclass under more than one congruence relation if and only if the zero idealof L(@[I]) is a congruence class under more than one congruence relation.Thus the proof of Theorem 4 is completed.

If L is distributive, then we get from Theorem 4 the above theorem ofof G. J. ARESKIN.8 On the other hand, we want to point out that every relatively complemented lattice with zero element is weakly complemented, so asa trivial special case of Theorem 4 we get a result of G. BIRKHOFF (see [2],p. 23):

COROLLARY. In a lattice with 0, where all closed intervals [0, a} arecomplemented, every congruence relation is determined by the ideal consistingof all x with x == 0.

We can get another answer to the above-mentioned problem.

7 J. JAKUBIK [15] and J. HASHIMOTO [14] have also formulated in such a way themore natural generalization.

8 Comparing Theorem 4 with Lemma 3 we get the following theorem of G. J. ARESKIN

[I]: A distributive lattice with zero is relatively complemented if and only if every homomorphic image of it is weakly complemented. (A far-reaching generalization of this theoremmay be found in our paper [11].)


If a, bEL, then Va, b will denote the ideal which is generated by allx with a, b-x,

THEOREM 5. In the lattice L every congruence relation Il'ls an ideal asa congruence class and every ideal is a congruence class under at most onecongruence relation if and only if L is a weakly complemented lattice withzero element and to all a, bEL there exist ayE Va, b and a sequence a U b =

do>d1 >· .. 2:d,,=anb with y,O-dH,di (i=I, ... ,n).

PROOF. We already know the necessity of the existence of a zero element and of weak complementedness. The third condition is necessary too,because if for a, b it did not hold, then Va,b would be a congruence class undermore than one homomorphism. Indeed, if Va,b were the kernel of preciselyone homomorphism, then a- b (@[Va,bJ) would be valid, and this means justby Lemma 6 the validity of the third condition.

The sufficiency of the conditions follows from the fact that under theseconditions

a b (@) if and only if Va, b C Ie,

that is, Ie determines the congruence relation. Indeed, if a 0" b (@), thena, -C:-O implies c==O (@), that is, Va,bCIe. On the other hand, ifVa, b C Ie, then there exist ayE Va, b and a finite sequence a Ub = Yo ~: Yl ~

>· .. >y,,=anb with y,O-YiobYi, but from yE Va,bCle it followsy (@) and so a_b (@), q. e. d.

Theorem 5 is a generalization of a theorem of J. JAKUBIK [15]. J. JAKUBIK

dealt with discrete lattices and he got the conditions of Theorem 5 with thesmall difference that the conditions on a, b are supposed only if a covers b.An easy computation shows that these conditions are equivalent in discretelattices, and what is more it becomes trivial that it is valid not only in discrete lattices but under that weakened condition, too, that L is semi-discrete.

§ 3. Minimal congruence relations generated by ideals

In this section we deal with the correspondence 1- @[I].

THEOREM 6. The congruence relation generated by the ideal VIa isVe[/a], that is,

(18) @[V/a]=V6[la].

PROOF. First we verify that if a~ b and a~ c, then

(19)

IDEALS AND CONGRUENCE RELATIONS fN LATTfCES 155

Since a::::::. b (e", bUG) and a c (ea, bUt), thus ea, b u ea, c:'S e a, buc; on theother hand, a b (ea, b) and a _c (@a,,), hence a - b uc (ea,bU@a,c), thatis, e a,buc2@a,cU(8)",,,. These inequalities prove (19).

By Lemma 2, (18) is equivalent to

(20) V eO',1/= V V @a,b.X,1/E V Ta aEA a, bETa

aEA

Let us suppose that ea, b occurs in the right side of (20), then a, bEla forsome a EA, hence a, b E V la, thus we obtain that @a, b occurs in the left

aE,4

side of (20), i. e. (20) holds with 2: instead of =.Conversely, let e.r,1I be a congruence relation which occurs in the left

side of (20). By Lemma 1 this means the existence of such iar (Elar , arEA;II.

r= 1,2, ... , n) that x, y:::: ia , U ••• U iar • Let u = /\ iar n(x ny). Obviously,,'=1

u E I a ,_ (r = 1, ... , n), hence ell, iar

occurs in the right side of (20). By (19)

(/a Ef),

@[ /\ la] = /\ @[/a]aEA "EA

V @X,1I <: V V @a,b,"·.lIE V I a aEA a, bETa

aEA

that is, (20) is valid.Let us denote by eo[I] the least congruence relation under which I is a

congruence class. Obviously, e[I] = @o[I] if eo[f} exists.9

COROLLARY. If @o[/a] exists for all a EA, and also @o[ VIa] exists, thenaEA

Veo[/a] = @o[V la].In Theorem 6 we have proved that the join of minimal congruence rela

tions of ideals is a minimal congruence relation of an ideal. The analogousassertion for the meet is not true as it may be shown by the example ofthe lattice S. It would have some interest to give conditions under which themeet of minimal congruence relations remains minimal. We assert only

LEMMA 8. Let L be a lattice with 0 on which every ideal is a congruence class under at most one congruence relation. Let £ denote the lattice ofall ideals in L which are congruence classes under some congruence relation.Then f is isomorphic to @(L), i. e.

@[ VIa] = V@[/a]aEA aEA

(21)

(22)

9 It would be of great interest to examine the lattice of all ideals for which eo[!]exists. One can easily prove that they form a distributive lattice.

156

let (I, K, X, }j Ef)

and

O. GRATZER AND E.T. SCHMIDT

T= Yo::::) Yl ::::) ... ::::) Ys = K,then there are refinements of these chains of common length.

PROOF. (21) was proved in Theorem 6. (22) follows from the fact thatthe existence of 0 EL implies the existence of A/a; furthermore Ala is a congruence class under Ae [Ia]. But Ala is a congruence class under at mostone congruence relation, hence Ae [Ia ] = e [Ala]. Thus we have provedthat the correspondence 1---+ e[I) (I Ef") is an isomorphism, between f" andeeL), and so f is distributive. Hence the JORDAN-DEDEKIND theorem isapplicable to £, and this assures the validity of the last statement.

Now we give a simple answer to the problem formulated above.

THEOREM 7. The congruence relations of the form e[I] form a sublattice of eeL) if

(23) ea, b n e a, e = e a, bne for all a <:: b, a:::: c.

PROOF. We must prove only

(24)

for the same statement for joins was proved in Theorem 6. Applying Lemma 2,(24) get the following form:

V ea, b n Vee, d = V e.T , y,a, bEl, c, dEI2 X, yEll nl2

thus from the infinite distributive law we conclude

(25)

e a, b nee, ,I:::: e avb, t n ecvd,t = e(avb)()(evd), t,

If ex, y occurs in the right side of (25), i. e., x, yEll n12 , then ex, y occursin the left side too, i. e.

V (ea, b n@e,d) ~ vex, y.a,bE~;c,dE~ x,YE~n~

On the other hand, we see that if t <:: a n b nend (a, bEll' C, dE 12), thenby (23)

where (a Ub) n (c Ud) Ell n12 and tEll n 12 , i. e. every member of the leftside is less than or equal to a suitable member of the right side, and sothe inequality holds in the reversed sense too, completing the proof of (25).

The condition (23) is not necessary, not even in modular lattices.

IDEALS AND CONGRUENCE RELATlONS IN LATTICES 157

As an easy consequence of Theorems 1 and 7 we get:

COROLLARY. In a distributive lattice the congruence relations of typeelI] form a sublattice of e(L).

PROOf. Let a<'b and a c, then u v (ea,b) and u v (ea,c) underthe condition u::: l' are equivalent to (Theorem 1)

(26) (auv)nu=v,

(27) (buv)nu=u,

(28) (c uv) nu = u.

From (27) and (28) by the distributive law

(29) u un u = (b uv) nun (c uv) nu = [(b nc) uv] nu.

(26) and (29) together mean by Theorem 1 that u = v (ea, bnc)' Thusea, b nO", c <. @a, bnc; the converse inequality is an immediate consequenceof e a,b, e a, c e a, bnc; the proof is completed.

We remark that this Corollary is an immediate consequence of condition (f) of Theorem 2 too.

The validity of (21) and (22) is assured under a lot of restrictions by

THEOREM 8. Let L be a dual infinite distributive lattice with zero element. Then the congruence relations elI] form a complete sublattice of e (L),that is, (21) and (22) are valid.

PROOF. It is enough to prove (22). This may be treated in a similarmanner as the Corollary of Theorem 7. It suffices to note that the zero element assures the existence of Ala' and the dual infinite distributive law isused in the proof of

which is analogous to (23). We omit here the detailed proof.

§ 4. Remarks on weak projectivity

Let four elements a, b, c, d be given in L such that a, b -+ c, d andc, d -+ a, b. Then a _ b is equivalent to c d under every congruence relation. The situation is the same if a, band c, dare projectivelO (which shallbe denoted by [a, b] lI[c, d] if a::::: band c <. d). The following problemarises: give a necessary and sufficient condition under which a, b is

10 In the literature one speaks about the projectivity of intervals. We say that a, band c, d are projective if the intervals [a n b, aU b] and [c nd, cUd] are projective in theusual sense. Our definition is more convenient in the sequel.


projective into c, d if and only if

(30) a, -+ c, d and c, a, b.

Now we consider two classes of lattices in which this condition holds.

THEOREM 9. Let L be aA) distributive, orB) discrete, modular

lattice, then aJ band c, d are projective if and only if (30) holds.

PROOF. Evidently, in both cases it is enough to verify that (30) impliesthe projectivity of a, band c, d.

A) By Theorem 1, (30) is equivalent to (we suppose that a <: b, c ::5 d)

(31) (auc)nd c;

(32) (buc)nd=d;

(33) (e ua) nb = a;

(34) (d ua) nb = b.

Let us prove the equation b U(a Uc) = d u (a Ue), i. e.

(35) b uc= d U a.From (32) b Uc~ d and from b ~ a we get b Uc :> d Ua and, on the otherhand, from (34) au d ~ b and from d ~ c we get au d ::: b Uc; these inequalities prove (35). The equations (31), (33), (35) show that the consecutivemembers of the sequence of intervals [a, b], [a Uc, b Uel, [e, dJ are transposed,that is, [a, b] nrc, d], q. e. d.

We see that we have proved more than it was required by Theorem 9.In addition we get

THEOREM 9'. @a, b @c, d in a distributive lattice L if and only ifa, band c, d are projective.

B) The proof may be decomposed into two assertions. These may beproved by trivial induction, hence the detailed proofs may be omitted.

LEMMA 9. If L is a modular, discrete lattice and a, b, eEL, a~ b, thenunder condition B we havell

I[a, b] l[a Uc, b uc] and I[a, b] l[a nc, b ncl,and if a sign of equality holds, then the corresponding intervals are transposed.

PROOF by induction on l[a, b].

11 1[a, b] denotes the length of a maximal chain from a to b.


LEMMA 10. If L is a modular, discrete lattice and a, b, c, dEL,a ::: b, c ~ d, a:ti- c, d, then

I[a, b] ~ I[c, d].

PROOF by Lemma 9 and by an induction on the number of steps inthe definition of weak projectivity (the number n in (15) and (16».

The proof of case B) may be completed as follows: if a -< band c ~ dand condition (30) holds, then I [a, b] = I[c, d] from Lemma 10, hence[a, b] n [c, d] from Lemma 9, q. e. d.

By repeated use of distributivity it is clear that if in a distributivelattice a, b-c, d (a ~ b, c >- d), then for suitable p and q(EL) the followingtwo equations hold (see Theorem 1 too): .

(36)

(37)

(a up) n q=c,

(b up) n q= d.

Now we prove that this property characterizes the distributivity of the lattice L.

THEOREM 10. The condition a, b - c, d (a 2: b, c::: d) is equivalent to(36) and (37) if and only if L is distributive.

PROOF. The sufficiency of the distributivity is obvious. Therefore we mayrestrict ourselves to the necessity.

Let us suppose that the stated condition holds. We prove that c > d ~ ais impossible. Indeed, if d ~ a, then b Up ::: a, and so a Up ~ b Up, that is,au p = b Up, consequently c == (a up) n q = (b up) n q = d, a contradiction.

It follows from Lemma 6, that c > d >- a, c= d (0a• b) is impossible,hence from condition (a) of Theorem 2 we get the distributivity of L.

III. SEPARABLE CONGRUENCE RELATIONS

§ 1. The definition of separable congruence relations; examples

In this section we introduce the notion of separable congruence relation. This notion will enable us to solve many problems.

DEFINITION 3. Let L be a lattice and 0 a congruence relation on L. 0is separable if to all a <: b in L there exists a chain a = Zo <: Z1 <: ••• ~ Zn = bsuch that for each i either Zi-1 Zi (0) or (Zi-1$Zi(0) and) X,yE[Zi-l, z;j,x-y (0) impJy x=y.

We also say that this chain {Zi} is separated modulo 0, or 0 separatesthe chain {Zi}' or a and b are separated modulo 0 by the chain {Zi}.


a = a1 < b1 < ... < ai <OJ

V0 ai • I)i is noti=1

We get immediately from the definition:

LEMMA 11. If the lattice L is semi-discrete, then all congruence relationson L are separable.

Now let us consider an exampleJ2 of a non-separable congruence relation. Let L be the chain of all positive integers, together with + 00. Wedefine x y (0) if and only if x=2i, y=2i+l for some i=I,2, ....Obviously, 0 is non-separable, e. g. no chain separates 1 and + <Xl.

From the definition it is also clear that if 0 is separable, then betweenall a, b (a 2:: b) there exists a maximal chain such that on this chain there isbut a finite number of congruence classes with more than one elementunder 0. Indeed, every maximal chain which refines a separating chain hasthe required property.

The converse statement is in general not true. It is neither true that ifo is a congruence relation such that between all a > b there is a maximalchain with the property described above, then 0 is necessarily separable.The statement: if 0 is separable, then all maximal chains between any a ::0; bhave the property described above, is also false. Counterexamples may be foundin § 4 of this Part, see examples (A) and (B).

Some typical examples on separable and non-separable congruencerelation wi11 be shown by the following lemmas.

LEMMA 12. Let L be a lattice with the greatest element 1, and 1 aneutral ideaP3 of L. 0[1] is separable if and only if 1 is a principal ideal.

PROOF. If 1 and Y(EI) are separated under 0[1] (I='FL) by the chain {Zi}(i = I, ... , n), then it may be supposed that 2 1 = y, 23 = 1 (n = 3). There isno subinterval of [22 , 1] which is congruent under 0[/], thus 2 2 is the generating element of 1. On the other hand, if 1= (a], then x::o; y may be separated under 0 [I] by the chain y:=: x n(y U a) -< x. 14

The following is a significant example of a non-separable congruencerelation of distributive lattices.

LEMMA 13. Let an infinite sequence of elements

< bi < ... < b be given in the distributive lattice L. Then

separable.

12 This example is generalized by Lemma 13.13 The ideal I of the lattice L is called neutral if for any ideals j, K of L, the sub

lattice of the lattice of all ideals of L generated by I, j, K is distributive. If I is neutral,then x -- y under 6i [I] if and only if (x ny) U i ~ xU y for some i EI. For this fact werefer to [2], pp. 28, 79, 119 and 124, or to [14], p. 167.

14 If in Lemma 14 we omit the condition that L has a unit element, then theassertion does not remain valid.


PROOF. Suppose that e = Ve ai , bi is separable, and let {Zi} be a chainwhich separates a and b (a = Zo < Zl < ... < Zn = b). If Zi-l~ Zi (e), then

n

Zi-l Zi (V ea. , b')' that is, already a finite number of the [ai, bi] generates all1=1 Jl Jl

congruences on the chain {Zi}' Let [at, btl be an interval different from theabove ones. Let e'Ot, /;t be the complement of eat, bt (see condition (d) ofTheorem 2), then a b (eat, bt u e'a.t, bt)' a:$ b (e'at, bt) (for at:$ bt (e'Ot, bt»·Hence for a suitable index j we have Zj-l:$ Zj (0'0t> bt). According to Lemma5 applied to Zj-l Zj (0'at, /;t U 0 at , bt), there is a pair of elements u, v suchthat Zj-l ~ u < v ~ Zj and u v (eat, bt). On the other hand, Zj-l Zj (0),

n

that is, U 11 (e) whence U v (V ea' b')' Comparing this with1=1 Jl' Jt

the above congruence we get that is,

n

U~ V (V (eait. bjl n 0 at , /;t»' From the conditions of the Lemma and from1=1

[ajl' bjz] =F [at, btl, we get for each I either ajz < bjl < at < bt or at < bt < ail < bi!.Thus by condition (c) of Theorem 2 we get @a .• b· n 0 at , bt = w. Hence the.n Jl

above congruence becomes U~v (w), i. e. U = v, in contradiction to u < v.The proof is completed.

Now we prove

LEMMA 14. The separable congruence relations on L form a sublattice0 s (L) of 0(L). 0 s (L) contains I and w.

PROOF. It is clear that I and ware separable, so @.(L) is not the voidset. Furthermore, let 0, rlJ E e.(L), and let a >- b (a, bEL). The chain {Z;}iseparates a and b modulo 0, and let {Uij}j be a chain which separates Zi andZi-l modulo rlJ. A rather simple computation shows that the chain {Ui,j}i,j separates a and b modulo 0 u rlJ as well as modulo 0 n rlJ, completing the proof.

e(L) is distributive, hence its center 0,(L) is the set of all congruencerelations having a complement. It is well known that e,(L) is a sublattice of 0(L). (It is trivial from the identities (0 Ucpr = @' n r/Y and(0 n r1J)' = 0' U rlJ'.)

LEMMA 15. If the congruence relation (;; has a complement, then it isseparable, that is, ez(L) C e.(L).

PROOF. By Lemma 5, to all a> b there exists a chain a = ZO?;· .. ?:;Zn = bsuch that' either Zi Zi-l (0) or Zi Zi-l (0') for every i. We assert thatthe chain {Zi} separates a and b modulo 0. Indeed, if x, y E [Zi, Zi-J] andZi :$ Zi-1 (0), furthermore x y (0), then from x:--- y (0') we getx~y (e n 0'), that is, x = y, q. e. d.

11 Acta Mathematica IXjl-2

162 G. GRATZER ,AND E.T. SCHMIDT

COROLLARY. in a distributive lattice all congruence relations of the formea,1I are separable.

This is an immediate consequence of condition (d) of Theorem 2 andof Lemma 15.

§ 2. Weakly modular lattices and O. Birkhoff's problem 72

First of all we introduce the notion of weakly modular lattices. It playsan important role in the discussion of problem 72 as well as in our researches concerning the so-called standard ideals (see [8]).

DEFINITION 4. The lattice L is weakly modular if a, -+ c,(a < b, c =1= d, a, 0, c, dEL) implies the existence of elements ai, 01 (a ~ a j < 01 ::S 0)such that c, d -+ a1 , b~.

The weakly modular lattices are a common generalization of the modu-lar and relatively complemented lattices 15 as it is assured by

LEMMA 16. If L is a

(a) modular, or(b) relatively complemented

lattice, then it is weakly modular.

PROOF. The case (a) is an immediate consequence of the isomorphismtheorem for modular lattices (we refer to [2J, p. 73). Now consider case (b).Let a > 0 and a nx > y ;s; 0 nx. Then denoting by z the relative complementof y in the interval [0 nx, a nxJ, we have a nx, y -+ lJ-;7)ITz and 0< 0 ufor {(a nx) niJ u0 = b uz and (y nz) u0 = O. In the same lines it may beproved that in case a>b,aux:::y>bux, we have uX,y-+aj,a for

15 The necessity of a common generalization of modular and relatively complemented lattices has arisen in many cases. Let us consider an illustrative example. DILWORTH

and HALL [4J proved - generalizing a theorem of O. BIRKHOl'!' - that every weakly atomic(a lattice is called weakly atomic. if any a >b implies the existence of c, d with a~c >- d'ii';, b)modular lattice is the subdirect product of projective lattices (a projective lattice is a lattice in which all prime intervals are projective). J. HASlilMOTO [13J proved a similar result forrelatively complemented lattices. Thus the necessity of a theorem arises which is a common generalization of the above mentioned ones. Let us call the lattice L weakly projective if for any pair of prime intervals p, q the relations p -+ q and q -+ p hold (the notationsare that of § 3). Obviously, any weakly projective modular or relatively complemented latticeis projective. We assert: Any weakly atomic, weakly modular lattice is a suMireet productof weakly projective lattices. The proof goes on the same lines as the proof of the assertion of DILWORTH and HALL, or, what is the same, the proof of J. HASHIMOTO. This is alsoa consequence of Lemma 18.

];)EALS ,\ND CONGRUENCE RELATiONS IN LATTiCES 163

suitable a > a j ~~ b. The proof may be completed by an easy induction on nof Definition 1.

From Lemma 16 it is also clear that the weakly modular lattices generalize the modularity in another way than the semi modularity. We remarkthat by the Corollary 2 of Lemma 6 all simple lattices are also weaklymodular.

An important property of weakly modular lattices is proved in

LEMMA 17. Let L be a weakly modular lattice and e a congruence relation on L. Define x =y (e*) if and only if in the interval [x ny, x uy] everycongruence class under e consists of a single element. Then e* is a congruence relation,furthermore e* is the pseudo-eomplement l6 of e in (5)(L).

PROOF. Owing to the definition of e*, it is reflexive and satisfies thecondition (b) of Lemma 4. Let u > v > w, u= v «(5)*) and v = W (e*) andlet us suppose that for some u 2:: x > Y .~ v we have x - y (e). Sincex y (eu,v u e,l, w), from Lemma 5 it follows the existence of Xl' Yl such thatx :::: Xl > Yl ~ Y and either u, v -.. Xl' Y1 or v, w........ x~y~. From the weak modularity it results that Xl' Y1 ........ 1'l , ~ for some 1) ::::/'1> Wl :::: w or X, y ........ ul , 1'1

for some u:::: u] > Vl 2:: v. But x y «(5) implies 1h - WI (e) or u1 1'1 «(5),in contradiction to v w (e*) or to u v «(5)*). The cases u = 1: and v = ware trivial. Finally, we prove that X::> Y and x:::::::: y (e*) imply xu t=Y u t (e*).Indeed, if x u Ut (e*) is not true, then u (e) is valid for someX Ut:::: u > 11~ Y Ut. From the weak modularity it follows that u, v ........ Xl' Yrfor some x?: Xl > Yl y, that is, x == y (e*) is false. Thus we have provedthe validity of the conditions (a)-(d) of Lemma 4, and so e* is a congruence relation. The last assertion of the lemma is clear.

COROLLARY. Any separable congruence relation of a weakly modular lat-tice has a complement, that is, es(L) (-iJz(L).

PROOF. Let {'1 be separable; we assert that the congruence relation e*of Lemma 17 is the complement of e. Indeed, if a ~ b (a, bEL), then leta = Zo ~ ZI ?: ... ?: z" b be a chain which separates a and b modulo e. IfZi$Zi-l (e), then by the definition of e* it follows Z,:-Zi+l (@*), whencea _ b «(5) u (5)*), 'completing the proof of (5) U{'1* t.

Now we proceed to problem 72 of O. BIRKHOFF (see [2], p. 153):Find necessary and sufficient conditions on a lattice L that its con

gruence relations should form a Boolean algebra.

16 Let Lbe a lattice with O. The element a* is called the pseudo-complement of aif x na = 0 is equivalent to x::o; a*.

11*

164 G. GR,\TZER AND E. T. SCHMIDT

First T. TANAKA [18] gave an answer to this question. t7 He got thefollowing interesting theorem which is a generalization of a theorem of R. P.DILWORTH [3]:

'The congruence relations of the lattice L form a Boolean algebra if andonly if L is a discrete subdirect product of simple lattices. (Discrete subdirect product is a subdirect product in which any two eh:~ments differ onlyin a finite number of components.)

The theorem of T. TANAKA may be considered as the structure theoremof lattices L for which e(L) is a Boolean algebra. However, in somerespects the following theorem is more applicable to interesting specialcases:

THEOREM 11. The congruence relations of the lattice L form a Booleanalgebra if and only if

(W) L is weakly modular

and

(5) all congruence relations on L are separable.

PROOF.

The necessity of the conditions. Let us suppose that e(L) is a Booleanalgebra for the lattice L. Then by Lemma 15 all congruence relations areseparable, hence (5) is necessary.

Let us suppose that a, b -. c, d (a> b, c =F d). ee, d has a complement, letus denote it by (jJ. Now, a_b (ee,aU (jJ), but in case a b «(jJ), it followsthat c d «(jJ) which is impossible, so a $= b «(j». Thus Lemma 5 impliesthat for some a ~ at > b1 ~ b the relation a1 - b1 (ee, a) holds. By Lemma 6this implies that for some a1~ a2 > b2~ b1 , C, d -. a2 , b2 is valid, thus in case0(L) is complemented, (W) holds.

The sufficiency of the conditions. By (W), ez(L) = es(L), as it wasproved in the Corollary of Lemma 17. Condition (5) is equivalent to e(L) =

= 0 s(L), thus @(L) = ez(L), as we wished to prove.

We get from Theorem 11 a lot of Corollaries.

COROLLARY 1. The lattice of all congruence relations of a

(a) modular, or

17 The result of T. TANAKA remains valid in abstract algebras, too, this explains thatfor lattices one can get sharper results. We note that while the result of T. TANAKA depends on the Axiom of Choice, our result does not use it.

IDEALS AND CONGRUENCE RELATIONS IN LATTICES 16l>

(b) relatively complemented lS

lattice is a Boolean algebra if and only if the condition (S) holds.It follows from Theorem 11 and from Lemma 16.In case the distributivity of L is assumed, we can get further improve

ments.

COROLLARY 2 (Theorem of J. HASHIMOTO [14]). The lattice of all congruence relations of a distributive lattice L is a Boolean algebra if and onlyif L is discrete.

PROOF. By Corollary 1 it is enough to prove that in distributive lattices(S) is equivalent to the discreteness of L. Indeed, if L is discrete, then byLemma 11 (S) holds. On the other hand, if L is not discrete, then by theusual method of bisection of intervals we get a sequence of elements required in Lemma 13, so that, by this lemma it follows the existence of anonseparable congruence relation, that is, condition (S) is false.19

In case of modular complemented lattices, SHIH-CHIANG WANG got acondition for the complementedness of 0(L).

COROLLARY 3 (Theorem of SHIH-CHIANG WANG [20]). The lattice of allcongruence relations of a complemented modular lattice is a Boolean algebraif and only if all neutral ideals are principal.

PROOF. By a theorem of O. BIRKHOFF, every congruence relation of acomplemented modular lattice is a minimal congruence relation of a neutralideaL By Lemma 12, the ,minimal congruence relation of a neutral ideal isseparable if and only if the ideal is principal, and so Corollary 3 follows.

It is surprising that Corollary 3 which seems to be true only in complemented modular lattices remains true after omitting the condition of modularity, provided that we replace neutral ideals by standard 20 ones. In [8] weproved that every congruence relation of a relatively complemented latticewith 0 and 1 is a minimal congruence relation of a standard ideal, thus theproof of Corollary 3 may be applied to establish

18 The results of this section were published in [7] in 1957. At the same time,J. HASHIMOTO published in [13J the following result: If in L the restricted chain conditionholds (that is, in every (closed) interval of L the maximum or the minimum conditionholds) and L is relatively complemented, then @(L) is a Boolean algebra. Indeed, therestricted chain condition is a special case of semi-discreteness, further, on any semidiscrete lattice all congruence relations are separable (Lemma 11), thus the assertion followsfrom the part (b) of Corollary 1.

19 For a direct proof of Corollary 2 see our paper [l2J.20 We have introduced the notion of standard ideals in [8J. Among the more than

seven equivalent definitions now we formulate only the following two: (a) the ideal I iscalled standard if for any idealsJ, K of L the relation] n(I UK) (J nI) U(J nK) holds;

166 G. GRATZE.R AND E. T. SCHMIDT

COROLLARY 4. The lattice of all congruence relations of a relativelycomplemented lattice with 0 and 1 is a Boolean algebra if and only if allstandard ideals are principal.

We get other types of Corollaries if we restrict our consideration todiscrete or semi-discrete lattices. In semi-discrete lattices (5) is valid (Lemma 11). We prove that in semi-discrete lattices (W) is equivalent to

0) weak projectivity between prime intervals is symmetric.

(The interval [a, b] is called prime if b covers a.) Indeed, if (W) holdsand b covers a, d covers c, li-:b ---+ c, d, then for some b bi > a l ~ a,c,d ---+ aI' bi is valid, but from the covering relations we infer b = bl anda = a j , hence (J) is valid. On the other hand, assume the validity of 0),and let a, b ---+ C, d, a < b. Let a = Xo< Xl < ...< Xn b be a finite maximal

n

chain between a and b. Then C=d (U 0J'i' J'i_)' so that by Lemmas 4 and 5,i=l

for some c n d <:: CI -< d l ;'? cud and for some i, Xi-I, Xi ---+ cl , dl is valid. Butthen by (J) cI , dl---+xi, Xi-l and the assertion follows. Thus we have

COROLLARY 5. The lattice of all congruence relations of a semi-discretelattice is a Boolean algebra if and only if the relation of weak projectivitybetween prime intervals is symmetric.

Corollary 5 in case of discrete lattices was firstly proved by J. JAKUBIK [15].We shall weaken the conditions of Corollary 5 in the following section.

§ 3. Special properties of 0(L)

If we can construct from the lattice L a new lattice, then it is alwaysinteresting to characterize those lattices for which the new lattice has somespecial properties. So, for instance, the characterization of those lattices forwhich 0(L) is a Boolean algebra was the content of § 2. Now we considerfurther problems of this kind.

(b) the ideal I is said to be standard if x =y under e [I] if and only if (x ny) ut = xU Yfor some t EI. From (a) it is clear that the notion of standard ideals is a generalizationof the neutral ideals; from (b) we see that the proof of Lemma 12 for standard idealsremains valid.

In [8J we have proved Corollary 4 in another way. We can sharpen Corollary 4,for in [8J we have proved that in a weakly modular lattice all standard elements are neutraland in a relatively complemented lattice all ideals which are congruence classes undersome congruence relation are standard, thus we get: the lattice of all congruence relationsof a relatively complemented lattice with 0 and I is a Boolean algebra if and only if everyideal which is a congruence class under some congruence relation is a principal ideal.

IDEALS AND CONGlH'ENCE RELATICNS IN LATTlCES 167

We know that in @(L) the infinite distributive law(ID) (-) nV@a= V(@ n 0 a)

holds, but, as it was pointed out by N. FUNAYAMA and T. NAKAYAMA [5],the dual law

(DID) @ u /\ 0 a = 1\(0 U 0 a )

does not hold in general. Let us consider the lattices in which (DID) doeshold. First we prove

LEMMA 18. Let 0 be a separable congruence relation of L. Then forany subset A of @(L)

is valid.

PROOF. Since 0u/\0a-<:/\(0u 0 a ) is true in any complete lattice, itis enough to verify that 0 U /\ 0 a?:;.I\(® u 0a). Let x~- y (1\(0 u 0 a»;since 0 is separable, there exists a chain xu y = zo:::: Zl 2.: ••• ?:;. Zn = x n yseparating x Uy and x ny modulo 0. If Zi Zi-l «(-» for some i, then fromZ;-l==Z; (/\(0 u0 a» we get Z; Zi-l (A0a ). Thus for every i either Zi-Zi-1 (0)or Zi == Z;-l (/\ 0 a), that is, x y (0 U /\ @a), which we intended to prove.

COROLLARY. If all congruence relations on L are separable, then (DID)holds unrestrictedly.

Lemma 18 or its Corollary may not be conversed, as it will be shownin § 4 by a counterexample (example (C».

Now we characterize the distributive lattices L such that in 0(L)(DID) holds.

THEOREM 12. In the lattice of all congruence relations of a distributivelattice (DID) holds unrestrictedly if and only if L is discrete.21

PROOF. If L is discrete, then all congruence relations on L are separableby Lemma 11, thus, by the Corollary of Lemma 18, (DID) holds in 0(L).

On the other hand, assume that (DID) holds in 0(L). Let @E @(L).then @ = V ('1a, b· By condition (d) of Theorem 2 any 0 a, b has a com-

a=b(e)

plement (j~, Ii. Put (fJ = /\ e~, b, then by (ID)a =iii (6)

@ n (fJ = Ve", b n(/\ 0~.1J) = V(ea, b n /\ e~. b)::S V(ea, b n @~, b) = VW = w,

hence e n (fJ w, and from (DID)

@ U (fJ yea, bU(/\ 0~, b) = 1\(0~, bUVea, b):::: 1\(0~,b U0 a,b) = I,

21 A simple proof of Theorem 12 was published in our paper [12].


(the A and V are extended to all a, b with a b (0) in all above formulae),thus 0 U (Jj = t, that is, (Jj is the complement of 0. Thus 0(L) is a Booleanalgebra, hence from Corollary 2 of Theorem 11 L is discrete and the theoremfollows.

*Now we consider a question related to problem 67 of G. BIRKHOFF [2].Let P be the set of all prime intervals of the lattice L; the elements

of P are denoted by p, q. If P = [a, b] and q = [c, d], furthermore a, b -+ c, d,then we write p -+ q. The elements of P under the relation -+ are quasiordered, thus if we identify those p, q for which p -+ q and q -+ p simultaneously, then we get a partially ordered set which will be denoted alsoby P. Now we are seeking for a condition under which (,,)(L) "'" 2P • (2 denotesthe lattice of two elements. The definition of 2P may be found in [2], p. 8.)

LEMMA 19. For any lattice L, 2P is a complete homomorphic image of@(L). .

PROOF. We say that the congruence relation @ collapses the primeinterval p, if P = [a, b] and a==::. b (("». We call a subset A of P s-ideal, ifPEA and p-+q imply qEA. We assert that every s-ideal may be regardedas the set of all prime intervals collapsed by some congruence relation.Indeed, if @ is a congruence relation, then the set of all collapsed primeintervals A form an s-ideal, for if pEA and P-+ q, then q is also collapsedby @. On the other hand, let A be an s-ideal of P, and let us define(>9 V @a,b' Under 0 the prime intervals of A are collapsed, further-

JI·~la, b]EA

more if q is collapsed by ("), then q = [a, b], a -- b (@), thus, by Lemmas 4and 5, p-+q for some pEA, hence qEA.

In a similar way we get that if under (Jj and (iJ the collapsed primeintervals are A p and Ae, respectively, then under @ U (Jj and @ n (Jj the collapsed prime intervals are A.p U Ae and At]) n Ae, where u and n denote theset theoretical meet and join. Thus the set B of all s-ideals of P, partiallyordered under set-inclusion is a homomorphic, moreover, a complete homomorphic image of @(L) (naturally the void set is also regarded as an s-ideal).It is evident that B is isomorphic to 2P , completing the proof of Lemma 19.

A trivial condition concerning the problem under discussion followsfrom Lemma 19.

THEOREM 13. The isomorphism @(L) "'"' 2P holds if and only if to anypair 0> (Jj (@, (JjE@(L» there exists some pEP collapsed bye but not by (Jj.

PROOF. Since 2P is a homomorphic image of (,,)(L), the condition ofTheorem 13 is necessary and sufficient in order that this homomorphism may bean isomorphism. Q. e. d.

IDEALS AND CONGRUENCE RELATIONS lN LATTICES 169

As a trivial consequence of Theorem 13 we get immediately a sharpenedform of a theorem of J. JAKUBIK [15] (he restricts himself to discrete lattices;in § 4 we prove by examples that the following Corollary is more sharpenthan JAKUBIK'S theorem):

COROLLARY 1. If L is a semi-discrete lattice, then @(L) '"" 2P•

Instead of proving it we shall verify a more general assertion.

COROLLARY 2. Let L be a weakly atomic lattice with separable congru-ence relations. Then@(L).......,2P •

PROOF. It is enough to prove that if @ > rJ>, then there exists a primeinterval p which is collapsed by @ but not by <P. As a matter of fact, thereexists a pair of elements a, b with a > b, a - b (@) and a b (<P), andthere exists a chain which separates aand b modulo <P; let a = Zo > ... ~ z" bbe this chain. Choose an index i for which Zi:$ Zi-1 (rJ». Then no subinterval of [Zi' Zi-J] is congruent under rJ>. By weak atomicity there is a primeinterval p in [Zi' z,:-J]; thus p is not collapsed by rJ> but is collapsed by @,

completing the proof.Theorem 13 and Corollary 2 may be regarded as a general solution of

G. BIRKHOFF'S problem 67.From Corollary 1 one can deduce Corollary 5 of Theorem II using

only the fact that 2P is a Boolean algebra if and only if P is unordered.:1'2Thus from Corollary 2 of Theorem 13 we get a generalization of Corollary5 of Theorem 11:

Let L be a weakly atomic lattice with separable congruence relations.@(L) is a Boolean algebra if and only if weak projectivity is a symmetricrelation among its prime intervals.

§ 4. Counterexamples

Now we construct some counterexamples to questions raised in Part Ill.(A) There is a lattice having a congruence relation @ and a maximal

chain C such that @ induces on C an infinity of congruence classesof more than one element.

22 Let us prove that 2P is a Boolean algebra if and only if P is unordered. Indeed, ifP is unordered and f E2P, i. e. f is an isotone function from P to 2, then define g byg(a) 0 if j(a) 1 and g(a) = 1 if f(a) O. Obviously, g is the complement of f in 2P•

On the other hand, if x, yEP and x > y, then consider the function f for which f(x) t,f(y) O. If g is the complement of f, then max (f(a), g(a» = 1 .for all a EP, that is.g(y) I, min (f(a), g(a» = 0 for all a EP, whence g(x) = 0, g is not isotone, a contradiction.

t70 G. GRATZER AND E. T. SCHMIDT

EXAMPLE. Let P be the chain of all non-positive integers together with- <Xl, with the natural ordering. In the cardinal product of P with itself letus consider the congruence relation @ = @\O,ol.(-en,O) and a maximal chain Cwhich consists of all elements of type (x, x). By the Corollary of Lemma 15(3J is separable, yet on the chain C it induces an infinity of congruenceclasses with more than one element.

(B) There exists a lattice L on which there is a non-separable congruencerelation @ with the property that any 0, b (o?': b) may be connected by amaximal chain on which there is but a finite number of congruence classesof more than one element.

EXAMPLE. Let P be the chain of all non-negative integers and let L been

the lattice p.p bounded with 1, and @ = V @(2i,O)(2i+l.O)' By Lemma 13, @i=l

is non-separable. Let 0> b. We may suppose 0 = I unless [b, oj is finite.If 0 = I, then a chain with the required properties is formed by the elements(br.x), where b=(b j ,b2) and x runs over the numbers b2 ,b2 +1,b2 +2, ....

It is of some interest that examples (A) and (B) could be constructedamong distributive lattices.

(C) There is a lattice L with the property that in @(L), although thedual infinite distributive law unrestrictedly holds, yet there are non-separablecongruence relations.

EXAMPLE. Let P be again the set of all non-negative integers and letL consist of P and of three new elements I, x, y. L will be a lattice if thepartial ordering of P remains the usual and the following relations hold:

x U i = Y u i x uY = x u1= Y u I = I,x n i = y n i =-~ I n0 = 0 for all i E P.

Let us have a look over the congruence relations of this lattice L. It is easyto verify that 1_ i and i 0 (i =f= 0) hold only under t. This implies thatwith the exception of t all congruence relations of L are those of P, in the sensethat the congruence relations of P are extended to congruence relations of Lsuch that the congruence classes outside P consist of one element only.In @(P) the law (DID) is satisfied as we proved it in Theorem 12, thus atrivial calculation shows its validity in @(L) too. Yet in eel) there are nonseparable congruence relations, for instance, let x .y (e) if and only ifx -- 2i+ 1 and y = 2i (i is arbitrary, i E P, i =1= 0), then one cannot separatee. g. 1 and I.

Naturally, all counterexamples of type (C) are non-distributive, for ifL were distributive, then by Theorem 12 it would follow that L is discrete,hence by Lemma 11 all congruence relations on L are separable.


(0) There is a semi-discrete but not discrete lattice L, with the property that 0(L) is a Boolean algebra.23

EXAMPLE. Let L be the set of all non-negative integers. We partiallyorder L by putting

2i-l<2i<O,{2i-l < 2i+ 1 \ (i= 1, 2, ...).

Then L is a lattice which is obviously non-discrete but semi-discrete, furthermore L is simple, that is, it has all the required properties.

(E) There is a weakly atomic, not semi-discrete lattice L with separablecongruence relations such that 0(L) is a Boolean algebra. 24

EXAMPLE. Let L be the lattice of all partitions of an infinite set. ThenL is a simple, weakly atomic lattice (for the proof we refer to O. ORE [171),thus it satisfies the required properties.

IV. BOOLEAN RING OPERATIONS ON DISTRIBUTIVE LATTICES

§ t. A characterization of relatively complemented distributivelattices

In this section we prove a theorem which enables us to prove themain theorem of this part without complicated computations.

Let

andfi(uu ... , Un, Xl>"" Xm) I'l/Ji(Ul1 ••• , Un, Xl"'" Xm) \

(i = 1,2, ... , k)

be lattice-polynomials with the variables X,i'

THEOREM 14. In a relatively complemented lattice L the system of equations

(38) f; = 'l/Ji (i = 1, 2, ... , k)

has a solution for any(aj EL;j = 1,2, ... , n)

if and only if (38) has a solution in 2 for any

U l bl , ... , Un bn (bj E2;j 1,2, ... , n).We remark that if m = 0 (that is the set of unknowns is void), then

(38) is a system of identities, the validity of which is in question.

23 Example (D) shows that Corollary 5 of Theorem 11 is applicable to more latticesthan the original theorem of J. JAKUBIK.

24 (E) shows that the assertion formulated at the end of § 3 is actually strongerthan Corollary 5 of Theorem It.


First we prove

LEMMA 20. Let L be a relatively complemented distributive lattice andXI> ••. , x" EL. L has a sublattice Bn which is a finite Boolean algebra containing Xl, ••• ,Xn (and 0(Bn)~4n).2['

PROOF. The assertion for n 1 is true. Now we make an inductionon n. Let us suppose that we have already constructed B"-1 which containsXl> ... ,Xu-1' Let 0,,-1, /n-1 be the least and greatest elements of Bn - 1 ,

respectively. Let us consider in the interval [0,,-1, Un U1"-1] the relative complement I~'-J of In - 1 and let Al and A2 be sublattices of L consisting of0,,-1,1;,-1 and Xu U In- 1 , Xn, respectively. Let Bn = (Bn- 1 • A,) 0 A2 where 0 denotes the cardinal product, but if Bn is regarded as a sublattice of L, then theembedding Bn in L is effected by (x, y) ---+ X ny. Obviously, Bn is a finiteBoolean algebra and Xl' ••• , X n EBn • The calculation on the number of theelements of B" is very easy by the construction.

Now we prove Theorem 14.

The necessity of the condition. Consider a finite Boolean algebra Bn+m

containing the elements XI, ... ,X"'; a], ... ,an. (38) is solvable in B,,+m, thusit is solvable in 2 too. Any choice of ai may be regarded as a homomorphicimage of a suitable chosen ai.

The sufficiency of the condition. Let us suppose that (38) may besolved for some Xi = bi in 2. Then (38) is solvable in all finite Boolean algebras, for (38) is solvable componentwise. Let B" be a finite Boolean algebracontaining a], ... , an. (38) is solvable in B,,, thus it is solvable in L too.

From Theorem 14 we get easily a theorem which characterizes therelatively complemented distributive lattices.

THEOREM 15. The solvability of (38) in L is equivalent to the solvability in 2 if and only if L is relatively complemented and distributive.

PROOF. The case "if" was proved in Theorem 14. Now prove the"only if". The identity

au (b n c) (a U b) n (a U c)

holds in 2, thus it must hold in L, that is, L is distributive. Furthermore,the equation system

(a U b) U X au b U c,

(a U b) nX = a

is solvable in 2, thus it must be solvable in L too, hence L is relativelycomplemented, q. e. d.

25 The number of the elements of the finite lattice L is indicated by O(L).


§ 2. Boolean ring operations

173

In Corollary of Theorem 3 we have shown that among distributivelattices just the relatively complemented ones have the property that everycongruence relation is determined by any congruence class of it. It is wellknown that the rings have the same property. We prove that this connectionbetween the rings and relatively complemented distributive lattices is notaccidental. We shall see that any relatively complemented distributive latticemay be regarded as a Boolean ring, hence the validity of the above statement becomes very natural.

DEFINITION 5. Let A be a set of equations on the distributive lattice L,containing a finite number of equations, parameters and the unknowns x, yand z. If A has a unique solution with respect to z for any fixed values of xand y in any homomorphic image of L, then we write z = x+y. If the operation + satisfies the group axioms, then we speak of a group operationdefined on the lattice L. If, furthermore, in a similar way (that is, with anequation system, having unique solution in any homomorphic image of L)there is defined another operation denoted by . such that + and . satisfythe ring axioms, then we speak of a ring operation defined on the lattice L.26

THEOREM 16. On the distributive lattice L one may define a Booleanring operation if and only if L is relatively complemented.27 All Boolean ringoperations may be defined in the following way:

Let a be a fixed element of L. Let x·y be equal to (x U a) n(x uy) n(a U y)and let x+ y be the relative complement of x·y in the interval [anxny, auxuy].

PROOF. First we prove that the operations defined in the Theorem arering operations. Applying Theorem 14 we get that it suffices to prove incase of the lattice 2.

In the lattice 2 the above operations may be given by the followingtables:

a=O+10 1

o I 0 11 1 0

o 10001 0 1

26 The conditions of Definition 5 are satisfied if we define + and . only withthe operations: join, meet, and taking the relative complement of an element in an interval.

27 One can easily show that if we restrict ourselves to that special case in Definition 5 in which the equations of A contain x and y only in the form xU y and x ny, thenthe existence of a ring operation characterizes not only the relative complementedness ofa distributive lattice, but even the distributivity of the lattice. It immediately follows fromthe fact that with the above definition + is ambiguous in the lattices Sand T.

174 G. GRATZER AND E. T. SCHMJDJ'

Thus it is clear that we get Boolean ring operations. The case a = 1 is thedual of the above.

Now we prove that in a relatively complemented distributive lattice onecannot define other Boolean ring operations. First we prove this for finitelattices. If the lattice considered is 2, then the assertion is trivial, there isonly two types of Boolean ring operations. Let us consider the Booleanalgebra Bn = 2". A system of equations is uniquely solvable in a latticewhich is a cardinal product of lattices if and only if the same is truein all of its cardinal-components. Thus the Boolean ring operations in Bn

are in a one-to-one correspondence with the Boolean ring operations of then components. In all of its components two operations may be defined, thusin B" the number of different operations is 2

n(these are all different from

each other, for the zero elements are unequal). On the other hand, the construction in the Theorem gives also 2n different operations, for a may bechosen in 2" different ways and these are also different from each other,for the zeros of the rings (a) are unequal. Thus the definition of Theorem 16exhausts all the Boolean ring operations in the finite case.

Now, let us turn to the general case. Let x, y and a, v be two pairs ofelements of L, and Bx, 11 ,Bu," will denote the finite Boolean algebras containingx, y resp. a, v and the parameters of the operations. BX'1I and Bu, v are finite,thus they have elements a and b which characterize the operations in BX '1I andin Bu, v, respectively. We get a contradiction from a+b, and this will complete the proof of the statement according to which all Boolean ring operations may be defined in the way described in the Theorem. Indeed,if a + b, then consider an element s common to BX '1I and to B u , v' Now,s s = a considered in Bx , 11 and s +s = b in Bu, v, thus necessarily a = b.

We shall use the following note: if L is a distributive lattice andXl' ••• , XnE L, then there exists a finite sublattice L" of L, containingXl"'" XI" Indeed, by an obvious induction (n = 1 is trivial) if Ln-l isalready constructed such that Xl"'" Xn-l E Ln-1 , then let L" consist of L"-l,from Xn and from the elements of the form Xn na and Xn Ua where u runsover the elements of Ln - 1 •

At last we prove that if on the distributive lattice L one may define aBoolean ring operation, then L is relatively complemented. Let L.r , 11 be a finitesublattice of L which contains X, y and all the parameters. Lx, y is a subdirectproduct of replicas of 2 and in 2 the operations are defined as in theTheorem, thus in Lx, 11 the operations are defined by the definition of ourTheorem too. From the fact that L.x, y is closed under the operations + and . ,it follows the relative complementedness of L X ,1I and in the same way therelative complementedness of L, q. e. d.

(Received 2 January 1958)

IDEALS AND CONGRDENCE RE:'ATIONS IN LATTiCES

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