Congruences and Ideals

CHAPTER

111

CONGRUENCES AND IDEALS

1. Weak Projectivity and Congruences Let a , b, c, and d be elenients of a lattice L ; if, for any congruence relation 0 of L, a = b (0)implies that c =d (O), then we can say that “a =b forces c = d”. It is necessary to understand “forcing” in order to study the structure of congruence relations of lattices; this will be accomplished in the present section. In Sect,ion 1.3 we proved that a = b (0)iff a n b E a v b (0) and so it is enough to deal with pairs of comparable elements. To simFlify our notation, let a/b denote an ordered pair of elements a , b of a lattice L satisfying b i a ; alb is called a quotient of L. (This notation obviously imitates quotient groups: GIH.) cld is a subquotient of a/b iff b d
<

Figure 1

Figure 2

either case we shall write alb-cld, and say that alb is perspective to cld. I f we want to show whether the perspectivity is “up” or “down”, we shall write a l b p c l d in the first case and alb \ cld in the second case. If for some natural number n there exist alb = eo/fo, el/fi, , e,/f,, =c/d such that e,lf,-ei+l/fz+,, i = O , . . . , n-1, then we shall say that alb is projective to cld, and write a l b = c / d . (Note that a/b z a l b with n = 1 and a/b-c/d implies that alb x c l d . ) Yrojectivity is the trnnsitive extension of perspectivity. Observe that alb Y c l d and a = b imply that c = d .

...

130

111. Congruelices end Idcrtls

Figure 3

Figure 4

The concept of projectivity is sufficient for the study of “forcing” in many large classes of lattices (for instance, in the class of modular lattices, see Section IV.1). I n general, however, we have to introduce somewhat more cumbersome notions : weak perspectivity and weak projectivity. Consider Figures 3 and 4 permitting the degenerate cases ai = c and b, =cl. If a = b (O), then al = b, ( 0 ) by Lemma 1.3.7; since aJbi -c/o?, a = b ( 0 )iniplies that c ~ (0). d Let us say that cld is weakly projective into alb iff we can get from c]d into a/b in finitely many steps as described in Figures 3 and 4. Figures 3 and 4 then should describe the concept cld weakly perspective into alb. It turns out however that a special case of Figures 3 and 4 suffices to describe the same. We write cld 4,alb iff b d and c = a v d ; similarly, eld p w a / b iff c 2 a and d = b Ac (see Figures 5 and 6). If c/d p alb or cld \, aJb, then c/d is weakly perspective into alb, in symbols, cld-, alb. If for some natural number n and cld =eo/fo, e i / f l . . . , en/fn=a/b we have eilflhWei+,lfi+l, i = O , . . , n - 1 , then c/d is weakly projective into alb, in notation, c l d z , alb. (Note that cld is weakly projective into cld with n = 1.) Weak projectivity is the transitive extension of weak perspectivity. Observe that neither weak perspectivity nor weak projectivity is a symmetric relation. The concept of weak perspectivity is due to R. P. Dilworth [1950 a]. The notational system we use is geometrically motivated. Many papers (starting with G . Gratzer and E. T. Schmidt [1958d]) use the notation [b, a ] -[d, c ] or alb -cld

.

cld -w alb

c/d \ alb Figure 5

.

131

1. Weak Projectivity end Congruences

for c/dZw a/b; t,hese notations appear to be more natural from a universal algebraic point of view and have the advantage of emphasizing the nonsymmetric nature of weak projectivity. The following elementary observation shows the equivalence of the various forms of the definition of weak projectivity.

LEMMA1. Let L be a lattice, a, b, c, d E L, b $ a, and d I c. Then the following conditions are equivalent: (i) cld i s weakly projective into alb. (ii) There i s nn integer m and there are elements eo, . . , ,em-l EL such that

pm(a, eo, . * * em-1) = C a d ~ m ( beo, , .. where the polynomial pm i s defined by 9

> em-1)

=d,

Pm(X, YO, * * * ~ m - 1 ) * . ( ( ( x A Y o ) v Y I ) ’ Y ~ ) ~* ** * (iii) There i s nn integer n and there are quotients cld = e& eJf& eilfl, e;/f; . ,eJf, =aJb such that e:/fl i s n subquotient of ei+llfi+i and ei/fi-ei/fi, for i = O , 1,. , . , 9

..

n-1.

REMARK.Condition (iii) is the definition of weak projectivity using Figures 3 and 4.

PROOF.Figures 5 and 6 are special cases of Figures 3 and 4 respectively; hence (i) implies (iii). Now let (iii) hold. Then for each i =0, 1, ,n - 1, either ei/fi 7 e;/fi or ei/fi \ e:/f:. In both cases,

. ..

ei=p4(ei+l, e:, f:, ei, f i ) , fi=P4(fi+1, e:, f:, ei, f i ) . Repeating these steps n times we get (ii) with m <4n. Thus (iii)implies (ii). Finally, let (ii) hold. Then aAeo/bAeo7 , alb, (aAeo)vel/(bAeo)VelL, aAeolbAeo, and so on; in m steps this yields that c/d=, alb. k

We shall write cld =, a/b iff Lemma 1 (ii) holds with rn = k. This is slightly artifioial. It corresponds to requiring that the series of k weak perspectivities end with 7 , and that 7 , and \, alternate throughout. Intuitively, “a = b forces c =d” iff cld is put together from pieces each weakly projective into alb. To state this more precisely, we describe @(a, b), the smallest congruence relation under which a = b (see Section 11.3).

THEOREM 2 (R. P. Dilworth [ 1950a]). Let L be a lattice, a,, b, c, d EL, b 5 a, d
ejlei+i=w alb, for 10 Grltzer

j=O,.

. .,m-1.

132

111. Congruences and Ideals

PROOF.Let @ denote the following relation on L: x = y (0) iff x v y =c and X A Y = d satisfy the condition of the theorem. We first prove that @ is a congruence relation by verifying the conditions of Lemma 1.3.8. 0 is reflexive since for any c L w e get c/c

p wavclbvc I,,alb. ,

It is also obvious that if al 2 a2 2 a3 and al = a2 (a),a2 = a3 (a),then al = u:)(a). Indeed, take the sequences establishing uI=a, (@) and a2= a3 (a); putting the two sequences together we get a sequence establishing ai = a 3 (@). Nowlet CEO!(@), c > d , and /EL. Let c = e o 2 e l > - - . 2 e m = d bethesequence d that is, e i / e i + l x w alb for i = O , , . . , m - - l . ThencAf=e,,Af> establishing c ~ (@), el/\/ 2 2 e,Af =dAf and eiAflei+lAf fweilei+lxw alb, hence eiAf/ei+lAfx,,, alb for i =0, . . . ,n - 1 ; this proves that cAf =dAf (@). Similarly, cvf E d v f (@). Thus by LemmaI.3.8, is a congruence relation. a EE b (.@) and so 0 is a congruence relation under which a = b. Now let 0 be any congruence relation satisfying a = b (0). It is easy to see that for x >y and u 2 w, x = y (0)and ulv w Wxly imply that u =v (0). By a trivial induction, x =y (0)and u/vxwx / y imply that u = w (0).So finally, let c = d (a), established by cvd = e0 2 el 2 * * 2 em =cAd. Since ei/ei+lxw a/b we conclude that ei=ei+l (@), for i = O , . , ,m -1. Therefore, by the transitivity of 0 , we obtain that c = d (0).This proves that (D is the smallest congruence relation under which u = b, and so @ =@(a, b). Let L be a lattice and H L? To compute @ ( H ) ,the smallest congruence relation 0 under which a =- b (0)for all ( a , b) E H , we use the formula (Lemma 11.3.2)

- -

.

-

= V ( @ ( a b) , I (a, b>EH),

and we need a formula for joins:

LEMMA 3. Let L be a lattice and let Oi, i C I , be congruence relations of L. Then a = b ( V ( O i I i E I ) ) i f f there i s a sequence zo = aAb I zl5 z, =avb such that for each j with 0
---

<

The proof of Lemma 3 is the same as that of Theorem 1.3.9, namely, a direct application of Lemma 1.3.8. By combining' Theorem 2 and Lemma 3 we get :

COROUARY4. Let L be a lattice, let H L2, and let a , b L with b< a. Then a ZE b ( 0 ( H ) )i f f for s m e integer n there exists a sequence a =co 2 ci 2 2 c, = b such that for each i, with 0 5 i < n , there exiats a (d, e) EN mtisfying

---

ci/c,i

d ve/dAe.

COROLLARY 5. Let L be a lattice, let I be a n idea,l of L, and let a , b E L , b
--

1. Wmk Projwtivit,y and

Congruences

133 0

Figure 7

Figure 8

Recall that, an ideal I is called the (ideal) kernel of a congruence relation 0 iff I is a congruence class modulo 0 (Section 1.3).

COROLLARY6. Let L be a lattice and let I be u n idenl of L. Then I i s a kernel of u congruence relation iff u/b=, c / d , u L, and

b, c, d I imply that a E I .

PROOF.Combine Corollary 5 with the observation that I is a congruence kernel of sotiie congruence relation iff I is the kernel of @ [ I ] . In distributive lattices every ideal is the kernel of some congruence relation, in fact this property characterizes distributivity. I n general lattices we shall introduce various classes of ideals that are congruence kernels for which @ [ I ] can be nicely described. This will be done and applied in Sections 2-4. In a sense, weak projectivity describes the structure of congruence relations of a lattice. It is not surprising, therefore, that many important classes of lattices can be described by weak projectivities. We give two examples. To introduce the first class, the class of weakly modular lattices, we need a lemma, for niotivation. LEMMA7. Let L be a lattice, ( I , b, c, d E L , b < a , d
--

PROOF.Let alb = eolfo-," ell/, -w * * en/fn = c / d ; we prove the statement by induction on n. By duality, we can assume that en-l/fa-l p w c / d and by theinduction hypothesis there exist eL-i >fk-i, such that e ~ - i l f ~ - i = walb and f,-tSfi-i

<

e:&-, i ert-110.

134

111. Congruences and Ideals

Let L be modular. As in Figure 7, define d'=dvf;-, and c' =dveL-,. Then d'veL-, = (dvfk-,)vei-, = d v ( f ~ - , v e ~ - ,=rive;-, ) =c'. By modularity, d'Aek-., = (dvfk-,)Ae;-, = (since &-,
,

,

DEFINITION 8 (G. Griitzer end E. T. Schmidt [1958d]). A lattice L i s called weakly modular i f f a, b, c, d L, b
<

COROLLBY 9. Every modular and every relutively complemented lattice i s weakly modular.

The import'ance of the class of weakly modular lattices will be illustrated in Sections 2 and 4. To introduce the second class of lattices we again need a lemma.

LEMMA10. Let L be a lattice

and let

ai, bi E L, bi


A(@(a. bi) I i= 1 , 2 , . . . ,n)+ w

i f f there exists a proper quotient alb of L such that

alb=:, ailbi, for

i = 1, 2, . . . , n.

PROOF.I f such a, b exist, then a = b ( @ ( a b, J ) , for i=1 , 2 , . . . , n. Hence u = b (A(@(ai, b,J I i = 1,2, . . . , n ) ) and so A(@(ai, bi) I i = 1, 2, . . . , n) w . We prove

+

<

the converse by induction on n in a somewhat stronger form: if v 2~ and u = v (A(O(ai,bi) I i = l , 2, , n)), then there exists e proper subquotient alb of u/v such that a l b z , a J b , for i = 1,2, . , n. For n = 1, apply Theorem 2. Assuming the statement proved for n 1 we get a'/b' a proper subquotient of ulv satisfying a'lb'z, ,n 1. Since u =w (@(a,, 6,)) we get a' = b' (@(a,, bJ). Hence ai/bi, for i = 1, by Theorem 2 we obtain a proper subquotient alb of a'lb' with a l b x , anlb,. Since a l b x , a'lb' we also have a/b=sw ai/bi, for i = 1 , 2 , , n - 1. Since in a subdirectly irreducible lattice w is meet-irreducible we conclude :

.. . .. . -

..

...

COROLLARY11. Let L be a subdirectly irreducible lattice and let be proper quotients of L, for i = 1, . . , n. Then there exists a proper quotient alb of L satisfying albw, ai/bi,i = 1, . . . , n.

.

I n fact, Corollary 11 holds for any lattice in which w is meet-irreducible.

1.

Weak Projrct,ivity and Congruelices

136

DEFINITION 12 (R. Wille [1972]). Let L be a lattice and let P be a finite nonvdd subset of L . Then P is called primitive iff A ( @ ( z , x v y I) G Y E P , x * . V Y ) * o . Combining Lemma 10 and Definition 12 we obtain COROLLARY13. P is priniitice iff there exists a proper quotient ap/bp of thut a.plbp=:tucvdlc for all c , d C P , c =i= cvd.

L, such

Observe also that by Corollary 11 every finite nonvoid subset of a subdirectly irreducible lattice is primitive. It will be pointed out in Chapter V that important classes of lattices can be defined by the property of not containing a primitive subset (poset) isomorphic t o a member of a given collection of posets. We study weak projectivities to enable us to obtain descriptions of quotient lattices such as L / O ( a , b). Sonietimes however L/0 can be identified very simply within L : if there is a sublattice Li of L having one and only one element in every congruence class, then LIO s L i . Such congruence relations are called representable. It happens much more often that we can get a meet-subsemilattice or a join-subsemilattice L, of L having one and only one element in every congruence class; in such cases we call 0 meet-representable or join-representable, respectively.

LEMMA14. Let L be a lattice and let 0 be a congruence relation o/ L. If every congruence class of @ has a mininial element, then 0 is join-representable. PROOF.Let Li be the set of minimal elements of congruence classes of 0 ;then Ll contains exactly one element of each congruence, class. Now let a, b c Ll. We show that avb is the snisllest element of [ a v b ] @ = [ a ] @ v [ b ] OIndeed, . if c < a v b and c E a v b (O), then aAC = a (0)and bAc = b (0).Since c < a v b we get that UAC < a or bAc b, say QAC < a . Then a ~
<

Exercises For all pairs of quotients xly, u1.u of investigate when “xzy forces u ~ v ” . Repeat the investigation for XU3. 2. Show that if L is sectionally complemented, that is, L has a zero and for every a E L [0, a] is complemented, then in order to learn the congruence structure of L it is sufficient t o consider ‘‘aE b forces c Ed’’ in the special case b =d =O. 3. Prove that, if a lattice is modular or relatively complemented, then alb x w c/d iff alb xc’ld’ for some subquotient c’ld’ of cld. 1.

136

111. Congruences and Ideals

Show, by an example, that the conclusion of Exercise 3 is false for lattices in gencAral. c / d . Prove that there exists a quotient 5. Let L be a distributive lattice and let a/b k W elf and a subquotient c'ld' of c/d such that

4.

a / b pelf \c'/d'.

.

.

6. Let p =p(z,, . . ,xn-l) be a lattice polynomial, and let L be a lattice and a l , . . , an-lEL. Consider the algebraic function q(z)= p ( z ,a l , . , a n - l ) . Under what conditions on p is it true that for all quotients a/b of L,

..

(This is true for p =. *

-

q(a)/n(b)=w a/b. ( ( ~ , A Z ~ ) V Z ~ ) A **

by Lemma 1.)

7. Referring to the proof of Lemma 1, axiomatize those properties ofp, which make

it possible to define pm from pz to obtain generalizations of Lemma 1 to some equational classes of algebras. 8. Show that pm(x,yo, ,yi, yi, yif2,. , ym-l) does not depend on z, yo, , yi-1 (that is, the value of pm(a,e,, ,ei, e i, e i + 2 , . . , e m - l ) is independent of a, eo, ,e i - 1 ) . .) =z for any d z 5 c. 9. Prove that p,(z, c, d , c , d , tn i 0. Rephrase and prove Lemma 1 using q m =. ((zv Y,)A yl)v - * *. Show that -",,

.. ...

.. .

...

..

m

11. 12.

13. 14. 16. 16.

17. 18. 19.

20. 21. 22. 23.

.

. ..

-

defined in terms of qrn, may differ from - w defined by pm. Show that in any sequence of weak perspectivities, any number of steps f w and h, can be put in. Observe that in any nonredundant sequence of weak perspectivities, f w and h uI have to alternate. Find examples to show that m cannot be bounded in Theorem 2. (Make all your examples planar modular lattices.) Find examples to show that in Theorem 2, a wcannot be replaced by E w for any . all your examples planar modular lattices.) natural number 7 ~ (Make Prove that a n ideal I is a kernel iff I is the kernel of @[I]. Let L be a lattice, and let I and J be ideals of L with J E I. Assume that J is a kernel in the lattice 1 and that I is a kernel in the lattice L. Is J a kernel in L ? Show that the ideal kernels of the lattice L form a sublattice I c ( L ) of I ( L ) , the lattice of all ideals of L. In fact, I c ( L ) is closed under arbitrary joins and under all meets existing in I ( L ) . Let L be a distributive lattice, let I be an ideal of L, and let a/b be a quotient of L. Then a = b (@[I]) iff a/b -a'/b' for some quotient a'lb' of I. Show that Exercise 6 characterizes distributivity. Show that Exercise 17 characterizes distributivity. In Exercises 20-23 (which are due to R. Wille [1972]) let L and L' be lattices and let cp be a homomorphism of L onto L'. Let a'lb' be a quotient of L' and let c/d be a quotient of L with a'/b' z Wccp/dcp. Prove that there is a quotient a/b of L with a/b =w c / d , acp =a', and b p =b'. Let ci/cli, . . . ,cn/dn be quotients of L such that cia, = C ~ Q = - * * =cnp and dlpl =dzcp = * * = d , p Show that there is a quotient a/b of L satisfying a/b =w cJdi and aglbcp = czp/dicp, for i = 1, 2, , ,n. Let P' be a primitive subset of L', and let P be a subset of L such that cp restricted to P is a (poset) isomorphism of P with P'. Show that P is primitive in L. Let P be a primitive subset of L, let ap, b p be given as in Corollary 13, and let app =+ b p p . Show that Pcp is primitive in L' and that cp restricted to P is a poset isomorphism of P with Pq.

..

1. Weak Projectivity and Congruences

24.

137

Let L be a lattice and let Y be a congruence relation of L. Verify the formula : @ ( [ a ] Y[,b ] Y )= ( @ ( ab, ) v Y ) / Y .

. . . ,PO. Show that

25.

Let ai, bicL, for i = 1,

26. 27.

Use Exercises 24 and 25 to verify Exercises 22 and 23. Show that for ideals I and J of a lattice L @ [ I v] @ [ J ]= @ [ I v J ] .

28.

Let L be a lattice and let J i , i€I,be ideals of L. Prove that V(@ [ J iI]i €1)= @ [ v ( J Ii i € I ) ] .

29. 30.

Show that @ [ I ] A @ [=J@ ] [ I h J ]does not hold in general. Verify @ [ I] A@ [= J ]@ [ I d ]

31. 32. 33.

34. 36. 36.

for ideals I and J of a distributive lattice. Show that the formula of Exercise 30 holds also under the condition that every ideal is the kernel of at most one congruence relation. Show that Corollary 11 does not hold for infinitely many ai/bi. Show that in a finite lattice (or in a lattice in which all chains are finite) every congruence relation is meet- and join-representable, Find a congruence relation 0 of the lattice of Figure 9 which is not representable. (Observe that 0 is both meet- and join-representable.) Let L be a finite lattice and let L / O ~'9.2~.Show that 0 is representable. Give a formal proof of the statement that if L,represents the congruence relat,ion 0 of L, then L/0 s Li.

Figure 9

111. Congruences and Ideals

138

37. A congruence relation 0 of a lattice L is order-repreeentableiff there exists a subset H L mtisfying [HI@=L and a 5 b in L iff [a105[b]O in L/0, for a, b CH.

Show that H as a subposet of L is a lattice and H EL/@. Prove that every meet-representable congruence relation is order-representable. 39. Let L be a lattice and let 0 be a congruence relation of L. Verify that if L / 0 is finite or countable, then 0 is order-representable. 40. Find a chain C, a lattice L,and a congruence relation 0 of L,such that L/O S C but 0 is not order-representable.

38.

2. Distributive, Standard, and Neutral Elements The three types of elements of a lattice mentioned in the title of this section were discovered by 0. Ore [1935], G. Griitzer [1969], and G. Birkhoff [1940a], respectively. It turned out later that all three can be defined using distributive equations only. This may be a coincidence, but it could be considered as a confirmation of the principle that distributive lattice theory provides the foundation of general lattice theory.

DEFINITION 1. Let L be a lattice and let a be an element of L. (i) The element a is called distributive iff

a v ( x A y ) =( a v z ) A ( a v y ) ,

for all x, y 6 L. (ii) The element a is called standard iff x A ( a v y ) =( X A U ) V ( X A Y ) ,

for all x , y C L . (iii) The element a is called neutral iff ( a A X ) V ( X A y ) V ( y A a )= ( a V X ) A ( z V y ) A ( y V a ) ,

lor all x , y EL. For instance, in 0,i, a, and c are distributive; 0, i, and a are standard but c is not standard; only o and i are neutral. I n %R3 only o and i are distributive; they are also standard and neutral. We can also dualize these definitions and define dually distributive elements and dually standard elements. Observe that the concept of neutrality is selfdual. Of course, every element of a distributive lattice is distributive, standard, and neutral. Various useful equivalent forms of these definitions are given in the three theorems that follow.

2. Distributive, Standard, and Neutral Elements

139

THEOREM 2 (0. Ore [1936]). Let L be a lattice and let a be a n element of L. The following conditions on a are equivalent: (i) a i s distributive. (ii) The map (xEL) y:x-.avx is a homomorphism of L onto [a). (iii) The binary relation 0, on L defined by

x i.s

(L

~ (0,) y iff

avx=avy

congruence relation.

REMARK.(i) and (ii) are equivalent since y is a homomorphism iff it preserves meets, while (ii) and (iii) are equivalent because 0,=Ker(y). A more formal proof follows.

PROOF. (i) implies (ii). y maps L into [ a ) for any element a of L ; in fact y is always onto since by = b for all b 2 a. The map y is always a join-honiomorphism since q v y y = ( a v x ) v ( a v y )= a v ( x v y ) = ( x v y ) y . In view of Definition l(i), if a is distributive, we also have

x y ~ y= y (avx)A(avy)= a v ( x ~ y= ) (xAY)~, and so y is a homomorphism. (ii) implies (iii). 0, is the kernel of the homomorphism 9 and therefore 0,isacongruence relation. @ (iii) implies (i). Since x v a = ( a v x ) v a , therefore x E a v x (0,). Similarly, y r a v y (0,). Therefore, X A Y E (avx)A(avy)(0,).

By the definition of 0, we get aV(xAy)=aV((aVx)A(avy))= (aVX)A(aVy), proving (i). THEOREM 3 (G. Griitzer and E. T. Schmidt [1961]). Let L be a lattice and let a be crn element of L. The following conditions are equivalent: (i) a i s standard. (ii) The binary relation 0, on L defined by

x EE y (0,) i f f

(XAy)Vai= x v y

for some ai

a

is u congruence relation. (iii) a i s distributive and, for x,y EL,

aAX = a ~ yand

a v x = a v y imply that x =y .

111. Congruences and Ideals

140

PROOF. (i) implies (ii). Let a be standard and let 0, be definedas in (ii). WeuseLemma 1.3.8 t o verify that 0, is a congruence relation. By definition, 0, is reflexive and x ~ (0,) y iff X A Y ~ X V Y(0,).If x l y l z x = y (@,), and y z z (0J, then xval = y and yvaz = z for some al, az
y At

y =xva, 5 x v a ,

and so, using the fact that a is standard we get

YAt = (YAt)A(XVU)= ((yAt)AX)V((yAt)Aa) = (XAt)VU*, where u2 = YAtAa
I aAc < (using c I b )
hence iiAc=uAb. To prove the second, we compute using the fact that a is distributive:

avb =aV((xA(avy)))= (UVX)A(aVy)=aV(XAy) =aV(xAa)V(xAy)=avc. THEOREM 4. Let L be a lattice and let a be a n element of L. The following conditions

are equivalent:

(i) a i s neutral. (ii) a i s distributive, a is dually distributive, and aAx=aAy and a v x = a v y imply x = y for any x,y C L . (iii) There i s a n embedding y of L into a direct p r d u c t A X B where A has a 1 und B has a 0 and a y = ( l , 0). (iv) For any x , y L, the sublattice generated by a, x , and y is distributive.

REMARK.The equivalence of (ii)-(iv) is due to G. Birkhoff [1940a]; in fact, G. Birkhoff used (iv) as a definition of neutrality. It was conjectured in G. Griitzer and E. T. Schmidt [1961] that (i) is equivalent to (ii)-(iv). This was proved in G. Grlitzer [1962], J. Hashimoto and S. Kinugawa [1963], and Iqbalunissa [1964].

2. Distribiit ive, Standard, and Neutral Elements

141

PROOF.

(i) implies (ii). Let a be neutral. Then (*)

UV(ZA~ =XA(UVY) ) for

x 2 u.

Indeed, c i v ( x ~ y= ) ( u A ~ ) v ( x A ~ ) v ( ~=A(by u ) Definition l(iii)) = ( U V X ) A ( X V ~ ) A (=xr\(avy). YV~)

To show that

(L

is distributive, compute:

UV(ZAY)

= w v ( a ~ x ) v ( x ~ y l ) v ( y=~(ab)y Definition l(iii)) =U V ( ( U V Z ) A ( W ~ ) A ( ~ V U ) )

(apply (*) to u,uvx, and ( x v ~ ) A ( ~ v ~ ) ) = (av 4 A(av((xvY 1A ( ? P a )1)

(apply (*) t o a, yva, and xvy) = ( a m )A (y va) A (UVZVY ) = (avx)A (avy ),

as claimed. By duality we get that a is dually distributive. Finally, let U A X = U A Y and uvx =avy. Then 2 = X A (a V X ) A ( U v y ) ~ ( x v y= ) X A (( u A ~ ) v ( ~ A ~ ) v ( u A ~ ) )

=Z A ( ( aAZ)V(X/\Y)) = ( ~ A Z ) V ( X A= Y() ~ A ~ ) v ( u A ~ ) v ( ~ A ~ ) .

Since the right-hand side is symmetric in x and y we conclude that x =y. (ii) implies (iii). Let (ii) hold for a and define A = (a]and B = [a). Let

0

g, : x +(.xAu, m a ) . Since (I is distributive and dually distributive, 9 is a homomorphism of L into A x B (by Theorem l(ii) and its dual). The map 91 is one-to-one, since if xg,=yrp for x, y E L, then ( x A U , xVa)

=(yAU, yVa),

that is, X A U =YAU and xva =yvu; thus x = y by (ii). So g, is a n embedding, ng, =(a, u ) , and a is the unit element of A and the zero of B. (iii) implies (iv). The following three statements are obvious: (iv) holds for the zero and the unit in any lattice; (iv) holds for (ao, at) in A o X A i iff it holds for ui in A , for i = 0, 1; if a is an element of Lo, Lo is a sublattice of L,, and (iv) holds for u in L , , then (iv) holds for a in Lo. (iii) along with these three statements proves (iv). Q (iv) implies (i). Obvious by Exercise 1.4.7. The results stated in Theorems 2-4 make it possible to verify the most important properties of distributive, standard, and neutral elements.

111. Congruences and Ideals

142

THEOREM 5. (i) Every neutral element i s standard. (ii) Every standard element i s distributive. (iii) Every standard bnd duully standard (dually distributive) element i s neutral. (iv) If a i s distributive or standard, then the 0, in Theorem 2(iii) and Theorem 3(ii), respectively, agrees with @[(a]].

PROOF. (i) and (iii). By Theorem 4(ii) and Theorem 3(iii). @ (ii) By Theorem 3(iii). (iv) If u = a (0)for any u 5 a and a congruence relation 0, then a v x =a v y implies that x = x v ( a n x )=xva = p a = y ~ ( a ~=yy) ( O ) , hence for the relation 0, given by Theorem 2(iii) we have a(, 0. Similarly, if (zr\y)vni = x v y for some ai a, then XAY = ( X A Y ) V ( U A X A Y )= (xr\y)va, = x v y (O), and so x = y ( 0 ) ;hence for the relation 0, of Theorem 3(ii) we obtain 0,s0. For a principal ideal (a]= I let @ [ I ]be denoted by 0,.Then by Theorem 5(iv), the 0,of Theorem 2(iii) and the 0, of Theorem 3(ii) are indeed the 0, just defined in the case a. is distributive or standard. Hence 2(iii) and 3(ii) are definitions of distributive and standard elements, respectively, in terms of the properties of 0,. As we have already seen, the converse of Theorem 5(i), as well as that of Theorem 5(ii), is false. Two wide classes of lattices in which the converse holds are the class of modular lattices and the class of relatively complemented lattices. THEOREM 6 (G. Griitzer and E. T. Schmidt [1961]). Let L be a weakly modular lattice. Then an element of L i s distributive i f f it is neutral. Before proving the theorem we verify a lemma connecting the distributivity of an element with weak projectivity. LEMMA^. Let L be a lattice and let a be a n element of L. The elementaisdistributive iffu
PROOF.Let a be a distributive element, u< z< a < y s x , and x/y=., zlu. Since u 2 z 5 a, we get u =z (0,), hence x E y By Theorem 2(iii), x =x v a = y v a = y. Let us now assume that a is not distributive. Then there exist b, c E L such that av(bAc)$ 5 ( a v b ) ~ ( a v c )Since . a = aAbAc (0,) we obtain

(en).

(UVb)A(aVC)= av((avb)A(avc))= U V ( ( ( U A ~ A C ) V ~ )AbAc)vc)) A((~ =UV(bAC) (0,) so by Corollary 1.5 there exist x, y , u EL satisfying, with z = a , u
(avb)A(avc)
2. Distributive, Standard, atid Neutral Elements

143

PROOF OF THEOREM 6. Let L be a weakly modular lattice, and let a be a distributive element. If a is not dually distributive, then by Lemma 7 there exist u z
<

and dually

x =y (@(a,aAx)) x = y (@(a,a v z ) ) .

Applying Theorem 1.2 to x = y (@(a,U A X ) ) we get a proper subquotient xi/yl of x v y / x ~ ysatisfying xi/yiXw n / U A X . Since x = y (@(a,a v x ) ) we also have xi=yi (@(a,a v x ) ) , and again by Theorem 1.2 weget aproper subquotient x2/y2of xJyi such that x2Jy2=:w.avxJa. Using the weak modularity of L, we obtain a proper subquotient x3/y3 of avxla satisfying x3/y3xWx2Jy2.Hence

xs/y3zw X2/y2~,uX ~

/ Y I X fl/aAX, , ~

contradicting Lemmrt 7. By Theorem 4(ii), a is neutral. COROLLARY8. I n a weakly modular lattice every standard element i s neutral.

THEOREM 9. Let L be a lattice. (i) Let D denote the set of all distributive elements of L. T h e n a and b c D imply that avb E D . (ii) Let S denote the set of all standard elements of L. Then a and b ES imply that aAb and avb €8. (iii) Let N denote the set of all neutral elements of L. Then a and b E N imply that aAb and a,vb E N . PROOF. (i) Let a , b E D and conipute: (avb)v(xAy) = LC v b v (a~y ) = rt v ( (bvx)A (bv y )) = (aVbvz)A (aVbV y) , soavbcD. 0 (ii) h t a, b € 8 .First we do the join: x / \ ( ~ V b V y= ) (xAa)V(xr\(bvy))= (X/\a)V (x/\b)V (XAy)= (ZA (av b ) )V ( xAy), proving that nvb E 8.Now we verify the formula @,A@,

= @,

,,,

where 8,,a,, and GIOhbare the relations described by Theorem 3(ii). Since @,A@* 2 a,,,, is trivial, let x G y (@,A@,). Then x G y (a,),and so (xAy)val = x v y for some a l g a . We also have X G (0,) ~ and therefore a l = a l ~ ( x v y = ) a,~x~y (ab).Thus for some bi
111. Congruences and Ideals

144

Figure 1

This formula shows that if a, b E S , then the relation OahBof Theorem 3(ii) is a congruence relation, hence aAb ES by Theorem 3. 0 (iii) Let a , b EN. By Theorem 5, a, b E S . Hence by (ii), aAb E S. By Theorems 3 and 4, t o show that aAb E N we have to prove only that aAb is dually distributive. Since a and b are dually distributive, they are distributive e1enient.s of the dual of L,hence by (i), a v b in the dual of L is dist.ributive and so aAb in L is dually distributive. Pigure.4 shows that a, b E D does not imply that arAb € D.

Exercises 1. Let L be 5 bounded lattice. Show that 0 and 1are distributive, standard, and neutral. 2. Let L be the lattice of Figure 2. Then (a]is the kernel of some congruence relation

but a is not distributive. 3. If a is a distributive element of L, then LIO, s [a).To what extent does L/O, z [a) characterize the distributivity of a? 4. Find distributive elements which are not dually distributive.

Figure 2

2. Distributive, Standard, and N~iit>ral Eltmmts 5.

6.

7. 8. 9. 10. 11.

12.

13. 14. 15.

145

Investigate the relation 0, of Theorem 2(iii) as a congruence relation of the joinsemilattice. Let L be a lattice, let p be a homomorphism of L onto L’, and let L, be a sublattice of L’. Let a be an element of L satisfying apEL,. Show that if a is distributive in L, then ap is distributive in L,. Prove the analogue of Exercise G for standard and neutral elements. Let a be a distribut’ive element of L. Show that ( a ]is a distributive element ofl(L). Prove the analogue of Exercise 8 for standard and neutral elements. Show bhat in Theorem 3(iii) (and in Theorem 4(ii)) the condit,ion can be weakened by assuming that IC 2 y. Verify that tho element a of a lattice L is standard iff x 2 a v y implies that x = ( z A ~ ) v ( x (G. A ~ Griitzer ) and E. T. Schmidt [196l]). Prove that t,he join of two distributive elements is again distributive by verifying the formula @ a v @ b = @ a v b . Find classes of lattices in which the meet of two distributive elements is distributive. Show that the map a - 4 3 , for standard elements is an embedding of the sublattice of standard elements into the congruence lattice. Let L be a lattice and let a, b, c L. Show that a, b, and c generate a distributive siiblattice iff for any permutation IC, y, z of a, b, c we have Z A ( ~ V Z )= ( Z A ~ ) V ( Z A Z ) .

SV(yA.2) =(ICVy)A(ZVZ), ( Z A ~ ) V ( ~ A Z ) V ( Z A Z= ) (

16.

17. 18. 19. 20.

21.

22.

z v y ) (~y v z ) (ZVZ) ~

(0.Ore [ 19401). Let L be a lattice and let a, b be standard elements of L. Show that foranycEL, a, b, and c generate a distributive sublattice. Do three distributive elements generate a distributive sublattice ? Verify directly that in a modular lattice distributive element =standard element = neut,ral element. Verify the conclusion of Exercise 18 in a relatively complemented lattice. An element of a modular lattice is neutral iff it has a t most one relative complement i n any interval containing it (G. Griitzer and E. T. Schmidt 119611). Let L be a modular lattice and let a E L. Then a is neutral iff a is neutral in every interval of the form [UAIC, a v z ] , for zEL. Show that the conclusion of Exercise 21 fails in weakly modular lattices. (See Figuru 3.)

Figure 3

111. Congruences atid Ideals

146 h

Figure 4

23. 24. 26. 26.

Show that Exercise 21 also fails for weakly modular lattices and standard elements. (See Figure 4.) Let L be a bounded relatively complemented lattice. Then a n element a of' L is neutral iff it has a unique complement (J.Hashimoto and S. Kinugawa [l963]). Show that the N of Theorem 9 is the intersection of the maximal distributive sublattices of L (G. Birkhoff [1940a]). Prove Theorem 9(iii) using Theorem 4(iii).

3. Distributive, Standard, and Neutral Ideals The t,hree types of ideals mentioned in the title of this section derive naturally from the concepts introduced in Section 2. of a lattice L is called distributive, standard, or neutral, respectively, iff I is distributive, standard, or neutral, respectively, as an element of I ( L ) , the lattice of all ideals of L.

DEFINITION 1. An ideal I

TO establish a connection between the type of elements and the type of ideals they generate we need a lemma: LEMMA 2. Let L be a lattice, let I be an ideal of L , and let p and q be wary polynomials.Let us assume that for all a E I there i s a b E I such that a 5 b and

PROOF.I n Section 1.4 (in the proof of Lemma 1.4.8) we proved the formula: p(Io, . . . ,In-l)={z I 2
3. Distributive, Standard, and Neutral Ideals

147

COROLLARY3. Let L be a lattice and let a E L . Then a is distributive, standard, or neutral, respectively, i f f (a] as a n ideal is distributive, standurd, or neutral, respectively. The main characterization theorems can be proved similarly to the proofs of Theorems 2.2-2.4. THEOREM 4. Let L be a lattice and let I be an ideal of L. T h e following conditions on I are equivalent: (i) I is distributive. (ii) @ [ I ]can be described as follows: s=y (@[I]) i f f

zvi=yvi

for some iCI.

PROOF.If in (ii) we put ( x ] v I= ( y l v l in place of zvi =pi,then the equivalence can be proved as in Section 2 since x =y ( @ [ I ]in ) L iff ( X I = ( y ] (0,)in I (L ).Now,if z v i = y v i for some i E I , then ( s ] v I = ( y ] v I is obvious. Conversely, if ( x ] v l = ( y l v l , then x( yvi0 and y < x v i , for some io,ilE I. Therefore, xvi= y v i with iovil=iEI. THEOREM 5.1 Let L be a luttice and let I be a n ideal of L. The followingconditions on I are equivalent: (i) I i s standard. (ii) The equality

( a ] A ( I v(b])= ( ( a ] ~ l ) V ( a A b ] holds for all a, b EL. (iii) For any ideal J of L, IvJ={ivjI icIandjEJ}. (iv) @ [ I can ] be described by z =y ( @ [ I ] ) i f f

( x n y ) v i=xvy

for some

i cI.

1 This result and all tho other unreferenced results in this section are based on G. [1959] and G. Grtitzer and E. T. Schmidt [l961].

11 Gratzer

Gratzer

111. Congruences and Ideals

148

(v) I is u distributive ideal and, for all J, K I(L),

IAJ= I A K and I v J = I v K imply that J = K .

PROOF.The equivalence of the five conditions can be verified as in Section 2. Only (iii) is new. But (iii) follows from (ii) : if I satisfies (ii) and a E I v J , then a i v j for

some iE I and j € J ; hence (a]= ( a ] ~ ( I v ( j ]=) (by (ii))= ((a]AI)V(aAj]. Therefore, a< ilvjl, where ii E ( a ] d and jl I at$ Consequently, a = ilvji. Finally, observe that when proving the analogue of “(i)implies (iv)” it is sufficient to use (iii).

TIIEOREM 6. Let L be a lattice and let I be an ideal of L . The following conditions on I are equiualent : (i) I is neutral. (ii) For all i, k E L,

~ 1v ((41\11= ( I v(il) I\((ilv (kl )A ((kIv4( I A(illv ( ( i l(kl (iii) For all J, K E I(L),I , J , and K generate a distributive sublattice of I(L). (iv) I is distributive and dually distributive, and, for all J , K E I ( L ) , IAJ=I A K and I v J =I v K

imply that J =K .

PROOF.We can verify that (i)is equivalent to (ii)by u&ng the argument of Lemma 2.

The rest of the proof is the same as in Section 2. Observe that every distributive ideal I of a lattice L is the kernel of @ [ I ] Indeed, . if i E I , a E L , and i = a ( @ [ I ] ) , then i v j = a v j for some $ € I , thus a S i v j E I , and so a E I. This explains why L / @ [ I ]can always be described. This description isespecially simple if I is principal, I = (a]. Then @[(a]]= 0, is a representable congruencerelation . and [ a ) represents a,, hence

L/O, s [ a ) . If I is not principal, then we describe I ( L / O [ I ] as ) follows:

THEOREM 7. Let I be a distributive ideal of a Za,ttice L . Then I ( L / @ [ I ]is) isomorphic with the lattice of all ideals of L containing I , that is, with [ I , L ] i n I ( L ) .

PROOF.Let

Q] be the homomorphism x + [ x ] 0 of L onto LIO. Then the map y: K -+Kq-l maps I ( L / @ )into [ I , L ] . To show that this map is onto it is sufficient to seethat [ J ] O = J forall J 2 I . Indeed,if j E J , a € L a n d j = a ( @ [ I ] )t,h e n j v i = a v i for some i E I and so a
CO~OLIARY8. LIO[I] is determined by the interval [I, L ] I is a distributive idenl of L.

of

I ( L ) , provided that

3. Diatrihutirc., St,andard, and Neutral Ideals

140

PROOF.We know (Section 11.3) that I ( L / O [ I ] determines ) L/O[I] up to isomorphism. Call a congruence relation 0 distributive, standard, or neutral, respectively, iff 0 = @ [ I ]where I is an ideal which is distributive, standard, or neutral, respectively. The following result was first established by J. Hashimoto [1952] for neutral congruences and by G. Gratzer and E. T. Schmidt [1961] in its present form. The congruences @ and Y of a lattice L are said to permute (or are permutable) iff @vY =@ * Y where @ * Yis the binary relation defined by and c =b (Y). a 3 b (0* Y) iff there exists a c E L with a = c (0)

-

-

An equivalent definition is that 0 a n d Y permute iff @ Y =Y 0,which is, in turn, equivalent t o @ 'V being a congruence relation.

-

THEOREM 9. A n y two standard congruences of a lattice permute. PROOF.Let @ and Y be standard congruences of the lattice L , that is, @ = @ [ I ] and Y = 0[J], where I and J are standard ideals of L. Let a G b (@) and b E C (Y), a, b, c EL. We want to show that there exists a d EL satisfying a =d (Y)and d =c (0). If x < y ( z , x = y (@), and y z (Y), then y = x v i for some i c I and z = y v j for and u z z (0). some j c J. With u =x v j we have z = uvi, .hence x E u (Y) Applying this observation to a = avb (@), avb ~ a v b v (Y), c and to c izbvc (Y), bvc = avbvc (a), we obtain elements e, f L satisfying a sze (Y), e = avbvc (a), (i
c ~ f = f ~ ( a v b v c ) r f r \ e =(@). d Theorem 9 is significant because in a wide class of lattices all congruences are standard. Recall, that a lattice L is sectionally complemented iff L has a zero, 0, end all intervals [0, a ] of L are complemented. 10. In a eectionally complemented lattice L all congruence relations are THEOREM standard.

Y a o o ~. Let 0 be a congruence relation of L and let I = [ 0 ] 0 be the ideal kernel of 0. Let a, b c L and a = b (0). Let c be the complement of ar\b in [0, avb]. Then c=cr\(avb)~cr\(ar\b)= O (@), hence c €1. Thus a = b (0) iff (ar\b)vi= avb for some i c I . By Theorem S(iv), this shows t,hat I is standard and 0 = @ [ I ] .

COROLLARY11. Let L be a weakly modular sectionally complemented lattice. If L satisfies the Ascending Chain Condition, then all congruences of L are neutral, in fact, of the form 0,where a i s a neutral element. 11.

111.Congruences and Ideals

150

PROOF.Let 0 be a congruence relation of L. By Theorem 10, 0 =@[I] where I is a standard ideal of L. By the Ascending Chain Condition, I = (a]. By Corollary 3, a is standard. Since L is weakly modular, by Theorem 2.6, a is neutral. Hence 0 = 0, with a neutral.

Exercises 1. Prove that an ideal generated by a set of distributive elements is distributive. 2. Show that an ideal generated by a set of standard elements is standard. 3. Do the analogues of Exercises 1 and 2 hold for neutral ideals ? 4. Verify that the converse of Exercise 2 does not hold. (Hint : Consider the lattice of Figure 1.)

0

0

0

Figure 1

6. Prove Corollary 3 directly. 6.

7. 8.

9. 10.

*ll. 12.

13. 14. 16.

16.

Consider the lattice L as a sublattice of I ( L ) under the natural embedding r -+(r]. Show that every congruence relation of L can be extended to I ( L ) . Characterize those congruence relations of I ( L )which are extensions of congruences of L. For any ideal I of a lattice L relate the congruence relation @ [ I ]of L with the congruence relation 01of I ( L ) . Characterize a standard ideal I of L in terms of the congruence relation 01on I ( L ) . Show that in Theorem 6 it is sufficient to assume that condition (iii) holds for principal ideals. Construct a lattice L and an ideal I of L such that Theorem S(v) holds for all principal ideals J and K yet I is not standard (Iqbalunissa [1966]). Show that in Theorem 6(v) end Theorem 6(iv) we can assume that J 3 K. Show that the congruence relation 0 and Q, of the lattice L permute iff, for all a,b,cELwitha
4. Structure Theorems

161

-. \

\

Figure 2 Let L be a lattice and let I and J be ideals of L. If I is standard and I A J , I v J are principal, then J is principal. 18. For a lattice L and standard ideal I of L , let L / I denote the quotient lattice L / O [ I ] . Verify the First Isomorphism Theorem for Standard Ideals: For any ideal J of L, I A J is a standard ideal of I and

17.

IvJlI e J / I A J . 19.

Prove the Second Isomorphism Theorem for Standard Ideals: Let L be a lattice, let I and J be ideals of L , J I, and let J be standard. Then I is standard iff IIJ is standard in LIJ, and in this case L / I EZ ( L / J ) / ( I / J ) .

20.

State and verify the Second Isomorphism Theorem for Neutral Ideals (J.Hashimot0 [1952]).

4. Structure Theorems Let A and B be bounded lattices and L = A x B. Then a = ( l ,0) a n d b =(O, 1) are neutral elements, and they are complementary. Conversely, if L is a bounded lattice, a , b L, a and b are complementary elements, and a is neutral, then, by Theorem 2.4, the map

y : 5 -+(XAa,

zVU)

embeds L into (a]x [a). Observe that y is also onto: if (a,w) (a]x [a), then zy = for x = u v ( w ~ b ) .Indeed, the first component of zy is (uv(wAb))Aa= ( u A n ) V ( v A a A b )= 21, since u a , aAb = 0 , and a is dually distributive. Similarly, xva=uv(wr\b)va=uv((wva)~(avb))=w, sinceaisdistributive, u l a < w , a n d a v b = l . ( u , w)

152

111. Congruences and Ideals

THEOREM1. The direct decompositions of a bounded lattice L into tuo factors are (up to i8omorphism) in one-to-one correspondence with the complemented neutral elements of L. The proof given above is slightly artificial. Its essence will come out better if we consider lattices with 0. Let L be a lattice with 0, and let L = A x B. Then both A and B have 0 so we can set

I ={(a, 0) I a € 4 , J ={(0, b) I b E B } .

It is easily seen that I and J are ideals of L, a +(a, 0) is an isomorphism of A and I, b +(O, b) is an isomorphism between B and J , IAJ ={O},

I v J = L.

The last equality holds because (a, b) =(a, O)v(O,b). For the same reason, every ideal K of L has a unique representation of the form

K = I i v J i , Ii E I , J i E J , I,, J i € I ( & ) (with I, = I n K and Ji=J n K ) and every such pair (I,,J,) occurs (indeed for I< =I,vJ1). Therefore, the map

K-(KnI, KnJ) is a direct product representation of I(L)and so I and J are neutral ideals by Theorem 2.4. Conversely, let I and J be complementary neutral ideals of L. Then, for any x EL,

x E (x]= ( x ] A ( I v J = ) ((XI n I ) v ( ( xn] J ) , and so x< ivj, i € (x]n I , j E (XI n J . But i< x, j < x, hence x =ivj. So every elernent x of L has a representation of the form x = i v j , i E I , j c J . This representation is unique since if x =ivj, i E I , j E J, then

(XI n I = (ivj]n I = ((i]v(j]) n I = ( i ] v ( ( fn I ) = (i], and similarly

(51 n J = ( j ] . Since every pair (i,j ) occurs in some representation we obtain that

x 4,j> is an isomorphism between L and I x J . Generalizing slightly we get the following:

THEOREM2. Let L be a lattice with 0. There is (up to isomorphism) a one-to-one correspondence between direct decomposition

L =A , X *

* *

X Aa-i

4.

Structure Theorems

153

and n-tuples of neutral ideals (Io, . . . ,In-l)which satisfy I , A I ~ = O for i + j

and

Iov-* * v I f i - =L. i

COROLLARY3. Let L be a lattice with 0. Let L have two direct decompositions:

L=Ao X . * * XAn-l, L = B o X - - * X Bm-l. Then L has a direct decomposition

L =Co,o X Co,1 x such that

*

x C 0 , n - l x CI.0 x

* * *

X C1,n-i X *

A,zC~,,~X .-xC,-,,,, .

.

X.Cm-l,o X *

* *

Xem-1,n-l

for O l i < n ,

u nd

Bi Y Ci,, X

* * *

X Ci,n-I,

for 0
...

PROOF.Let ( I o , . . . ,I n - 1 ) and (Jo, ,J,-J be the neutral ideals associated with the two decompositions as in Theorem 2. Then

--

( I o ~ J Ioi, ~ J O , * I ~ - ~ A J* o* *, , I o A J m - 1 ,

* *

,In-lAJm-1)

is again a sequence of neutral ideals satisfying the conditions of Theorem 2. (We only need Theorem 2.9 to help verify this.) The direct decomposition associated with this sequence will yield the decomposition required. A lattice L is called directly indecomposable iff L has no representation in the forin L = A x B where both A and B have more than one element. COROLLARY4.

Let L be a lattice with 0. If L has a representation

L =A , x * rohere each Ai, 0 < i position


*

X An-1,

i s directly indecomposable, then for any other decom-

L = BOX*.* X Bm-i of L into directly indecomposable factors we have that n = m and that there e&ts a permutation a of (0, . . . , n - l} such that Ai B,, for all 0 2 i n.

<

Results analogous to Corollaries 3 and 4 also hold for general lattices, see the exercises. In trying to sharpen Corollary 4 to a structure theorem there are two difficulties we have to overcome. A lattice need not have a decomposition into directly indecomposable factors. Directly indecomposable lattices are hard.to accept as " building blocks "; one would rather have simple lattices, that is, lattices with only the two trivial congruences,w and L.

154

111. Congruences and Ideals

The first difficulty is easy to overcome by chain conditions. Observe that if A and B are lattices with 0, C = { O , ai, a2, ,an} is a chain in A, and D ={O, b,, ,b,} is a chain in B, then

...

= { ( O J O), (ai,O),

* * * >

(an,

.. .

O),

‘I), ‘ ’ ’ >

(afi>

b,)}

+

is a chain in A x B of length n +m. This easily implies that l ( A x B ) = l ( A ) l ( B ) for lattices A and B of finite length. LEMMA 5. Let L be a bounded lattice. If L is of finite length, then L is isomorphic to a direct product of directly &decomposable lattices. The passage from ‘I directly indecomposable ” to ‘ I simple ” requires much stronger hypotheses. THEOREM 6. Let L be a sectionally complemented weakly modular lattice. If L is of finite length, then L can be represented as a direct product of simple lattices.

PROOF.First observe that if L z A x B and L satisfies the hypotheses of this theorem then so do A and B ; only the statement about weak modularity has to be verified and it follows from the observation that (q,a2)/(b ~b2) , xW(ci c2)/(4,d2) implies that q l b l x w Gildl. J

By Lemma 5, L 2 Lt X * X L, where all Li are directly indecomposable. By the observation above, all Li srttisfy the hypotheses of this theorem. So it is sufficient to prove that if L satisfies the hypotheses of the theorem and L is directly indecomposrtble, then L is simple. Indeed, if 0 is a congruence relation of L, then 0 = @[Z] where I is a standard ideal by Theorem 3.10. By the chain condition, I.= (a], and a is standard by Corollary 3.3. By Theorem 2.6 a is neutral. Since L is complemented, by Theorem 1 (a] is a direct factor of L. But L is directly indecomposable, hence a = O or a = 1, yielding 0 = w or 0 = I , verifying that L is simple. Since modular lattices and relatively complemented lattices are special cases of weakly modular lattices we obtain two famous structure theorems at3 special cases: COROLLARY 7 (The Birkhoff-Menger Theorem, G. Birkhoff [1935b] and K. Menger [1936]). Let L be a complemented modular lattice. If L is of finite length, then L is isomorphic to a direct product of simple lattices, We shall see in Chapter I V that these simple lattices are exactly g2and the nondegenerate projective geometries of finite dimension.

COROLLARY8 (R. P. Dilworth [1950]). Let L be a relatively complemented lattice. If L i s of finite length, then L is isomorphic to a direct product of simple lattices.

4.

Structure Theorems

155

. . . ,L,,then by Theo-

If L is a direct product of finitely many simple lattices Li, rem 1.3.13

C(L)s C(LJ x *

* *

x C(L,)z ( Q k ,

that is, C ( L ) is a Boolean lattice. This led G. Birkhoff to suggest (in G. Birkhoff [1948]) the study of lattices L for which C( L )is Boolean. We are going t o present a solution t o this problem. Let us call a congruence relation 0 of a lattice L separable iff, for all a, b EL, a b, there exists a sequence a =xo
<

< --

THEOREM 9 (G. GriCtzer and E. T. Schmidt [1958d]). For a lattice L, C ( L ) i s Booleait iff L is weakly niodular and all congruences of L are separable.

PROOF.Let C ( L )be Boolean. Let a/b=., cld, a , b, c, d E L, a + b ; we wish to find a proper subquotient c’ld’ of c/o? satisfying c’/d’xW a/b. Let 0 =@(a,b ) and let 0’ be the complement of 0. Then 0v0’ =L and so c G d (Ova’).B y Theorem 1.3.9, there is a sequence d =xo < x i * x, = c such that, for each 0
< --<

x.=x. $-

or

<

X~=X~+~(@’).

If, for all O < i < n , X~=X~+~(@’), then c ~ (0’); d since wlb=.,c/d, a ~ (0’). b But 8 = @ ( a ,a) and so n b (0). We conclude that a = b (@A@’), that is, a = b (w), contradicting u =kb. Therefore, xi=xi+l (0) for some 0 I i n . Applying Theorem 1.2 we obtain a proper subquotient c’/d‘ of satisfying to xi=xifI( @ ( ab, ) ) , c ‘ / d ’ = , u/b, proving that L is weakly modular. Now let 0 be a congruence relation of L and let a , b E L, a b. Then a G b ( O V ~ ’ ) , where 0’ is the complement of 0, and so by Theorem 1.3.9 there exists a sequence u = x o <.r,
<

<

<-- -

x.=x.2 + 1 (0)or xi=xi+i (0’). $-

We clsini that the saine sequence establishes the separability of 0. Indeed, if for some i , ~ ~ + x(a), & +and ~ u , w E [ x ~ , x ~ + ~ ] , u =(a), w then xi=xi+l (0’) and so u ~ v ( ( O ’ ) , implying that u e w (@A@’), that is u = w. To prove the converse, let us start out with a weakly modular lattice L and a congruence relation 0 of L. We claim: The binary relation Q, defined on L by u

= b ( 0 ) iff

ZL

= w (0) for no proper subquotient u/w of avb/aAb

is a congruence relation; in fact, CD is the pseudocoinplement of 0. We prove the first part of this claim by verifying that 0 satisfies conditions (i)-(iii) of Lemma 1.3.8. Condition (i) is clear by the definition of 0.To verify (ii),let a , b,

166

111. Congruences and Ideals

c E L, a
(a),

+

<

(a),

(a),

<

< <- - <

THEOREM10. Let L be a complete, sectionally complemented, dually sectionully complemented, weakly modular lattice. Then the centre of L is a complete sublattice of L.

PROOF.Let XECen(L). Set u = r \ X . Then (a] is the kernel of the congruence relation

A (0, I .2: E X). Hence, by Theorem 3.10, (a] is standard; by Corollary 3.3, a is standard. Now Corollary 2.8 tells us that a is neutral. Since a is complemented, a E Cen ( L ) . By duality, we obtain V X ECen(L).

4. Structure Theorems

COROLLARY11 (M.F. Janowitz [1967]). mented lattice i s n complete sublattice.

157

The centre of a complete relatively comple-

This result has soiiie interesting implications concerning direct decompositions of complete relatively conipleiriented lattices; see S. Maeda [1966].

Exercises 1.

2.

3. 4.

6.

6.

7. 8.

9. 10.

11.

12. 13. 14.

15. 16.

17. 1

Show that the representations of a lattice L with 0 as a direct product of two lattices are (up to isomorphism) in one-to-one correspondence with pairs of ideals (I,J) satisfying I n J = (0) and every element a of L has exactly one representation of the form a = i v j , i €1,j E J. Let L = A , x A 2 . Define the binary rulation Oi on L by (al, a,)=@*,b,) (0i) iff ai=bi (i =1, 2 ) . Show that 0 1 and 02 are congruence relations of L, 0 1 ~ 0 =2 w , and O l v 0 2 =L. Show that 0,and O2of Exercise 2 are permutable. Prove that the representat,ions of a lattice L as a direct product of two lattices are (up to isomorphism) in a one-t,o-one correspondence with pairs of congruence relations (01,0,) which are complementary and permutable. Show that if in Exercise 4 we pass from two to n factors, then we get a n n-tuple of congruences (a,, . . . ,on)such that @ ' A * * ~0~= w , (@,A. - A @ ; - I ) V @ ~ = ~ , and @ , A . * . A @ ~ - I and 0i permute, for i =2, . . . , n. Can we replace in Exercise 5 the condition " @,A * m ~ 0 i - i and 0i permute " by any two distinct Oi and Oj permute " 1 Use Exercise 5 to verify Corollary 3 for arbitrary lattices. Verify Corollary 4 for arbitrary lat,tices. Relate Exercise 5 to Theorem 2. Let B be the Boolean lattice R-generated by the rational interval [0, 11. Show that, for any natural number n, B has a representation as a direct product of n lattices L,, . . . , L,, ILil > 1 for i = 1, . , , ,n, but B has no representation as a direct product of directly indecomposable lattices. Construct a lattice which is not of finite lengt'h but every chain in the lattice is finite. Statements 5-8 of this section deal with lattices of finite length. Which of these statements remain valid for lattices in which every chain is finite? Prove that every congruence relation of a finite lattice or of a lattice of finite length is separable.' Verify that if in a lattice L, for every a, b EL, a < b, there is a finite maximal chain in [a, b], then all congruence relations of L are separable. This holds, in particular, if the lattice is locally finite, t>hatis, if all intervals are finite. < b. Then Let L be a distributive lattice, a, b, a l , a,, . . . EL, a =al < a, < a3 < * 0 = V ( O ( a 2 i - i , ati) I i = 1, 2, . . .) is not a separable congruence relation. For a distributive lattice L, C ( L )is Boolean iff L is locally finite. (Use Exercises 14 and 16. J. Hashimoto [1952].) The separable congruences of a lattice L form a sublattice of C ( L ) .

-

Exercises 13-19 are based on G. Griitzer and E. T. Schmidt [1958d].

--

158

111. Congruences and Ideals

Let L be a lattice with 1. A neutral congruence reletion @ [ I ]is separable iff I is principal. 19. Let L be a complemented modular lattice. Then C(L)is Boolean iff all neutral ideals of L are principal (Shih-chiangWang [1963]). 20. Find a complete lattice L and a sublattice K of L such that K is a complete lattice but not a complete sublattice of L. 18.

Further Topics and References The properties of weak projectivity and of the congruences @[I]are discussed in greater detail in G. Gratzer and E. T. Schmidt [1958d]. For universal algebras A. I. Mal'cev introduced similar concepts; the difference is that while for lattices it is sufficient to consider " unary algebraic functions " of a special form (namely, . (( (2AaO)Vai)Aa2) * .), for universal algebras in general we have to consider arbitrary unary algebraic functions (see, for instance G. Griitzer [1968]). The polynomial p 2 is also of special interest ; identities of p z that hold for lattices imply that the congruence lattices of lattices are distributive. A general condition for the distributivity of congruence lattices of algebras in a given equational class of algebras can be found in B. Jbnsson [1967]. Weak modularity is a rather complicated condition. It can be somewhat simplified for finite lattices: a finite lattice is weakly modular iff a l b x , cJd and a b 2 c > d imply the existence of aproper subquotient c f / d fof cfd satisfying c p / d ' X walb (G. Gratzer [1963a]). In general, if in a lattice L any interval contains a finite maximal chain, then it is sufficient to consider weak projectivities of prime quotients, see for instance N. Funayama [1942], J. Jakubik [1956a] and D. T. Finkbeiner [1960]. Under this condition, L is simple iff any pair of prime quotients are projective. In general, no bound can be put on the number of perspectivities. However, if L is a relatively complemen-

..

-

>

1:

+

1

ted lattice of length n, a, b >.0 and a10 x b/O, then a/O x b/O, where 12 < 2 - (n 1) , see J. E. McLaughlin [1951] and [1953]. The investigation of the number of projectivities becomes important in the study of equational classes of lattices, see qection V.3 and C. Herrmann 119731. Local conditions implying the projectivity of any pair of prime quotients can be found in F. A. Smith [1974]. Modular lattices and relatively complemented lattices share a property stronger than weak modularity. In P. Crawley and It. P. Dilworth [1973] a lattice 'is said to have the Projectivity Property iff whenever a / b is weakly projective into cld, then alb is projective to some subquotient c'ld' of cld. Weak modularity seems to be a very natural concept. It appears quite surprisingly in some results. The following result of Iqbalunissa [1966] is a good illustration of this point: if in the lattice L every congruence relation is neutral, then L is weakly niodular. A lot more information on standard elements and ideals can be found in G. Griitzer [1959] and G. Griitzer and E. T. Schmidt [1961]. Among the topics discussed in k

169

Problems

these papers are the Isomorphism Theorems, the Zassenhaus Lemma, the JordanHolder Theorem for standard ideals, and the Schreier extension problem. It is also shown that in a finite modular lattice exactly the neutral ideals satisfy the First Isomorphism Theorem. Some of these results were inspired by J. Hashimoto [1952]. In the proof of Theorem 4.9 we describe the pseudocomplement 0 of a congruence relation 0 of a weakly modular lattice. Iqbalunissa [1966] observes that the converse also holds: if in a lattice L the relation 0 given by any congruence relation 0 (as described in the proof of Theorem 4.9) is always a congruence relation, then L is weakly modular. For some additional results on standard ideals see M. F. Janowitz [1964], [1964a], and [1965], in which some types of ideals, more general than standard ideals, are discussed. Congruences of relatively complemented lattices, in particular a description of @(a,b), is given in M. F. Janowitz [1968]. Some counterexamples are given in Iqbalunissa [1965] and [1965a]. Lattices whose congruences form a Boolean lattice are discussed also in T. Tanaka [1952], J. Hashinioto [1957], P. Crawley [1960]. See also D. T. Finkbeiner [1960] and J. Jakubik [1955]. Lattices whose congruences form a Stone lattice are studied in M. F. Janowitz [1968] and [1968a], and Iqbalunissa, [1971]. Standard elements in the lattice of all subsemigroups of a group are studied in S. G. Ivanov [1966]. It is pointed out in S. Maeda [1974], that if L is the dual of the lattice of ell Titopologies on an infinite set, then L has infinitely many standard elements, but only the elements 0 and 1 are neutral. The concept of a standard ideal has recently been extended to convex sublattices in E. Fried and E. T. Schmidt [1976]. Distributivity of a pair of lattice elements is investigated in P. G. KontoroviE, S. G . Ivanov, and G. P. Kondrajov [1965]; see also F. and S. Maeda [1970]. We shall consider modular pairs of elements in Section IV.2. A great deal of useful information on congruence relations of lattices, in particular about regularity and permutability, can be found in J. Hashimoto [1963]. There is a connection between projectivity and representability. If L / @ is projective (in a class containing L), then @ is representable. A. Day [1973] considers a weaker concept implying the representability of @ for finite L.

Problems 111. 1. Generalize the concepts of distributive and neutral ideals to convex sublattices. (For standard convex sublattices, see E. Fried and E. T. Schmidt [1975].) 111.2. Let L = I o 2 I , 2 * * * 3 I,, = I be a descending sequence of ideals. If Ii+i is a standard ideal of I j for j =0, , n, then I E ic a standard ideal of order n of L. Investigate standard ideals of order 2 (order n). Under what conditions do they forni a sublattice?

...

160

111. Coiigruences and Ideals

111.3. An ideal I of a lattice L is said to satisfy the First Isomorphism Theorem (G. Gratzer and E. T. Schmidt [19&3d]) iff (IvJ)/@[I]s J / @ [ In J ] for any ideal J of L (under the natural isomorphism). Investigate this concept and relate it to standard ideals. 111.4. Under what conditions do the ideals of a weakly modular lattice form a weakly modular lattice? 111.6. Investigate lattices whose congruences form a Stone lattice. 111.6. For n 1, investigate lattices whose congruence lattices, as distributive 1a.ttices with pseudocomplementation, belong to B,. 111.7. Develop structure theorems for lattices all of whose congruences are standard (distributive, neutral).

>