Congruence Lattices of Lattices

App end ix

Congruence lattices of lattices

G. Gratzer and E. T. Schmidt

In the early sixties, wecharacterized congruence lattices of universal algebras as

algebraic lattices; then our interest turned to the characterization of congruence

lattices of lattices. Since these lattices are distributive, we gured that this must

b e an easier job. After more then 35 years, we know that wewere wrong.

In this App endix, we give a brief overview of the results and metho ds. In

Section 1, we deal with nite distributive lattices; their representation as congru-

ence lattices raises manyinteresting questions and needs sp ecialized techniques.

In Section 2, we discuss the general case.

A related topic is the lattice of complete congruences of a complete lattice.

In contrast to the previous problem, we can characterize complete congruence

lattices, as outlined in Section 3.

1. The Finite Case

The congruence lattice, Con L, of a nite lattice L is a nite distributive lattice

N. Funayama and T. Nakayama, see Theorem I I.3.11. The converse is a result

of R. P. Dilworth see Theorem I I.3.17, rst published G. Gratzer and E. T.

Schmidt [1962]. Note that the lattice we construct is sectionally complemented.

In Section I I.1, we learned that a nite distributive lattice D is determined

by the p oset J D of its join-irreducible elements, and every nite p oset can b e

represented as J D , for some nite distributive lattice D . So for a nite lattice L,

we can reduce the characterization problem of nite congruence lattices to the

representation of nite p osets as J Con L. 545

546 C. Congruence lattices of lattices

Planar lattices, small lattices

We start with the following result G. Gratzer, H. Lakser, and E. T. Schmidt

[A118]:

Theorem 1 Let D b e a nite distributive lattice with n join-irreducible ele-

ments. Then there exists a planar lattice L of O n elements with Con L D .

The original constructions R. P. Dilworth's and also our own pro duced

lattices of size O 2 and of order dimension O 2n. In G. Gratzer and H. Lakser

[C11], this was improved to size O n and order dimension 2 planar.

Wesketch the construction for Theorem 1.

Let P =JD, P = fp ;p ;:::;p g, and we takeachain

1 2 n

C = fc ;c ;:::;c g; c c c :

0 1 2n 0 1 2n

Toevery prime interval [c ;c ], we assign an elementofP as its \color", so that

i i+1

each elementof P is the color of two adjacent prime intervals: let the color of

[c ;c ] and [c ;c ]bep ;of[c ;c ] and [c ;c ]be p , and so on, of [c ;c ]

0 1 1 2 1 2 3 3 4 2 2n2 2n1

and [c ;c ]be p .Follow this on the two examples in Figures 1 and 2; in

2n1 2n n

Figure 1, P = fp ;p g and p

1 2 1 2 1 2 3

1 2 3 2

2 2

In C ,we ll in a \covering square" C with one more element so that we

obtain an M , if the two sides have the same color, see Figure 3. Moreover, if p,

q 2 P and p

3 2

longer side has two prime intervals of color q and the shorter side is of color p,

and we add one more element, as illustrated in Figure 3, to obtain the sublattice

N .

5;5

It is an easy computation to show that jLj kn , for some constant k , and

that D Con L; this isomorphism is established by assigning to p 2 P the

congruence of L generated by collapsing any all prime intervals of color p.

For a natural number n, de ne cr n as the smallest integer such that, for

any distributive lattice D with n join-irreducible elements, there exists a nite

lattice L satisfying Con L D and jLj cr n. From the construction sketched

ab ove, it follows that

crn < 3n +1 :

In Section A.1.7, we discussed the lower b ound for cr n. Here is how

16 log n

it evolved: G. Gratzer, I. Rival, and N. Zaguia [C18] proved that Theorem 1 is

\b est p ossible" in the sense that size O n cannot b e replaced by size O n ,

for any < 2; that is,

kn < cr n;

1. The Finite Case 547

p1 p2 p2

p2 p2 p2

p1 p1 p1 C C p1 p1 p1 P C L Figure 1

p2 p3 p3

p3 p3 p1 p2 p2

p2 p2 p2 p1 C p1 p1 C

p1 p3 p1 p1

P C L

Figure 2

for any constant k , for any <2, and for any suciently large integer n.

Y. Zhang [C39] noticed that the pro of of this inequality can b e improved to

obtain the following result: for n 64,

1 n

< cr n:

64 log n

The lower b ound for cr n is, of course, much stronger than the last one.

16 log n 2

548 C. Congruence lattices of lattices

p p p p

M 3

q q q p q p

N 5,5

Figure 3

A di erent kind of lower b ound is obtained in R. Freese [C5]; it is shown that

if JCon L has e edges e>2, then

jLj:

2 log e

R. Freese also proves that JCon L can b e computed in time O jLj log jLj.

Consider the optimal length of L. E.T.Schmidt [1975] constructs a nite

lattice L of length 5m, where m is the numb er of dual atoms of D ; S.-K. Teo

[C36] proves that this result is b est p ossible. For nite chains, this was done in

J. Berman [1972].

Sp ecial classes of lattices

Mo dular and semimo dular lattices

E. T. Schmidt [1974] proves that Every nite distributive lattice D can berep-

resented as the congruence latticeofa mo dular lattice M . It follows from The-

orem I I I.4.9 that the congruence lattice of a nite mo dular lattice is a Bo olean

lattice, therefore, we cannot exp ect M to b e nite. For a short pro of of this

result, see [C35].

Amuch deep er result was proved in E. T. Schmidt [C33]:

Theorem 2 Every nite distributive lattice can b e represented as the con-

gruence lattice of a complementedmodular lattice.

It was p ointed out byF.Wehrung that the ring theoretic and ordered group-

theoretic results of G. A. Elliott [C4], P. A. Grillet [C28], E. G. E ros, D. E.

Handelman, and Chao Liang Shen [C3] along with some elementary results in

F. Wehrung [A274] contain this theorem; see K. R. Go o dearl and F. Wehrung

[C7] for more detail.

In G. Gratzer, H. Lakser, and E. T. Schmidt [A119], we constructed a nite

semimodular lattice L:

1. The Finite Case 549

Theorem 3 Every nite distributive lattice D can b e represented as the con-

gruence lattice of a nite semimodular lattice S . In fact, S can b e constructed as

a planar lattice of size O n , where n is the numb er of join-irreducible elements

of D .

Congruence-preserving extensions

In Section A.1.7, we discussed the concept of congruence-preserving extensions

and some of the ma jor result concerning it. Many of the older results use the

following construction, see E. T. Schmidt [1968]. Let D b e a b ounded distributive

lattice; let M [D ] the extension of M by D consist of all triples hx; y ; z i2D

3 3

satisfying x ^ y = x ^ z = y ^ z . Then M [D ] is a mo dular lattice, x 7! hx; 0; 0i,

x 2 D ,isanemb edding of D into M [D ] and by identifying x with hx; 0; 0i,we

obtain that M [D ] is a congruence-preser ving extension of D . See a variantof

this construction in Section 2.

In G. Gratzer, H. Lakser, and R. W. Quackenbush [C12], it is proved that the

M [D ] construction is a sp ecial case of tensor products.IfAand B are lattices

with zero, then A B , the tensor pro duct of A and B , is the join-semilattice

freely generated by the p oset A f0gB f0g sub ject to the relations:

ha ;bi_ha ;bi = ha _ a ;bi and ha; b i_ha; b i = ha; b _ b i a, a , a 2 A, b,

0 1 0 1 0 1 0 1 0 1

b , b 2 B .

0 1

Let A and B b e nite lattices. Then A B is obviously a lattice, and the

following isomorphism holds see [C12]:

Theorem 4

Con A Con B ConA B :

For any nite simple lattice S , this isomorphism implies that Con A

ConA S . The case A = D , S = M is the M [D ] result.

3 3

Theorem 4 has b een extended to wide classes of in nite lattices substituting

Con for Con in G. Gratzer and F. Wehrung [C26] and to arbitrary lattices

with zero using \b ox pro ducts" a variant of tensor pro ducts in G. Gratzer and

F. Wehrung [C27].

Many of the newer results utilize a new technique in G. Gratzer and E. T.

Schmidt [A123] applied also in G. Gratzer and E. T. Schmidt [A124] and [C25].

The rectangular extension RK of a nite lattice K is de ned as the direct

pro duct of al l sub direct factors of K , that is,

RK = K= j 2 MCon K ;

where MCon K is the set of all meet-irreducible congruences of K .

K has a natural emb edding into RK by

: a7! a = h[a] j 2 MCon K i:

550 C. Congruence lattices of lattices

Let K = K .

Theorem 5 Let K b e a nite lattice. Then K has the Congruence Extension

Prop ertyinRK that is, every congruence of K can b e extended to RK .

p-algebras

T. Katrin ak [C31]characterized the congruence lattices of nite p-algebras that

is, lattices with pseudo complementation, see Section I I.6:

Theorem 6 Every nite distributive lattice is isomorphic to the congruence

lattice of a nite p-algebra.

In [C22], we give a more elementary pro of of this theorem. In fact, we prove

the following generalization:

Theorem 7 Let D b e an algebraic distributive lattice in which the unit ele-

mentofD is compact and every compact elementofD is a nite join of join-

irreducible compact elements. Then D can b e represented as the congruence

lattice of a p-algebra P .

Simultaneous representation

It is well-known that, given a lattice L and a convex sublattice K , the restriction

map Con L ! Con K is a f0; 1g-preserving lattice homomorphism. InG.Gratzer

and H. Lakser [C9], see also E. T. Schmidt [C34], the converse is proved: any

f0; 1g-preserving homomorphism of nite distributive lattices can b e realized as

such a restriction and, indeed, as a restriction to an ideal of a nite lattice.

If the sublattice K is not a convex sublattice, then the restriction map

Con L ! Con K need not preserve join, but it still preserves meet, 0, and 1.

Similarly,we can extend congruences from the sublattice K to L by minimal

extension. This extension map need not preserve meet, but it do es preserve

join and 0. Furthermore, it separates 0, that is, nonzero congruences extend to

nonzero congruences. Wenow formalize this.

Let K and L b e lattices, and let ' b e a homomorphism of K into L. Then '

induces a map ext ' of Con K into Con L: for a congruence relation of K , let

the image under ext ' b e the congruence relation of L generated by the set

' = fha'; b'ija b g.

The following result was proved byA.P. Huhn in [C29] in the sp ecial case

when is an emb edding and was proved for arbitrary in G. Gratzer, H. Lakser,

and E. T. Schmidt [C14]:

1. The Finite Case 551

Theorem 8 Let D and E b e nite distributive lattices, and let

: D ! E

be a f0; _g-homomorphism. Then there are nite lattices K and L, a lattice

homomorphism ' : K ! L, and isomorphisms

: D ! Con K;

: E ! Con L

with

= ext ':

Furthermore, ' is an emb edding i separates 0.

Theorem 8 concludes that the following diagram is commutative:

D ! E

? ?

= =

y y

ext '

Con K ! Con L

In G. Gratzer, H. Lakser, and E. T. Schmidt [C15] the following stronger

version is proved see G. Gratzer, H. Lakser, and E. T. Schmidt [C13] for a short

pro of :

Theorem 9 Let K b e a nite lattice, let E b e a nite distributive lattice, and

let : Con K ! E be a f0; _g-homomorphism. Then there is a nite lattice L,

a lattice homomorphism ' : K ! L, and an isomorphism : E ! Con L with

ext ' = .Furthermore, ' is an emb edding i separates 0.

If L is a lattice and L , L are sublattices of L, then there is a map

1 2

Con L ! Con L

1 2

obtained by rst extending each congruence relation of L to L and then restrict-

ing the resulting congruence relation to L . All we can say ab out this map is that

it is isotone and that it preserves 0. The main result of G. Gratzer, H. Lakser,

and E. T. Schmidt [C14] see G. Gratzer, H. Lakser, and E. T. Schmidt [C13] for

a short pro of is that this is, in fact, a characterization of 0-preserving isotone

maps b etween nite distributive lattices:

552 C. Congruence lattices of lattices

Theorem 10 Let D and D b e nite distributive lattices, and let

1 2

: D ! D

1 2

b e an isotone map that preserves 0. Then there is a nite lattice L with sublat-

tices L and L and there are isomorphisms

1 2

: D ! Con L ;

1 1 1

: D ! Con L

2 2 2

such that the diagram

D D

1 2

? ?

1 2

= =

y y

extension restriction

Con L ! Con L ! Con L

1 2

is commutative.

G. Gratzer, H. Lakser, and E. T. Schmidt [C16] and [C17] prove that if is

a 0-preserving join-homomorphism of D into E D and E are nite distributive

lattices and n = maxjJD j; jJE j, then we can construct the nite lattices

K and L and the lattice homomorphism ' : K ! L that represent so that

the size of K and L is O n and the order dimensions of K and L are 3. We

conjecture that this result is the b est.

The D-relation

In a nite lattice L,every element a is determined by the set of join-irreducible

elements contained in a. Therefore, it is quite natural that we can characterize

congruences in terms of the join-irreducible elements. This idea was develop ed

in the pap ers B. J onsson and J. B. Nation [C30], A. Day [A42], and R. Wille

[C38]. On the set JL, a binary relation D is de ned as follows: for p, q 2 JL,

let pD q i there exists an x 2 L such that p q _ x and p q _ x, where

q denotes the lower cover of q . This relation D de nes a closure op erator D as

follows: for A JL, let A be D-closed i pD q and q 2 A implies that p 2 A.

The following result describ es congruence lattices.

Theorem 11 The congruence lattice of a nite lattice L is isomorphic to the

lattice of D -closed subsets of J L. This isomorphism is given by

7! f p 2 JL j p ;p g:

This topic is treated in depth in the b o ok R. Freese, J. Jezek, and J. B.

Nation [A65]; see also M. Tischendorf [A267].

2. The General Case 553

2. The General Case

In the general case, there are two approaches to try to solve the characterization

problem of congruence lattices of lattices. The rst is due to the second author

and the second was suggested byP. Pudl ak [C32].

The rst approach: distributive join-homomorphisms

Let Con L denote the semilattice of compact congruences of the lattice L. Let

us call a semilattice S representable i there is a lattice L with Con L S .

Let D b e a nite distributive lattice, and take the Bo olean lattice B generated

by D see Section I I.4. Then for every d 2 B , there exists a smallest element

sd 2 D such that d sd. The map s : B ! D is a closure op erator see

De nition IV.3.8i. We construct a lattice K , whose congruence lattice is D

with a new unit element adjoined. Take B C and its meet-subsemilattice K ,

consisting of all elements of the form hb; 0i, b 2 B and hd; 1i, d 2 D . Then K is

a lattice and B can b e identi ed with an ideal of K under the map b 7! hb; 0i,

b 2 B . Consider a congruence hx; 0i h0;0iin B . Joining b oth sides with h0; 1i,

we get hsx; 1i = hx; 0i_h0;1i h0;0i_ h0;1i = h0;1i.Thus hsx; 0i h0;0i.

It is easy to see now that the congruence relation hsx; 0i; h0; 0iofB can

b e extended to K and h0; 1i; h0; 0i is the unit congruence. Nowwehaveto

identify the new unit element with the unit of Con K . The following construction

solves this problem see E. T. Schmidt [1968].

Let L consist of all triples hx; y ; z i2B satisfying x ^ y = x ^ z = y ^ z and

x 2 D . Then L is a lattice and D Con L.

Note that the elements h0; 0;zi, z 2 B, and hx; 1;xi, x 2 D, form a sublattice

of L isomorphic to K .

This construction also works for certain in nite distributive lattices, namely,

for dually relatively pseudo complemented lattices; indeed, for every d 2 D , then

there exists a smallest sd 2 B such that d sd.

Let B b e a generalized Bo olean lattice and let h : B ! S b e an onto join-

homomorphism.We will say that h is weakly distributive i for all x, y , z 2 B

with x _ y z and hx _ y =hz, there are x , y 2 B such that x _ y = z ,

1 1 1 1

hx=hx , and hy =hy . If B is the Bo olean lattice generated by D and

1 1

sx exists for every x 2 D , then the map h : x 7! sxisaweakly distributive

join-homomorphism.

We call h distributive i Ker h = Ker s j i 2 I , where each s , i 2 I ,isa

i i

closure op erator on B .

Theorem 12 Let S b e a semilattice with zero. If there is a generalized

Bo olean lattice B and a distributive join-homomorphism h : B ! S , then S

is representable.

We apply the previous construction for every closure op erator s to obtain

the lattices L , i 2 I .We glue these lattices to each other to construct L with i

554 C. Congruence lattices of lattices

Con L S .

By construction, every L has an ideal B isomorphic to B . We de ne a

i i

lattice M whose elements are vectors, h:::;x ;:::i, x 2 B, i 2 I, satisfying the

I i i

condition that for three distinct indices i, j , and k , x ^ x = x ^ x = x ^ x .

i j i k j k

For every i 2 I , the elements hx;:::;x;1;x;:::;xi, where the 1 is i-th entry, form

a dual ideal B of M , which is isomorphic to B and, consequently, to the ideal

B of L .Nowwe glue together M and L by identifying B and B . This way,

i i I i i

we get a meet semilattice. The lattice L is the lattice of all nitely generated

ideals of this semilattice.

Theorem 12 is sucient to obtain most representation theorems:

Theorem 13 Let S b e a semilattice with zero. Each one of the following

conditions implies that S is representable:

i Id S is completely distributive equivalently, S is isomorphic to the semi-

lattice of all nitely generated hereditary subsets of some partially ordered

set.

ii S is a lattice.

iii S is lo cally countable that is, for every s 2 S ,s] is countable.

iv jS j @ .

i was rst obtained byR.P. Dilworth. Pro ofs can b e found in G. Gratzer

and E. T. Schmidt [1962], H. Dobb ertin [C1], and P. Crawley and R. P. Dilworth

[1973].

ii was proved in E. T. Schmidt [A247]. P. Pudl ak [C32] provides another

pro of, in a more general, categorical context. Another pro of can b e found in

H. Dobb ertin [C2].

iii was rst obtained in A. P. Huhn [C29] under the condition that jS j @ .

The general result for lo cally countable semilattices was obtained in H. Dobb ertin

[C1]. Dobb ertin proved that if B is a lo cally countable generalized Bo olean

semilattice and if S is a distributive semilattice, then every weakly distributive

homomorphism from B to S is distributive. Further, he proved that every lo cally

countable distributive semilattice with zero is the weakly distributive image of

some lo cally countable generalized Bo olean algebra.

iv was obtained in A. P. Huhn [C29]. One of the main to ols used by Huhn

is the notion of frame intro duced in H. Dobb ertin [C1], which is a sp ecial sort

of lattice with zero used for building trans nite direct limits of direct systems

having up to @ ob jects.

Some of these results are presented in an axiomatic form in M. Tischendorf

[C37].

3. Complete Congruences 555

The second approach

Wehave seen that the representation is relatively easy for nite distributive

lattices. Let D b e an arbitrary distributive semilattice with zero; P. Pudl ak

[C32] proved that for each nite subset F of D , there is a nite subsemilattice of

D containing F , therefore, D is a direct limit of all the nite distributive lattices

contained in it as distributive join-semilattices with zero.

Let S b e a distributive semilattice with zero, and let S b e the set of nite

subsets of S f0g.P. Pudl ak's approach see [C32] is the following. Cho ose

an order preserving function that assigns to each F 2S a nite distributive

subsemilattice S of S containing F ;thus S is the direct limit of the S , F 2S.

F F

For each F 2S, construct a nite atomistic lattice L whose congruence lattice

is isomorphic to S and such that if F G, then L emb eds into L with the

F F G

Congruence Extension Prop erty. Then if L is the limit of the L , F 2S, then

S F

the congruence lattice of L is isomorphic to Id S .

Applying this metho d, P. Pudl ak [C32]gave a new pro of of Theorem 13ii.

Negative results

There are a numb er of related negative results.

H. Dobb ertin [C2] constructs a distributive semilattice with zero S and a

semilattice homomorphism f of a Bo olean algebra B of size @ onto S such

that f is weakly distributive but not distributive.

F. Wehrung [A273] constructs a b ounded distributive semilattice S of size

@ that is not isomorphic to anyweakly distributive image of a generalized

Bo olean algebra. Note that the @ size is optimal. This shows that we cannot

obtain a p ositive solution of the congruence lattice characterization problem of

lattices by the rst approach.

The second approachwas also ruled out in F. Wehrung [A274]. The semilat-

tice S of previous paragraph is not isomorphic to Con L, for any lattice L which

is a direct limit of lattices that are either atomistic or sectionally complemented.

M. Ploscica, J. Tuma, and F. Wehrung [A234] prove that there exists a

distributive semilattice S that is representable as Con L, for a lattice L, but S

cannot b e represented using the rst or the second approach, that is, S is not a

distributive join-homomorphic image of a generalized Bo olean lattice nor is S a

direct limit of lattices that are either atomistic or sectionally complemented. In

fact, one can take S = Con F , where F is the free lattice on @ generators in

c 2

any nondistributivevariety of lattices.

3. Complete Congruences

In Section A.1.8, we brie y mentioned the result G. Gratzer [A100]:

Theorem 14 Every complete lattice K can b e represented as the lattice of

complete congruence relations of a complete lattice L.

556 C. Congruence lattices of lattices

In a series of pap ers, much sharp er results have b een obtained.

G. Gratzer and H. Lakser [A116] had the rst published pro of of Theorem 14;

in fact, it already contained more: L was constructed as a planar lattice.

Let m b e an in nite regular cardinal, and let K be an m-complete lattice.

Then the lattice Con K of all m-complete congruence relations of K is m-

algebraic this concept is the obvious mo di cation of De nition I I.3.12.

G. Gratzer and H. Lakser [C10] proved a partial converse:

Let m bearegular cardinal > @ , and let L bean m-algebraic lattice with

an m-compact unit element. Then L is isomorphic to the latticeof m-algebraic

congruences of an m-algebraic lattice K .

Amuch sharp er form of the original result was proved in the pap er R. Freese,

G. Gratzer, and E. T. Schmidt [C6]:

Every complete lattice L is isomorphic to the latticeofcomplete congruence

relations of a complete mo dular lattice K .

The m-algebraic direction and the mo dular direction were combined by the

present authors in [C19]:

Let m bearegular cardinal > @ . Every m-algebraic lattice L is isomorphic to

the latticeof m-complete congruencerelations of a suitable m-complete mo dular

lattice K .

G. Gratzer, P. Johnson, and E. T. Schmidt [C8] presents the same construc-

tion with a simpli ed pro of.

The sharp est result is the following G. Gratzer and E. T. Schmidt [C23]:

Theorem 15 Let m b e a regular cardinal > @ .Every m-algebraic lattice

L can b e represented as the lattice of m-complete congruence relations of an

m-complete distributive lattice K .

In the construction, we use in nite complete-simple complete distributive lat-

tices a complete lattice is complete-simple if it has only the two trivial complete

congruences. Such lattices were constructed in G. Gratzer and E. T. Schmidt

[C20] and G. Gratzer and E. T. Schmidt [C21]. It can b e shown, see G. Gratzer

and E. T. Schmidt [C24], that the representation of the three-elementchain must

contain such a lattice.

Bibliography

[C1] H. Dobb ertin, Vaught measures and their applications in lattice theory,J.

Pure Appl. Algebra 43 1986, 27{51.

[C2] , Boolean Representations of Re nement Monoids and their Appli-

cations, manuscript.

[C3] E. G. E ros, D. E. Handelman, and Chao Liang Shen, Dimension groups

and their ane representations, Amer. J. Math. 102 1980, 385{407.

[C4] G. A. Elliott, On the classi cation of inductive limits of sequences of

semisimple nite-dimensional algebras, J. Algebra 38 1976, 29{44.

[C5] R. Freese, Computing congruence lattices of nite lattices, Pro c. Amer.

Math. So c., 1997 125, 3457{3463.

[C6] R. Freese, G. Gratzer, and E. T. Schmidt, On complete congruence lattices

of complete modular lattices,Internat. J. Algebra Comput. 1 1991, 147{

160.

[C7] K. R. Go o dearl and F. Wehrung, Representations of distributive semilat-

tices by dimension groups, regular rings, C*-algebras, and complemented

modular lattices, manuscript, 1997.

[C8] G. Gratzer, P. Johnson, and E. T. Schmidt, Arepresentation of m-

algebraic lattices, Algebra Universalis 32 1994, 1{12.

[C9] G. Gratzer and H. Lakser, Homomorphisms of distributive lattices as re-

strictions of congruences, Can. J. Math. 38 1986, 1122{1134.

[C10] , On congruence lattices of m-complete lattices, J. Austral. Math.

So c. Ser. A 52 1992, 57{87.

[C11] , Congruence lattices of planar lattices, Acta Math. Hungar. 60

1992, 251{268. 557

558 BIBLIOGRAPHY

[C12] G. Gratzer, H. Lakser, and R. W. Quackenbush, The structure of tensor

products of semilattices with zero,Trans. Amer. Math. So c. 267 1981,

503{515.

[C13] G. Gratzer, H. Lakser, and E. T. Schmidt, Congruencerepresentations of

join homomorphisms of distributive lattices: A short proof, Math. Slovaca

46 1996, 363{369.

[C14] , Isotone maps as maps of congruences. I. Abstract maps, Acta

Math. Hungar. 75 1997, 81-111.

[C15] , Isotone maps as maps of congruences. II. Concrete maps,

manuscript.

[C16] , Congruencerepresentations of join-homomorphisms of distribu-

tive lattices. I. Size and breadth, manuscript.

[C17] , Congruencerepresentations of join-homomorphisms of distribu-

tive lattices. II. Order dimension, manuscript.

[C18] G. Gratzer, I. Rival, and N. Zaguia, Smal l representations of nite dis-

tributive lattices as congruence lattices, Pro c. Amer. Math. So c. 123

1995, 1959{1961.

[C19] G. Gratzer and E. T. Schmidt, Algebraic lattices as congruence lattices:

The m-complete case, Lattice theory and its applications Darmstadt,

1991, pp. 91{101, Res. Exp. Math., 23, Heldermann, Lemgo, 1995.

, \Complete-simple" distributive lattices, Pro c. Amer. Math. So c. [C20]

119 1993, 63{69.

[C21] , Another construction of complete-simple distributive lattices, Acta

Sci. Math. Szeged 58 1993, 115-126.

, Congruence lattices of p-algebras, Algebra Universalis 33 1995, [C22]

470{477.

[C23] , Complete congruence lattices of complete distributive lattices,J.

Algebra 170 1995, 204{229.

[C24] , Do we needcomplete-simple distributive lattices? Algebra Uni-

versalis 33 1995, 140{141.

[C25] , Sublattices and standardcongruences, manuscript.

[C26] G. Gratzer and F. Wehrung, Tensor products of lattices with zero, revis-

ited, AMS Abstract 97T-06-190.

[C27] , A new latticeconstruction: the box product, manuscript.

BIBLIOGRAPHY 559

[C28] P. A. Grillet, Directedcolimits of freecommutative semigroups, J. Pure

Appl. Algebra 9 1976/77, 73{87.

[C29] A. P. Huhn, On the representation of distributive algebraic lattices. I{III,

Acta Sci. Math. Szeged 45 1983, 239{246; 53 1989, 3{10, 11{18.

[C30] B. J onsson and J. B. Nation, Areport on sublattices of a free lattice, Con-

tributions to universal algebra Collo q., J ozsef Attila Univ., Szeged, 1975,

pp. 223{257. Collo q. Math. So c. J anos Bolyai, Vol. 17, North-Holland,

Amsterdam, 1977.

[C31] T. Katrin ak, Congruence lattices of nite p-algebras, Algebra Universalis

31 1994 475{491.

[C32] P. Pudl ak, On congruence lattices of lattices, Algebra Universalis 20

1985, 96{114.

[C33] E. T. Schmidt, Congruence lattices of complementedmodular lattices, Al-

gebra Universalis 18 1984, 386{395.

[C34] , Homomorphisms of distributive lattices as restrictions of congru-

ences, Acta Sci. Math. Szeged 51 1987, 209{215.

[C35] , Congruence lattices of modular lattices, Publ. Math. Debrecen 42

1993, 129{134.

[C36] S.-K. Teo, On the length of the congruence lattice of a lattice,Perio d.

Math. Hungar. 21 1990, 179{186.

[C37] M. Tischendorf, On the representation of distributive semilattices, Algebra

Universalis 31 1994, 446{455.

[C38] R. Wille, Subdirect decomposition of concept lattices, Algebra Universalis

17 1983, 275 { 287.

[C39] Y. Zhang, A note on \Smal l representations of nite distributive lattices

as congruence lattices", Order 13 1996, 365{367.