E. T. Schmidt

On congruence lattices of lattices

If K is a lattice, then let | denote the lattice of all congruence relations of K. It is known (see [1]) that O(K) is a distributive lattice satisfying some continuity properties (see below). It is natural to ask about the lattice-theoretical characterization of O(K). I f K is finite, then | is also finite, and conversely, every finite distributive lattice L is isomorphic to a O(K) where Kis finite too. This theorem is due to R. P. DILWORTH and is mentioned in [I] without proof. No proof of this theorem has been published as yet. In this note we give a proof of this theorem; some generalizations are also mentioned. Before stating the results some notions are needed. A lattice K is called section complemented if K has a least element 0, and if every x with x_-< y has a complement z in [0, y], i. e. xP, z=O, x U z = y . The length of a chain C o f n + l elements is n, and the length of a finite lattice K is n if K contains a subchain of length n but no subchain of length n + 1.

Congruence lattices of finite semimodular lattices

We prove that every finite distributive lattice can be represented as the congruence lattice of a finite (planar) semimodular lattice.

Weakly independent subsets in lattices

D E F I N I T I O N . A subset H of a lattice L is called weakly independent iff for all h, hl, h2 . . . . . h ~ e H which satisfy h < h l v h 2 v . . . v h , there exists an i (1--<i-< n) such that h-<h~. A maximal weakly independent set is called a basis of L. Every subchain of L is a weakly independent subset and any maximal chain is a basis. In this paper L denotes a finite distributive lattice and let Jo(L) denote the set of all join-irreducible elements of L. By using L e m m a 1 (cf. later) it is easy to show that Jo(L) is a basis of L.

Congruence lattices of small planar lattices

For a finite distributive lattice D with n join-irreducible elements, we construct a finite (planar) lattice L with 0(n2) elements such that the congruence lattice of L is isomorphic to D. This improves on an early result of R. P. Dilworth (around 1940) and G. Gratzer and E. T. Schmidt (1962) constructing such a (nonplanar) lattice L with 0(22n) elements, and on a recent construction of G. Gratzer and H. Lakser which yields a finite (planar) lattice L with 0(n3) elements.

CONGRUENCE-PRESERVING EXTENSIONS OF FINITE LATTICES TO SECTIONALLY COMPLEMENTED LATTICES

In 1962, the authors proved that every finite distributive lattice can be represented as the congruence lattice of a finite sectionally complemented lattice. In 1992, M. Tischendorf verified that every finite lattice has a congruence-preserving extension to an atomistic lattice. In this paper, we bring these two results together. We prove that every finite lattice has a congruence-preserving extension to a finite sectionally complemented lattice.

A lattice construction and congruence-preserving extensions

A chopped lattice is a partial lattice we obtain from a bounded lattice by removing the unit element. Under a very natural condition, (FG), the nitely generated ideals of a chopped lattice M form a lattice which is a congruence-preserving extension of M; that is, every congruence of M has exactly one extension to this lattice. In this paper, we investigate how we can obtain from a pair of lattices A and B by amalgamation a chopped lattice. We establish a set of six su±cient conditions. We then investigate when the chopped lattice so obtained will satisfy Condition (FG). A typical result is the following: if C =A\B is a principal ideal of both A and B and A is modular, then Condition (FG) holds. We apply this to prove that if L is a lattice with a nontrivial distributive interval, then L has a proper congruence-preserving extension.

An extension theorem for planar semimodular lattices

We prove that every finite distributive lattice

A representation of m-algebraic lattices

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