G. Grätzer

Lattice theory : foundation

Preface.- Introduction.- Glossary of Notation.- I First Concepts.- 1 Two Definitions of Lattices.- 2 How to Describe Lattices.- 3 Some Basic Concepts.- 4 Terms, Identities, and Inequalities.- 5 Free Lattices.- 6 Special Elements.- II Distributive Lattices.- 1 Characterization and Representation Theorems.- 2 Terms and Freeness.- 3 Congruence Relations.- 4 Boolean Algebras.- 5 Topological Representation.- 6 Pseudocomplementation.- III Congruences.- 1 Congruence Spreading.- 2 Distributive, Standard, and Neutral Elements.- 3 Distributive, Standard, and Neutral Ideals.- 4 Structure Theorems.- IV Lattice Constructions.- 1 Adding an Element.- 2 Gluing.- 3 Chopped Lattices.- 4 Constructing Lattices with Given Congruence Lattices.- 5 Boolean Triples.- V Modular and Semimodular Lattices.- 1 Modular Lattices.- 2 Semimodular Lattices.- 3 Geometric Lattices.- 4 Partition Lattices.- 5 Complemented Modular Lattices.- VI Varieties of Lattices.- 1 Characterizations of Varieties 397.- 2 The Lattice of Varieties of Lattices.- 3 Finding Equational Bases.- 4 The Amalgamation Property.- VII Free Products.- 1 Free Products of Lattices.- 2 The Structure of Free Lattices.- 3 Reduced Free Products.- 4 Hopfian Lattices.- Afterword.- Bibliography.

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.

Stone lattices. II. Structure theorems

Using the triple associated with a Stone algebra L , as introduced in the first part of this paper ( 1 ), we will investigate certain problems concerning the structure of a Stone lattice. The following topics will be discussed: prime ideals, topological representation, completeness, relative Stone lattices, and the reduced triple. It is assumed that the reader is familiar with §§ 2–4 of ( 1 ). For the sake of convenience, we will write L = 〈 C, D, ϕ 〉 to indicate that 〈 C, D, ϕ 〉 is the triple associated with L , and whenever convenient we will write the elements of L as ordered pairs 〈 x, a 〉, as it is given in ( 1 , § 4, the Construction Theorem).

The structure of tensor products of semilattices with zero

If A and B are finite lattices, then the tensor product C of A and B in the category of join semilattices with zero is a lattice again. The main result of this paper is the description of the congruence lattice of C as the free product (in the category of bounded distributive lattices) of the congruence lattice of A and the congruence lattice of B. This provides us with a method of constructing finite subdirectly irreducible (resp., simple) lattices: if A and B are finite subdirectly irreducible (resp., simple) lattices then so is their tensor product. Another application is a result of E. T. Schmidt describing the congruence lattice of a bounded distributive extension of M3.

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.

Boolean extensions and normal subdirect powers of finite universal algebras

(i) ~B is subdirectly representable in {91o . . . . ,91,_2}c_K if and only if satisfies the identities of 91o x ... x 91n2. (ii) I f I f3 ] > 1 and ~ is subdirectly representable in K, then ~ has a unique set of factors in K. We shall generalize the concept of normal subdirect power in a way that will enable us to prove Theorems 1 * and 2* for arbi t rary (rather than binary or f-) algebras; Theorem 3 will be proved without recourse to the theory of f algebras.

Equational classes of lattices

In this section we shall discuss the basic properties of equational classes of lattices. Of the four characterizations and descriptions given, three apply to arbitrary equational classes of universal algebras; the fourth is valid only for those equational classes of universal algebras that are congruence distributive (that is, the congruence lattice of any algebra in the class is distributive). For the sake of simplicity, all these results are stated and proved only for lattices.

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.

PROPER CONGRUENCE-PRESERVING EXTENSIONS OF LATTICES

We prove that every lattice with more than one element has a proper congruence-preserving extension.