If x, y and z are elements of a lattice, then x ∨ (y ∨ (x ∧ z)) = x ∨ y is always true, while x ∨ y = z is usually not true. Is there an algorithm that, given two lattice expressions p and q, determines whether p = q holds for every substitution of the variables in every lattice? The answer is yes, and finding this algorithm (Corollary to Theorem 6.2) is our original motivation for studying free lattices. We say that a lattice L is generated by a set X ⊆ L if no proper sublattice of L contains X. In terms of the subalgebra closure operator Sg introduced in Chapter 3, this means Sg(X) = L. A lattice F is freely generated by X if (I) F is a lattice, (II) X generates F, (III) for every lattice L, every map h 0 : X → L can be extended to a homomorphism h : F → L.

Abstract. This paper studies the complexity of determining if a finite algebra generates a variety that satisfies various Maltsev conditions, such as congruence distributivity or modularity. For idempotent algebras we show that there are polynomial time algorithms to test for these conditions but that in general these problems are EXPTIME complete. In addition, we provide sharp bounds in terms of the size of two-generated free algebras on the number of terms needed to witness various Maltsev conditions, such as congruence distributivity.

A topological quasi-variety Q + T (M ∼ ) := IScP + M ∼ generated by a finite algebra M ∼with the discrete topology is said to be standard if it admits a canonical axiomatic description. Drawing on the formal language notion of syntactic congruences, we prove that Q + T (M ∼ ) is standard provided that the algebraic quasi-variety generated by M ∼ is a variety, and that syntactic congruences in that variety are determined by a finite set of terms. We give equivalent semantic and syntactic conditions for a variety to have Finitely Determined Syntactic Congruences (FDSC), show that FDSC is equivalent to a natural generalisation of Definable Principle Congruences (DPC) which we call Term Finite Principle Congruences (TFPC), and exhibit many familiar algebras M ∼ that our method reveals to be standard. As an application of our results we show, for example, that every Boolean topological lattice belonging to a finitely generated variety of lattices is profinite and that every Boolean topological group, semigroup, and ring is profinite. While the latter results are well known, the result on lattices was previously known only in the distributive case. Background, motivation and overview of resultsAn algebra M = M ; F with finite underlying set M and operations F generates an (algebraic) quasi-variety Q(M) := ISP M consisting of all isomorphic copies of subalgebras of direct powers of M. Similarly a structure M ∼ = M ; G, H, R, T with finite underlying set M , operations G, partial operations H, relations R and discrete topology T generates a topological quasi-variety Q + T (M ∼ ) := IS c P + M ∼ consisting of all isomorphic copies of topologically closed substructures of non-zero direct powers, with the product topology, of M ∼ . Interest in topological quasi-varieties stems from the fact that they arise as the duals to algebraic quasi-varieties under natural dualities. The general theory of natural dualities provides methods to Presented by R. W. Quackenbush.

No abstract

No abstract

In his thesis and [24], J. B. Nation showed the existence of certain lattice identities, strictly weaker than the modular law, such that if all the congruence lattices of a variety of algebras satisfy one of these identities, then all the congruence lattices were even modular. Moreover Freese and Jónsson showed in [10] that from this “congruence modularity” of a variety of algebras one can even deduce the (stronger) Arguesian identity.These and similar results [3; 5; 9; 12; 18; 21] induced Jónsson in [17; 18] to introduce the following notions. For a variety of algebras , is the (congruence) variety of lattices generated by the class () of all congruence lattices θ(A), . Secondly if is a lattice identity, and Σ is a set of such, holds if for any variety implies .

Abstract. It is shown that the lattice of subspaces of a finite dimensional vector space over a finite prime field is projective in the class of modular lattices provided the dimension is at least 4.In this paper it is shown that the lattice of subspaces of an «-dimensional vector space over the field with p elements (i.e., a projective geometry of dimension n -1 over Zp) is a projective modular lattice for 4 < n < w and p a prime. This answers problem 9 of [15].Recall that a modular lattice L is a projective modular lattice if for any modular lattices M and TV and any lattice homomorphisms h of L into N and f of M onto TV, there is a homomorphism g of L into M such that f(g(a)) = h(a) for all a E L. This is equivalent to the existence of a homomorphism / of a free modular lattice FM^) onto L and a homomorphism g of L to FM(X) such that /( g(a)) = a for all a E L. The map g ° / is a retraction, i.e., it is an endomorphism of FM(A') which is point-wise fixed on its image. Thus projective modular lattices are the retracts (images of retractions) of free modular lattices. In particular, every projective modular lattice is a sublattice of a free modular lattice. Thus as a corollary to our result we obtain that every finite planar modular lattice can be embedded into a free modular lattice (in fact, into FM(4)), since all these lattices can be embedded into the subspace lattices described above (cf. [3]).The first section of this paper reviews the definition and important results on von Neumann «-frames of characteristic r. It is shown in [4] the free modular lattice generated by an «-frame of characteristic r, which we denote FM(P(«, /•)), is a projective modular lattice for 3 < n < w and r > 1. In the second section a review of von Neumann's coordinatization is given. We prove the main result in the third section by showing that FM(P(«,p)) is isomorphic to the lattice of subspaces of an «-dimensional vector space over Z , for 4 < « < w andp a prime.A subdirectly irreducible modular lattice L is a splitting modular lattice if there is a lattice equation e such that each variety of modular lattices either

Abstract. Lattice diagrams, known as Hasse diagrams, have played an ever increasing role in lattice theory and fields that use lattices as a tool. Initially regarded with suspicion, they now play an important role in both pure lattice theory and in data representation. Now that lattices can be created by software, it is important to have software that can automatically draw them. This paper covers:-The role and history of the diagram.-What constitutes a good diagram.-Algorithms to produce good diagrams. Recent work on software incorporating these algorithms into a drawing program will also be covered.An ordered set P = (P, ≤) consists of a set P and a partial order relation ≤ on P . That is, the relation ≤ is reflexive (x ≤ x), transitive (x ≤ y and y ≤ z imply x ≤ z) and antisymmetric (x ≤ y and y ≤ x imply x = y). If P is finite there is a unique smallest relation ≺, known as the cover or neighbor relation, whose transitive, reflexive closure is ≤. (Graph theorists call this the transitive reduct of ≤.) A Hasse diagram of P is a diagram of the acyclic graph (P, ≺) where the edges are straight line segments and, if a < b in P, then the vertical coordinate for a is less than the one for b. Because of this second condition arrows are omitted from the edges in the diagram.A lattice is an ordered set in which every pair of elements a and b has a least upper bound, a ∨ b, and a greatest lower bound, a ∧ b, and so also has a Hasse diagram.These Hasse diagrams 1 are an important tool for researchers in lattice theory and ordered set theory and are now used to visualize data. This paper deals the special issues involved in such diagrams. It gives several approaches that have been used to automatically draw such diagrams concentrating on a three dimension force algorithm especially adapted for ordered sets that does particularly well.We begin with some examples.1 In the second edition of his famous book on lattice theory [3] Birkhoff says these diagrams are called Hasse diagrams because of Hasse's effective use of them but that they go back at least to H. Vogt, Résolution algébrique deséquation, Paris, 1895.