Ralph Freese

Inequalities for generalized matrix functions based on arbitrary characters

where ai~ denotes the (i, ])th entry of A. Using the techniques of multilinear algebra Marcus E2J and others have been able to prove important inequalities for these generalized matrix functions in the case that the degree of • is one. We extend these results to the case where 9f is an arbitrary character. In doing so we obtain an interesting decomposition of the tensor space [Eq. (7)]. Also the proof of a key lemma is simplified by the use of algebraic derivations.

Congruence modularity implies the Arguesian identity

It is shown that if is any variety of algebras all of whose congruence lattices are modular, then the congruence lattice of every algebra in satisfies the Arguesian law.

Automated Lattice Drawing

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.

An application of dilworth's lattice of maximal antichains

Recently there has been a good deal of interest in the maximal sized antichains of a partially ordered set [1-8]. A theorem of Dilworth states that under the natural ordering these antichains form a distributive lattice. This paper outlines a proof of this theorem and applies it to strengthen the results and shorten the proofs of [4].

On the Complexity of Some Maltsev Conditions

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.

The structure of modular lattices of width four with applications to varieties of lattices

Apparatus for finishing the edge of particleboard, plywood and the like by smoothing and sealing the edge with a fast-drying material providing a firm base on which a woodgrain pattern or the like may be printed. The finish developed by extruding and shaping a fast-drying material to a thickness of approximately 5/1000 inch along the edge of the board. Switches are employed to precisely control the application of material when the board is in the vicinity of the applicator. A heating cycle is included in which the edge material is dried and cured after application. The apparatus admits of automatic operation at substantial rates of application with virtually no waste or trimming of the edging material after application.

ON COMPLETE CONGRUENCE LATTICES OF COMPLETE MODULAR LATTICES

The lattice of all complete congruence relations of a complete lattice is itself a complete lattice. In 1988, the second author announced the converse: every complete lattice L can be represented as the lattice of complete congruence relations of some complete lattice K. In this paper we improve this result by showing that K can be chosen to be ac ompletemodular lattice. Internat. J. Algebra Comput., 1 (1991) 147{160 c 1991 World Scientic Pubishing

Finitely presented lattices: canonical forms and the covering relation

A canonical form for elements of a lattice freely generated by a partial lattice is given. This form agrees with Whitmans canonical form for free lattices when the partial lattice is an antichain. The connection between this canonical form and the arithmetic of the lattice is given. For example, it is shown that every element of a nitely presented lattice has only nitely many minimal join representations and that every join representation can be rened to one of these. An algorithm is given which decides if a given element of a nitely presented lattice has a cover and nds them if it does. An example is given of a nontrivial, nitely presented lattice with no cover at all. Trans. Amer. Math. Soc. 312 (1989), 841{860 c 1989 Amer. Math. Soc. This paper studies nitely presented lattices. We introduce a canonical form for the elements of such a lattice which agrees with Whitmans canonical form for free lattices in the case the nitely presented lattice is free. This canonical form, like Whitmans, has several nice theoretical properties. In addition it allows one to eciently calculate in such a lattice. We have programs, written in both Common Lisp and muLisp, for dealing with nitely presented lattices. We also investigate the covering relation in nitely presented lattices. We show that there is an eective proceedure for determining if an element of such a lattice has any lower covers and for nding them if there are any. Alan Day (2) has shown that every nitely generated free lattice is weakly atomic, i.e., every interval contains a covering. It was conceivable that such a theorem could extend to all nitely presented lattices. We show that this is not the case. In fact we show that there is a (nontrivial) nitely presented lattice without any coverings. There is a close connection between nitely presented lattices and lattices freely generated by a nite partial lattice (see below). Because of this, most of our theorems will be phrased in terms of FL(P), the free lattice generated by the partial lattice P.

Term Rewrite Systems for Lattice Theory

It is shown that, even though there is a very well-behaved, natural normal form for lattice theory, there is no finite, convergent AC term rewrite system for the equational theory of all lattices.

Computing congruence lattices of finite lattices

An inequality between the number of coverings in the ordered set J(Con L) of join irreducible congruences on a lattice L and the size of L is given. Using this inequality it is shown that this ordered set can be computed in time O(n2 log2 n), where n = ILl. This paper is motivated by the problem of efficiently calculating and representing the congruence lattice Con L of a finite lattice L. Of course Con L can be exponential in the size of L; for example, when L is a chain of length n, Con L has 2n elements. However, since Con L is a distributive lattice, it can be recovered easily from the ordered set of its join irreducible elements J(Con L). Indeed any finite distributive lattice D is isomorphic to the lattice of order ideals of J(D) and this lattice is in turn isomorphic to the lattice of all antichains of J(D), where the antichains are ordered by A ? B, i.e., for each a E A there is a b E B with a < b. If P is an ordered set of size n which has N order ideals, then there are straightforward algorithms to find the order ideals of P which run in time 0(nN); see, for example, [5]. In [10] Medina and Nourine give an algorithm which runs in time 0(dN), where d is the maximum number of covers of any element of P. Thus we will concentrate on the problem of efficiently finding J(Con L).