Robert W. Quackenbush

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.

A survey of minimal clones

SummaryA clone is a set of composition closed functions on some set. A non-trivial fact is that on a finite set every clone contains a minimal clone. This naturally leads to the problem of classifying all minimal clones on a finite set. In this paper I survey what is known about this classification. Rather than repeat the arguments used in the original papers, I have tried to use known results about finite algebras to give a more coherent and unified description of the known minimal clones.

Near vector spaces over GF(q) and (v,q+1,1)-BIBD's

Abstract The usual construction of ( v , q +1,1)−BIBDs from vector spaces over GF ( q ) is generalized to the class of near vector spaces over GF ( q ). It is shown that every ( v , q +1,1)−BIBD can be constructed from a near vector space over GF ( q ). Some corollaries are: Given a ( v 1 , q +1,1)−BIBD 〈 P 1 , B 1 〉 and a ( v 2 , q +1,1)−BIBD 〈 P 2 , B 2 〉, there is a (( q −1) v 1 v 2 + v 1 + v 2 , q +1,1)−BIBD 〈 P 3 , B 3 〉 containing 〈 P 1 , B 1 〉 and 〈 P 2 , B 2 〉 as disjoint subdesigns. If there is a ( v , q +1,1)−BIBD then there is a (( q −1) v +1, q ,1)−BIBD. Every finite partial ( v , q ,1)−BIBD can be embedded in a finite ( v ′, q +1,1)−BIBD.

Enumeration in Classes of Ordered Structures

Let K be a class of finite structures. There are two obvious enumeration questions to ask of K: what is the set of cardinalities of members of K (the spectrum problem) and, for each n ≥ 1 what is the number of pairwise non-isomorphic members of K of cardinality n (the fine spectrum problem). If we restrict our attention to classes of partially ordered structures, the spectrum problem is usually trivial since the standard classes considered usually contain the class of chains, while the fine spectrum problem is usually hopeless since it seems impossible to enumerate ordered sets much more complicated than chains or antichains.

Algebraic Speculations About Steiner Systems

Publisher Summary This chapter provides the algebraic speculations about Steiner systems. The chapter discusses how algebraic ideas are useful in telling more about combinatorial designs in general and Steiner systems in particular. The chapter presents two examples of algebraic techniques that provided the initial proofs of significant results in combinatorial design theory and one case where more algebraic knowledge are prevented an embarrassing mistake.

Pseudovarieties of finite algebras isomorphic to bounded distributive lattices

Let K be a class of finite algebras closed under subalgebras, homomorphic images and finite direct products. It is shown that K is isomorphic to the class of bounded distributive lattices if and only if K is generated by a lattice-primal algebra.

Nilpotent block designs I: Basic concepts for Steiner triple and quadruple systems

This paper discusses the concepts of nilpotence and the center for Steiner Triple and Quadruple Systems. The discussion is couched in the language of block designs rather than algebras. Nilpotence is closely connected to the well known doubling and tripling constructions for these designs. A sample result: a point p in an STS is projective if every triangle containing p generates the 7-element Fano plane; the p-center of the STS is the set of all projective points and is a projective geometry over GF(2).

A NOTE ON A PROBLEM OF GORALČÍK

This chapter describes the problem of Goralcik. The problem is to find a finite algebra whose congruence lattice is isomorphic to M ( M n is the ( n + 2)-element lattice with n atoms). It is found that if q is a prime power, then the congruence lattice of the two-dimensional vector space over GF ( q ) is M q +1 . It is assumed that q ≥ 7 and then, one can be sufficiently clever in introducing new operations so as to kill off all but 7 of the atoms. One would wind up with an algebra u such that u is a one-dimensional vector space over GF ( q ) with some added operations, and the congruence lattice of u 2 is M 7 .