Kirby A. Baker

Finite equational bases for finite algebras in a congruence-distributive equational class

Abstract Does every finite algebraic system A with finitely many operations possess a finite list of polynomial identities (laws), valid in A , from which all other such identities follow? Surprisingly, no ( R. C. Lyndon, 1954 ). The answer is, however, affirmative for various particular kinds of algebraic systems, such as finite groups (Oates and Powell), finite lattices, and even finite lattice-ordered algebraic systems (McKenzie). The purpose of the present paper is to provide a sufficient condition that guarantees an affirmative answer without referring to any particular kind of operation: It is sufficient for A to be a finite member of an equational class of algebraic systems whose congruence lattices are distributive. The proof is constructive. Applications include the case of lattice-ordered algebraic systems.

Growth problems for avoidable words

Abstract A word W is said to avoid a word U if no block (subword, factor) of W is the image of U under a homomorphism of free semigroups without unit. The theory of words avoiding xx (square-free words) has been much studied. The word U is said to be avoidable on n letters if there are arbitrarily long words on an n-letter alphabet that avoid U. If U is avoidable on n letters for some n, let μ(U) be the minimum possible such n. We show that μ(U) has a linear bound in terms of the alphabet size of U. We further show that there exists a word that is avoidable on four letters but not on three letters. Moreover, if U is this word, the number of words of length L on a μ(U)-letter alphabet that avoid U has a polynomial bound in terms of L, so that the question of the existence of such an example is resolved in the affirmative. In contrast, for xx the bound is known to be exponential.

A generalization of Sperner's lemma

Abstract A brief proof is given of a generalization of Sperners lemma to certain finite partially ordered sets.

Strong shift equivalence of 2 × 2 matrices of non-negative integers

The concept of strong shift equivalence of square non-negative integral matrices has been used by R. F. Williams to characterize topological isomorphism of the associated topological Markov chains. However, not much has been known about sufficient conditions for strong shift equivalence even for 2×2 matrices (other than those of unit determinant). The main theorem of this paper is: If A and B are positive 2×2 integral matrices of non-negative determinant and are similar over the integers, then A and B are strongly shift equivalent.

Strong shift equivalence and shear adjacency of nonnegative square integer matrices

Abstract For two square matrices A, B of possibly different sizes with nonnegative integer entries, write A ≈ 1 B if A = RS and B = SR for some two nonnegative integer matrices R , S . The transitive closure of this relation is called strong shift equivalence and is important in symbolic dynamics, where it has been shown by R.F. Williams to characterize the isomorphism of two topological Markov chains with transition matrices A and B . One invariant is the characteristic polynomial up to factors of λ. However, no procedure for deciding strong shift equivalence is known, even for 2×2 matrices A , B . In fact, for n × n matrices with n > 2, no nontrivial sufficient condition is known. This paper presents such a sufficient condition: that A and B are in the same component of a directed graph whose vertices are all n × n nonnegative integer matrices sharing a fixed characteristic polynomial and whose edges correspond to certain elementary similarities. For n > 2 this result gives confirmation of strong shift equivalences that previously could not be verified; for n = 2, previous results are strengthened and the structure of the directed graph is determined.

Nondefinability of projectivity in lattice varieties

The relationship of projectivity between two quotients in a lattice is shown not to be first-order definable, in any nondistributive lattice variety. The proof depends on a special kind of subdirect power construction that shows the existence of arbitrarily long non-shortenable projectivities in such a variety. A similar result holds for weak projectivities. Even so, weak projectivities of bounded length do suffice to determine principal congruences in any variety generated by a finite lattice.