John Rhodes

ALGEBRAIC THEORY OF MACHINES, LANGUAGES, AND SEMIGROUPS,

Abstract : The book is an integrated exposition of the algebraic, and especially semigroup-theoretic, approach to machines and languages. It is designed to carry the reader from the elementary theory all the way to hitherto unpublished research results.

Almost finite expansions of arbitrary semigroups

Abstract We describe algebraic techniques that enable us to apply methods of finite semigroup theory to arbitrary infinite semigroups. This will be used in later papers to put global coordinates on semigroups and their morphisms, and to study languages and automata that are not finite-state.

Complexity of Finite Semigroups

All semigroups considered are of finite order. The results of this paper were announced in [4]. For extensive background see [5]. This paper, however, is reasonably self-contained. (X, S) denotes the finite semigroup S acting faithfully on the right of the finite set X. One of the main problems in the study of finite semigroups is to determine all ways in which coordinates can be entered into X so that the action of S on X is in triangular form. (Precise definitions given below.) An important class of semigroups, namely wreath products, are by definition already in triangular form. Let (Xj, S,) be given for j = 1, n. Let X = X, x ... x X1. Let S be the semigroup of all functions A: X X satisfying the following conditions. Triangular action (1.1). If pk: X Xk denotes the klth projection map, then for each k = 1, *n , n, there exists fk: Xk x * x X, X Xk such that

New Techniques in Global Semigroup Theory

Herein I state new results of several researchers (including myself) and present some new conjectures. Almost no proofs are given, but instead, references and (hopefully) helpful remarks. The new techniques are to be learned by studying the references. This paper is just a signpost.

Decidability of complexity one-half for finite semigroups

ResumeAll semigroups considered are finite. The semigroup C is by definition combinatorial (or group free or aperiodic) iff the maximal subgroups of C are singletons. Let S2oS1 denote the wreath product of S1 by S2, so P1: S2oS1 → S1 with P1 the projection surmorphism. S<T, read S divides T, iff S is a homomorphic image of a subsemigroup of T.In this paper we give necessary and sufficient conditions in terms of a homomorphic image of S (resp. a subsemigroup of S) so that S<GoC (resp. S<CoG), with C a combinatorial semigroup and G a group. This requires an earlier results by Bret Tilson and John Rhodes in [8].

Finite Semigroups Whose Idempotents Commute or Form a Subsemigroup

We give a new proof that every finite semigroup whose idempotents commute divides a finite inverse semigroup (Ash’s theorem), and, more generally, we prove that every finite semigroup whose idempotents form a subsemigroup divides a finite orthodox semigroup.

SURVEY OF GLOBAL SEMIGROUP THEORY

What is global semigroup theory? Is it ‘truly’ global? What is its relation to other areas? For other surveys of global semigroup theory, see [Mar], [Chico], [B-Mar].

Transformations, Semigroups, and Metabolism

A discussion is given of the application of methods and results from the theory of finite semigroups to the analysis of systems which may be described by transformations on a finite state space. A model for metabolism is developed as an example.