Global structure theories for finite semigroups—Introduction

Abstract

The following four papers develop the global structure theory for finite semigroups. A global structure theory for the finite semigroup S can be carried out by embedding1 S in a long semidirect or wreath product of many finite groups alternated with many elementary (combinatorial order 3) semigroups. That such embeddings exist was proved ten years ago, initiating the global theory. The Cartesian coordinates of the semidirect or wreath product restricted to the image of the embedding gives rise to global (sequential or “triangular”) coordinates for S from which the global properties of the multiplication table can be viewed and computed. For example a global property is complexity, introduced in 1963, being by definition the minimal number of group coordinates possible in all such embeddings for S. In many applications we find complexity in this sense corresponds to complexity in the crude pragmatic sense. Alternatively, a global theory for S can be given by reconstructing S (up to embedding) from its maximal subgroups as follows. Take a maximal subgroup of S, G,, , and form a Rees matrix semigroup over G,, , denoted R(G,). Next add a maximal subgroup G, of S to R(G,) as a group of units (equals the invertible elements) yielding S, =