Syntactic and Global Semigroup Theory: A Synthesis Approach

Abstract

This paper is the culmination of a series of work integrating syntactic and global semigroup theoretical approaches for the purpose of calculating semidirect products of pseudovarieties of semigroups. We introduce various abstract and algorithmic properties that a pseudovariety of semigroups might possibly satisfy. The main theorem states that given a finite collection of pseudovarieties, each satisfying certain properties of the sort alluded to above, any iterated semidirect product of these pseudovarieties is decidable. In particular, the pseudovariety G of finite groups satisfies these properties. J. Rhodes has announced a proof, in collaboration with J. McCammond, that the pseudovariety A of finite aperiodic semigroups satisfies these properties as well. Thus, our main theorem would imply the decidability of the complexity of a finite semigroup. Their work, in light of our main theorem, would imply the decidability of the complexity of a finite semigroup.