On Straight Words and Minimal Permutators in Finite Transformation Semigroups

TL;DR

This paper studies the structure of certain words in finite transformation semigroups, revealing unique factorizations and minimal permutators that can aid in automata decomposition, with applications in biological systems.

Contribution

It introduces the concept of minimal permutators as a code in transformation semigroups and shows how straight words form a finite code for permuting subsets.

Findings

Minimal permutators form a code in the semigroup

Straight words constitute a finite code for permutators

Unique factorizations enable hierarchical automata analysis

Abstract

Motivated by issues arising in computer science, we investigate the loop-free paths from the identity transformation and corresponding straight words in the Cayley graph of a finite transformation semigroup with a fixed generator set. Of special interest are words that permute a given subset of the state set. Certain such words, called minimal permutators, are shown to comprise a code, and the straight ones comprise a finite code. Thus, words that permute a given subset are uniquely factorizable as products of the subset's minimal permutators, and these can be further reduced to straight minimal permutators. This leads to insight into structure of local pools of reversibility in transformation semigroups in terms of the set of words permuting a given subset. These findings can be exploited in practical calculations for hierarchical decompositions of finite automata. As an example we consider groups arising in biological systems.