A Theory of Transformation Monoids: Combinatorics and Representation Theory

TL;DR

This paper develops a comprehensive theory of finite transformation monoids, introducing new concepts like orbitals and orbital digraphs, and explores their representations, probabilistic applications, and connections to primitivity.

Contribution

It introduces the notion of orbitals and orbital digraphs for transformation monoids and characterizes primitivity through connectedness, also computing module covers and applying probability theory.

Findings

Characterization of primitive transformation monoids via orbital digraphs.

Explicit computation of projective covers of transformation modules.

Application of Markov chains and ergodic theorems to transformation monoids.

Abstract

The aim of this paper is to develop a theory of finite transformation monoids and in particular to study primitive transformation monoids. We introduce the notion of orbitals and orbital digraphs for transformation monoids and prove a monoid version of D. Higman's celebrated theorem characterizing primitivity in terms of connectedness of orbital digraphs. A thorough study of the module (or representation) associated to a transformation monoid is initiated. In particular, we compute the projective cover of the transformation module over a field of characteristic zero in the case of a transitive transformation or partial transformation monoid. Applications of probability theory and Markov chains to transformation monoids are also considered and an ergodic theorem is proved in this context. In particular, we obtain a generalization of a lemma of P. Neumann, from the theory of synchronizing groups, concerning the partition associated to a transformation of minimal rank.