Wreath product decompositions for triangular matrix semigroups

TL;DR

This paper presents explicit wreath product decompositions for semigroups of upper triangular matrices, optimizing the decomposition for finite fields and extending results to more general rings and semirings.

Contribution

It provides a new explicit wreath product decomposition for triangular matrix semigroups, achieving optimality over finite fields and extending to broader algebraic structures.

Findings

Decomposition is optimal for finite fields.

Explicit wreath product decomposition for all upper triangular matrices.

Extensions to semigroups over general rings and semirings.

Abstract

We consider wreath product decompositions for semigroups of triangular matrices. We exhibit an explicit wreath product decomposition for the semigroup of all n-by-n upper triangular matrices over a given field k, in terms of aperiodic semigroups and affine groups over k. In the case that k is finite this decomposition is optimal, in the sense that the number of group terms is equal to the group complexity of the semigroup. We also obtain some decompositions for semigroups of triangular matrices over more general rings and semirings.