Minimal generating sets for matrix monoids

TL;DR

This paper identifies minimal generating sets for various classes of matrix monoids over semirings, including boolean, max-plus, min-plus, and modular matrices, for specific dimensions and conditions.

Contribution

It provides the first comprehensive determination of minimal generating sets for several well-known matrix monoids over different semirings and conditions.

Findings

Minimal generating sets for boolean matrices up to size 8.

Minimal generating sets for reflexive and Hall boolean matrices.

Minimal generating sets for 2x2 matrices over max-plus, min-plus, and modular semirings.

Abstract

In this paper, we determine minimal generating sets for several well-known monoids of matrices over semirings. In particular, we find minimal generating sets for the monoids consisting of: all $n\times n$ boolean matrices when $n\leq 8$; the $n\times n$ boolean matrices containing the identity matrix (the reflexive boolean matrices) when $n\leq 7$; the $n\times n$ boolean matrices containing a permutation (the Hall matrices) when $n \leq 8$; the upper, and lower, triangular boolean matrices of every dimension; the $2 \times 2$ matrices over the semiring $\mathbb{N} \cup \{-\infty\}$ with addition $\oplus$ defined by $x\oplus y = \max(x, y)$ and multiplication $\otimes$ given by $x\otimes y = x + y$ (the max-plus semiring); the $2\times 2$ matrices over any quotient of the max-plus semiring by the congruence generated by $t = t + 1$ where $t\in \mathbb{N}$; the $2\times 2$ matrices over the min-plus semiring and its finite quotients by the congruences generated by $t = t + 1$ for all $t\in \mathbb{N}$; and the $n \times n$ matrices over $\mathbb{Z} / n\mathbb{Z}$ relative to their group of units.