Minimal generating sets for matrix monoids

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.