Minimal generating sets for matrix monoids

Abstract

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