Minimal generating sets for matrix monoids

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 × n boolean matrices when n ≤ 8; the n × n boolean matrices containing the identity matrix (the reflexive boolean matrices) when n ≤ 7; the n × n boolean matrices containing a permutation (the Hall matrices) when n ≤ 8; the upper, and lower, triangular boolean matrices of every dimension; the 2 × 2 matrices over the semiring ℕ ∪ {−∞} with addition ⊕ defined by x ⊕ y = max(x, y) and multiplication ⊗ given by x ⊗ y = x + y (the max-plus semiring); the 2 × 2 matrices over any quotient of the max-plus semiring by the congruence generated by t = t + 1 where t ∈ ℕ; the 2 × 2 matrices over the min-plus semiring and its finite quotients by the congruences generated by t = t + 1 for all t ∈ ℕ; and the n × n matrices over ℤ/nℤ relative to their group of units.