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 × 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.

Generators of matrix monoids over commutative semirings

“…Moreover, as φ 2 (Γ ′ ) = GL 2 (B) and GL 2 (B) is generated by φ 2 (A), we can see that φ 2 (A) ∪ φ 2 (γ) is a generating set for M 2 (S). However, this gives a contradiction as M 2 (S) is minimally generated by 3 matrices [9]. Therefore, M 2 (S) is minimally generated by these |X| + 2 matrices.…”

“…East, Jonušas and Mitchell [6] found generating sets for 2 × 2 full matrix monoids over the min-plus natural number semiring, max-plus natural number semiring, and their finite quotients. Moreover, Hivert, Mitchell, Smith, and Wilson [9] found minimal generating sets for a number of submonoids of the monoid of boolean matrices and showed that the generating sets given in [6] are minimal generating sets.…”