2020/12/18 by Hivert, F., Mitchell, J. D., Smith, F. L. +1
#20M20 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2012.10323
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× 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.