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arXiv · 2012.10323

Minimal generating sets for matrix monoids

F. Hivert·J. D. Mitchell·F. L. Smith·W. A. Wilson
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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.

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F. Hivert, J. D. Mitchell, F. L. Smith, W. A. Wilson. 2020-12-18. Minimal generating sets for matrix monoids. https://arxiv.org/abs/2012.10323

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