A combinatorial property for semigroups of matrices.
A semigroup \(S\) has the weak permutation property if there exists some integer \(k\geq 2\) such that for any sequence \(s_1,s_2,\dots,s_k\) of elements of \(S\), there are distinct permutations \(\sigma\) and \(\tau\) of the set \(\{1,2,\dots,k\}\) such that \(s_{\sigma(1)}s_{\sigma(2)}\cdots s_{\sigma(k)}=s_{\tau(1)}s_{\tau(2)}\cdots s_{\tau(k)}\). In this article, the authors study the semigroups of matrices over a commutative semiring, proving that a semigroup of matrices \(S\) over a tropical semiring has the weak permutation property and that if \(S\) is finitely generated then the growth function of \(S\) is polynomially upper bounded. They also find an application of this result to the Burnside problem for semigroups.
scientific article; zbMATH DE number 1972508