Matrix semigroups over semirings
The multiplicative semigroup $M_n(F)$ of $n\times n$ matrices over a field $F$ is well understood, in particular, it is a regular semigroup. This paper considers semigroups of the form $M_n(S)$, where $S$ is a semiring, and the subsemigroups $UT_n(S)$ and $U_n(S)$ of $M_n(S)$ consisting of upper triangular and unitriangular matrices. Our main interest is in the case where $S$ is an idempotent semifield, where we also consider the subsemigroups $UT_n(S^*)$ and $U_n(S^*)$ consisting of those matrices of $UT_n(S)$ and $U_n(S)$ having all elements on and above the leading diagonal non-zero. Our guiding examples of such $S$ are the 2-element Boolean semiring $\mathbb{B}$ and the tropical semiring $\mathbb{T}$. In the first case, $M_n(\mathbb{B})$ is isomorphic to the semigroup of binary relations on an $n$-element set, and in the second, $M_n(\mathbb{T})$ is the semigroup of $n\times n$ tropical matrices. Il'in has proved that for any semiring $R$ and $n>2$, the semigroup $M_n(R)$ is regular if and only if $R$ is a regular ring. We therefore base our investigations for $M_n(S)$ and its subsemigroups on the analogous but weaker concept of being Fountain (formerly, weakly abundant). These notions are determined by the existence and behaviour of idempotent left and right identities for elements, lying in particular equivalence classes. We show that certain subsemigroups of $M_n(S)$, including several generalisations of well-studied monoids of binary relations (Hall relations, reflexive relations, unitriangular Boolean matrices), are Fountain. We give a detailed study of a family of Fountain semigroups arising in this way that has particularly interesting and unusual properties.