Matrix semigroups over semirings

Abstract

We study properties determined by idempotents in the following families of matrix semigroups over a semiring [Formula: see text]: the full matrix semigroup [Formula: see text], the semigroup [Formula: see text] consisting of upper triangular matrices, and the semigroup [Formula: see text] consisting of all unitriangular matrices. Il’in has shown that (for [Formula: see text]) the semigroup [Formula: see text] is regular if and only if [Formula: see text] is a regular ring. We show that [Formula: see text] is regular if and only if [Formula: see text] and the multiplicative semigroup of [Formula: see text] is regular. The notions of being abundant or Fountain (formerly, weakly abundant) are weaker than being regular but are also defined in terms of idempotents, namely, every class of certain equivalence relations must contain an idempotent. Each of [Formula: see text], [Formula: see text] and [Formula: see text] admits a natural anti-isomorphism allowing us to characterise abundance and Fountainicity in terms of the left action of idempotent matrices upon column spaces. In the case where the semiring is exact, we show that [Formula: see text] is abundant if and only if it is regular. Our main interest is in the case where [Formula: see text] is an idempotent semifield, our motivating example being that of the tropical semiring [Formula: see text]. We prove that certain subsemigroups of [Formula: see text], including several generalisations of well-studied monoids of binary relations (Hall relations, reflexive relations, unitriangular Boolean matrices), are Fountain. We also consider the subsemigroups [Formula: see text] and [Formula: see text] consisting of those matrices of [Formula: see text] and [Formula: see text] having all elements on and above the leading diagonal non-zero. We prove the idempotent generated subsemigroup of [Formula: see text] is [Formula: see text]. Further, [Formula: see text] and [Formula: see text] are families of Fountain semigroups with interesting and unusual properties. In particular, every [Formula: see text]-class and [Formula: see text]-class contains a unique idempotent, where [Formula: see text] and [Formula: see text] are the relations used to define Fountainicity, but yet the idempotents do not form a semilattice.