MATRIX SEMIGROUPS OVER SEMIRINGS

We study properties determined by idempotents in the following families of matrix semigroups over a semiring S: the full matrix semigroup Mn(S), the semigroup UTn(S) consisting of upper triangular matrices, and the semigroup Un(S) consisting of all unitriangular matrices. Il’in has shown that (for n > 2) the semigroup Mn(S) is regular if and only if S is a regular ring. We show that UTn(S) is regular if and only if n = 1 and the multiplicative semigroup of S 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 Mn(S), UTn(S) and Un(S) admits a natural antiisomorphism 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 Mn(S) is abundant if and only if it is regular. Our main interest is in the case where S is an idempotent semifield, our motivating example being that of the tropical semiring T. We prove that certain subsemigroups of Mn(S), including several generalisations of well-studied monoids of binary relations (Hall relations, reflexive relations, unitriangular Boolean matrices), are Fountain. We also consider the subsemigroups UTn(S∗) and Un(S∗) consisting of those matrices of UTn(S) and Un(S) having all elements on and above the leading diagonal non-zero. We prove the idempotent generated subsemigroup of UTn(S∗) is Un(S∗). Further, UTn(S∗) and Un(S∗) are families of Fountain semigroups with interesting and unusual properties. In particular, every e R-class and e L-class contains a unique idempotent, where e R and e L are the relations used to define Fountainicity, but yet the idempotents do not form a semilattice. 1