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In this paper we characterize those linear bijective maps on the monoid of all $n times n$ square matrices over an anti-negative semifield which preserve and strongly preserve each of Greens equivalence relations $mathcal{L}, mathcal{R}, mathcal{D}, mathcal{J}$ and the corresponding three pre-orderings $leq_mathcal{L}, leq_mathcal{R}, leq_mathcal{J}$. These results apply in particular to the tropical and boolean semirings, and for these two semirings we also obtain corresponding results for the $mathcal{H}$ relation.
The multiplicative semigroup $M_n(F)$ of $ntimes n$ 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 $UT_n(S)$
Greens relations on the deformed finite inverse symmetric semigroup $mathcal{IS}_n$ and the deformed finite symmetric semigroup $mathcal{T}_n$ are described.
We study the free objects in the variety of semigroups and variety of monoids generated by the monoid of all $n times n$ upper triangular matrices over a commutative semiring. We obtain explicit representations of these, as multiplicative subsemigrou
We investigate ideal-semisimple and congruence-semisimple semirings. We give several new characterizations of such semirings using e-projective and e-injective semimodules. We extend several characterizations of semisimple rings to (not necessarily subtractive) commutative semirings.
In this paper, we introduce and study V- and CI-semirings---semirings all of whose simple and cyclic, respectively, semimodules are injective. We describe V-semirings for some classes of semirings and establish some fundamental properties of V-semiri