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Register a new userLinear isomorphisms preserving Greens relations for matrices over semirings
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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.
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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 $UT_n(S)$ 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 $ntimes n$ tropical matrices. Ilin 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.
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 subsemigroups of quiver algebras over polynomial semirings. In the $2 times 2$ case this also yields a representation as a subsemigroup of a semidirect product of commutative monoids. In particular, from the case where $n=2$ and the semiring is the tropical semifield, we obtain a representation of the free objects in the monoid and semigroup varieties generated by the bicyclic monoid (or equivalently, by the free monogenic inverse monoid), inside a semidirect product of a commutative monoid acting on a semilattice. We apply these representations to answer several questions, including that of when the given varieties are locally finite.
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-semirings. We show that all Jacobson-semisimple V-semirings are V-rings. We also completely describe the bounded distributive lattices, Gelfand, subtractive, semisimple, and anti-bounded, semirings that are CI-semirings. Applying these results, we give complete characterizations of congruence-simple subtractive and congruence-simple anti-bounded CI-semirings which solve two earlier open problems for these classes of CI-semirings.
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