Even Covers for Left Ample Semigroups

In this paper we describe the covers of a left ample semigroup that arise from strict (2,1)-embeddings in left factorizable left ample monoids.     

Proper Covers for Left Ample Semigroups

A left ample semigroup is a semigroup with a unary operation ⁺ which has a (2,1)-algebra embedding into a symmetric inverse monoid I(X), the operation ⁺ on I(X) being defined by α⁺ = αα^-1. We consider some analogues for left ample semigroups of results on E-unitary covers of inverse semigroups due to McAlister and Reilly. The analogue of an E-unitary cover is a proper cover, and we discuss the construction of proper covers in terms of relational homomorphisms, and of dual prehomomorphisms. We observe that our construction gives an E-dense proper cover for an E-dense left ample semigroup. We also consider proper covers constructed from strict embeddings into factorisable left ample monoids. In contrast to the inverse case, not all proper covers arise in this way. However, in the E-dense case, we characterise those E-dense proper covers which can be constructed from such embeddings.     

On Locally Inverse Semigroups Having Weakly E-unitary Covers

Recently, Billhardt has characterized the locally inverse semigroups embeddable in Rees matrix semigroups over generalized inverse semigroups. We prove here that these semigroups are just the locally inverse semigroups having weakly E-unitary covers.     

The Word Problem in the Variety of Inverse Semigroups with Abelian Covers

The variety of inverse semigroups which possess E-unitary coversover Abelian groups coincides with the Mal'cev product of thevariety of semilattices and the variety of Abelian groups,andalso with the variety generated by semidirect products of semilatticesand Abelian groups. We show that this variety (and any varietyof inverse semigroups that contains this variety) has undecidableword problem.     

Finite Proper Covers in a Class of Finite Semigroups with Commuting Idempotents

   Abstract. Weakly left ample semigroups are a class of semigroups that are (2,1) -subalgebras of semigroups of partial transformations, where the unary operation takes a transformation α to the identity map in the domain of α . It is known that there is a class of proper weakly left ample semigroups whose structure is determined by unipotent monoids acting on semilattices or categories. In this paper we show that for every finite weakly left ample semigroup S , there is a finite proper weakly left ample semigroup

Locally Closed Semirings and Iteration Semirings

Locally closed semirings, iteration semirings and Conway semirings play an important role in the algebraic theory of semirings and theoretical computer science. Z. ésik and W. Kuich showed that a locally closed commutative semiring is an iteration semiring (is also a Conway semiring). By study of polynomial semirings and matrix semirings, we obtain new expressions of certain polynomials and show that all matrix semirings over a locally closed semiring are also locally closed, and so a locally closed semiring (which need not be commutative) is an iteration semiring.     

Locally Closed Semirings and Iteration Semirings

Locally closed semirings, iteration semirings and Conway semirings play an important role in the algebraic theory of semirings and theoretical computer science. Z. ésik and W. Kuich showed that a locally closed commutative semiring is an iteration semiring (is also a Conway semiring). By study of polynomial semirings and matrix semirings, we obtain new expressions of certain polynomials and show that all matrix semirings over a locally closed semiring are also locally closed, and so a locally closed semiring (which need not be commutative) is an iteration semiring.     

G-Prime Ideals in Semirings and Their Skew Group Semirings

Semirings have been studied by various researchers either in an attempt to broaden the techniques coming from semigroup theory or generalization of group theory and ring theory. From an algebraic point of view, semirings provide the most natural generalization of the theory of rings. In this article, we generalize some of the results of Lorenz and Passman (1979 Lorenz , M. , Passman , D. S. ( 1979 ). Prime ideals in crossed products of finite groups . Israel J. Math. 33 ( 2 ): 89 – 132 .[Crossref], [Web of Science ®] , [Google Scholar]) for semirings with finite group action on them.     

Congruence-Simple Semirings

Several classes of congruence-simple semirings are characterized and various further examples are constructed. Among others, it is shown that every congruence-simple semiring fits into one of the following three classes: additively idempotent semirings, additively cancellative semirings, additively nil-semirings of index 2.     

Semirings and path spaces

This paper develops a unified algebraic theory for a class of path problems such as that of finding the shortest or, more generally, the k shortest paths in a network; the enumeration of elemementary or simple paths in a graph. It differs from most earlier work in that the algebraic structure appended to a graph or a network of a path problem is not axiomatically given as a starting point of the theory, but is derived from a novel concept called a “path space”. This concept is shown to provide a coherent framework for the analysis of path problems, and hence the development of algebraic methods for solving them.     

Semirings of cyclic types

The authors investigate semirings of cyclic types from the algebraic point of view. To simplify and facilitate the analysis, the local Fourier transform of these semirings is introduced. The authors describe zero divisors, nilpotent elements, invertible elements, idempotents, and the Jacobson radical. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 2, pp. 175–192, 2006.     

半环上的粗糙集

利用同余关系把粗糙集理论引入到半环里,给出了半环中的一个子集的上,下近似的概念,并研究了其一系列性质;另外我们还讨论了半环中的粗糙理想及性质.     

Semirings on cones

A topological semiring is a triplet (S, +, ?) where S is a Hausdorff topological space, “+” and “?” are jointly continuous associative binary operations on S and “?” distributes across “+” on both sides. Recent work by J. Selden [16], K. R. Pearson [13], and Paul H. Karvellas [7] has provided information about and, in some cases, complete characterizations of (S, +, ?) when (S, +) or (S, ?) are specified. Herein, we consider the case in which (S, ?) is the one-point compactification of a closed proper cone in Eⁿ with vector addition extended so that the point at infinity is a zero. Further, if (S, +) is assumed to be a semilattice, we give a complete characterization of (S, +, ?).     

Locally Closed Semirings

 We call a semiring S locally closed if for all a ∈ S there is some integer k such that 1 + a + ⋯ + a ^k  =1 + a + ⋯ + a ^k + 1 . In any locally closed semiring we may define a star operation a ↦ a ^*, where a ^* is the above finite sum. We prove that when S is locally closed and commutative, then S is an iteration semiring. Partially supported by grant no. T30511 from the National Foundation of Hungary for Scientific Research and the Austrian–Hungarian Bilateral Research and Development Fund, no. A-4/1999, and by the Austrian–Hungarian Action Foundation. Partially supported by the Austrian–Hungarian Bilateral Research and Development Fund, no. A-4/1999, and by the Austrian–Hungarian Action Foundation. Received March 16, 2001     

Locally Closed Semirings

 We call a semiring S locally closed if for all a ∈ S there is some integer k such that 1 + a + ⋯ + a ^k  =1 + a + ⋯ + a ^k + 1 . In any locally closed semiring we may define a star operation a ↦ a ^*, where a ^* is the above finite sum. We prove that when S is locally closed and commutative, then S is an iteration semiring.     

DG-Injective Covers, #-Injective Covers

Let R be a left noetherian ring. We show that every complex of left R-modules has a #-injective cover. We use this result to prove that the class of DG-injective complexes is covering if and only if the ring R is regular.