FP-injective semirings,semigroup rings and Leavitt path algebras

In this paper we give characterisations of FP-injective semirings (previously termed “exact” semirings in work of the first author). We provide a basic connection between FP-injective semirings and P-injective semirings, and establish that FP-injectivity of semirings is a Morita invariant property. We show that the analogue of the Faith-Menal conjecture (relating FP-injectivity and self-injectivity for rings satisfying certain chain conditions) does not hold for semirings. We prove that the semigroup ring of a locally finite inverse monoid over an FP-injective ring is FP-injective and give a criterion for the Leavitt path algebra of a finite graph to be FP-injective.     

Idempotent distributive semirings II

In [2] it is shown that every idempotent distributive semiring is the P?onka sum of a semilattice ordered system of idempotent distributive semirings which satisfy the generalized absorption law x+xyx+x=x. We shall show that an idempotent distributive semiring which satisfies the above absorption law must be a subdirect product of a distributive lattice and a semiring which satisfies the additional identity xyx+x+xyx=xyx. Using this, we construct the lattice of all equational classes of idempotent distributive semirings for which the two reducts are normal bands.     

The atomic theory of left division of two-sided ideals of semirings with unit

We consider two-sided ideals of semirings with unit. We study the theory of two-sided ideals in the signature consisting of the binary predicate symbol ? and a binary function symbol that denotes the left division of ideals. We prove the polynomial-time decidability of the problem of deciding whether a given atomic formula in this signature is valid for all semirings with unit and all valuations. A similar result holds for atomic formulas with the right division, but without the left division.     

Reduced Rickart semirings and their functional representations

In this paper, we consider reduced semirings with supplementary conditions of annihilators, namely these are Rickart and weakly Rickart semirings. The main aim of the paper is to study functional representations of semirings. We build two sheaves of semirings and prove that a reduced Rickart semiring is presented by sections of these sheaves. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 2, pp. 205–215, 2007.     

Interval-valued fuzzy k-ideals and k-regularity of semirings

In this paper, we characterize k-regularity of semirings and study some important properties of k-regularity of semirings in terms of interval-valued fuzzy k-ideals of semirings.     

Fuzzy semirings

In this paper we initiate the study of fuzzy semirings and fuzzy A-semimodules where A is a semiring and A-semimodules are representations of A. In particular, semirings all of whose ideals are idempotent, called fully idempotent semirings, are investigated in a fuzzy context. It is proved, among other results, that a semiring A is fully idempotent if and only if the lattice of fuzzy ideals of A is distributive under the sum and product of fuzzy ideals. It is also shown that the set of proper fuzzy prime ideals of a fully idempotent semiring A admits the structure of a topological space, called the fuzzy prime spectrum of A.     

On semirings satisfying the Baer criterion

It is well-known that for modules over rings the Baer injectivity criterion takes place. In this paper we prove that under one additional condition this criterion is also valid for modules over semirings. We prove that a semiring S satisfies the Baer criterion if and only if all injective (with respect to one-sided ideals of S) semimodules satisfy the above condition. We propose a newmethod for constructing semirings satisfying the Baer criterion.     

Pseudocomplementation and minimal prime ideals in semirings

In Section 2 of the present paper, we introduce the concept of pseudocomplementation for semirings and show the semiring version of some known results in lattice theory. We also introduce semirings with pc-functions and prove some interesting results for minimal prime ideals of such semirings. In Section 3, some classical results for minimal prime ideals in ring theory are generalized in the context of semiring theory.     

几类加法非正则半环

In this paper, we introduce Green＇s .-relations on semirings and define [left, right] adequate semirings to explore additively non-regular semirings. We characterize the semirings which are strong b-lattices of [left, right] skew-halfrings. Also, as further generalization, the semirings are described which are subdirect products of an additively commutative idempotent semiring and a [left, right] skew-halfring. We extend results of constructions of generalized Clifford semirings （given by M. K. Sen, S. K. MaRy, K. P. Shum, 2005） and the semirings which are subdirect products of a distributive lattice and a ring （given by S. Ghosh, 1999） to additively non-regular semirings.     

半环半直积的同构定理

半群半直积及封闭性刻划于文[1],为将半直积的研究推广至半环中,本文首先在交换半环与交换半群的直积上定义加法及乘法运算,从而构造了一类半环;半环半直积．并讨论了半环半直积的同构等问题．     

Convolution algebras over complete semirings

Constructions of complete semirings via categorical algebra are provided, that is, we construct convolution algebras of given complete semirings via so-called action networks (i.e., small covariant categories). We aim to get some structural insight into the principle of unrestricted convolution.     

半环的L-模糊理想的范畴性质

首先给出了半环的L-模糊理想同态的定义,在此基础上较系统地讨论了半环的L-模糊理想范畴的性质,证明了此范畴是半环范畴上的一个拓扑结构,并探讨了其中的等子、拉回和乘积等性质.另一方面,给出了半环的L-模糊理想范畴的逆系统的定义,建立了半环的L-模糊理想范畴中逆系统的逆极限结构.特别是在引入两个逆系统之间映射的基础上,得到了两个逆系统的逆极限之间的极限映射.     

Some homological characterizations of semigroups and semirings

In this paper we characterize semirings all of whose p-injective semimodules are injective. We also classify monoids all of whose p-injective acts are injective.     

幺半环上模糊有限状态机的一些性质

状态机的很多性质在计算机等方面有着广泛的应用,因此对状态机的研究具有重要的意义.本文给出了幺半环上模糊有限状态机的概念,对状态之间的等价进行了定义,引入了同态的概念,得到同态定理和满同态分解定理,讨论了幺半环上模糊有限状态机在同态下的交换性质和连通性以及子状态机的可分离性.     

On the additive structure of a class of ordered semirings

In this paper we shall determine the additive structure of totally ordered semirings if its multiplicative structure is a right naturally ordered o-Archimedean semigroup.     

Cyclic semirings with idempotent noncommutative addition

The article discusses the structure of cyclic semirings with noncommutative addition. In the infinite case, the addition is idempotent and is either left or right. Addition of a finite cyclic semirings can be either idempotent or nonidempotent. In the finite additively idempotent cyclic semiring, addition is reduced to the addition of a cyclic subsemiring with commutative addition and an absorbing element for multiplication and the addition of a cycle that is a finite semifield.     

The semiring of continous [0, 1]-valued functions

In this paper, we study the properties of prime ideals in semirings of continuous functions with values in the unit interval [0, 1] on topological spaces. We describe maximal and pure ideals of such semirings. We study homomorphisms of semirings of continuous [0, 1]-valued functions. In terms of semirings of functions we characterize some properties of compacta. We show that the theory of ideals in these semirings differs from the case of rings of continuous functions.     

Subdirectly irreducible commutative multiplicatively idempotent semirings

Commutative multiplicatively idempotent semirings were studied by the authors and F. ?vr?ek, where the connections to distributive lattices and unitary Boolean rings were established. The variety of these semirings has nice algebraic properties and hence there arose the question to describe this variety, possibly by its subdirectly irreducible members. For the subvariety of so-called Boolean semirings, the subdirectly irreducible members were described by F. Guzmán. He showed that there were just two subdirectly irreducible members, which are the 2-element distributive lattice and the 2-element Boolean ring. We are going to show that although commutative multiplicatively idempotent semirings are at first glance a slight modification of Boolean semirings, for each cardinal n > 1, there exist at least two subdirectly irreducible members of cardinality n and at least 2ⁿ such members if n is infinite. For \({n \in \{2, 3, 4\}}\) the number of subdirectly irreducible members of cardinality n is exactly 2.