Regular Semirings Research Articles

Right $\pi$-Regular Semirings

We prove the following results (1) If $R$ is a right and left $\pi$-regular semiring then $R$ is a $\pi$-regular semiring. (2) If $R$ is an additive cancellative semiprime, right Artinian or right $\pi$-regular right Noetherian semiring then $R$ is semisimple. (3) Let $I$ be a partitioning ideal of a semiring $R$ such that $Q=(R-I)\cup \{0\}$. If $I$ is a right regular ideal and the quotient semiring $R/I$ is right $\pi$-regular then $R$ is a right $\pi$-regular semiring. 2000 Mathematics Subject Classification. 16Y60

Fuzzy Soft Tri-quasi Ideals of Regular Semirings

In this paper, we study some of the properties fuzzy soft tri-quasi ideals of a semiring. Tri-quasi ideals generalize, bi-ideals, quasi-ideals and interior ideals. We characterize regular semiring using fuzzy soft tri-quasi ideals and prove that, if [Formula: see text] is a fuzzy soft tri-quasi ideal over a regular semiring [Formula: see text] then [Formula: see text] is a fuzzy soft tri-ideal over [Formula: see text]

Subdirect product decompositions of additively regular semirings

Subdirect product decompositions of additively regular semirings

Structural Properties of Completely Regular Semiring

Structural Properties of Completely Regular Semiring

Retractive nil-extensions of completely regular semirings

We extend the concept of nil-extensions and retractive nil-extensions of different classes of semigroups to semirings and find a semiring analog of structure theorems on nil-extensions of different classes of additively regular semirings.

Characterizations of Left H-Clifford Semirings by Their H-Ideals

The main aim of this research is to introduce Lefth−Clifford Semi-rings. Using some basic properties ofh−regular semi-rings we shall investigate several properties of Lefth−Clifford semi-rings and their characterizations. We will also establish that a semi-group Q will be a Left Clifford Semi-group iff the semi-group P (Q) of all subsets of Q is a Lefth−Clifford Semiring. Also, this research will investigate the distributive congruences of regular semi-rings.

On a Matrix over NC and Multiset NC Semigroups

In this paper, we define a matrix over neutrosophic components (NCs), which was built using the four different intervals (0,1), [0,1), (0,1], and [0,1]. This definition was made clear by introducing some examples. Then, the study of the algebraic structure of matrices over NC under addition modulo 1, the usual product, and product by using addition modulo 1 was introduced, from which it was found that the matrix over NC built using interval [0,1) happens to be an abelian group under addition modulo 1. Furthermore, it is proved that the matrix over NC defined on the interval [0,1) is not a regular semiring. Also, we define a matrix over multiset NC semigroup using the interval [0,1). Moreover, we define a matrix over m‐multiplicity multiset NC semigroup for finite m. Several interesting properties are discussed for the three structures. It was concluded that the last two structures are semigroups and semirings under addition modulo 1 and usual product, respectively.

A New Approach to Evaluate Regular Semirings in terms of Bipolar Fuzzy k‐Ideals Using k‐Products

In this paper, we provide a generalized form of ideals that is k‐ideals of semirings with the combination of a bipolar fuzzy set (BFS). The BFS is a generalization of fuzzy set (FS) that deals with uncertain problems in both positive and negative aspects. The main theme of this paper is to present the idea of (α, β)‐bipolar fuzzy k‐subsemiring (k‐BFSS), (α, β)‐bipolar fuzzy k‐ideals (k‐BFIs), and (α, β)‐bipolar fuzzy k‐bi‐ideals (k‐BFbIs) in semirings by applying belongingness (∈) and quasi‐coincidence (q) of the bipolar fuzzy (BF) point. After that, upper and lower parts of k‐product of BF subsets of semirings are introduced. Lastly, the notions of k‐regular and k‐intraregular semirings in terms of (∈, ∈∨q)‐k–BFIs and (∈, ∈∨q)‐k–BFbIs are characterized.

Additively orthodox semirings with special transversals

<abstract><p>A semiring $ (S, +, \cdot) $ is called additively orthodox semiring if its additive reduct $ (S, +) $ is a orthodox semigroup. In this paper, by introducing some special semiring transversals as the tools, the constructions of additively orthodox semirings with a skew-ring transversal or with a generalized Clifford semiring transversal are established. Meanwhile, it is shown that an additively orthodox semiring with a generalized Clifford semiring transversal is a b-lattice of additively orthodox semirings with skew-ring transversals. Consequently, the corresponding results of Clifford semirings and generalized Clifford semirings in reference (M. K. Sen, S. K. Maity, K. P. Shum, Clifford semirings and generalized Clifford semirings, Taiwan. J. Math., 9 (2005), 433–444) and completely regular semirings in reference (S. K. Maity, M. K. Sen, K. P. Shum, On completely regular semirings, Bull. Cal. Math. Soc., 98 (2006), 319–328) are extended and strengthened.</p></abstract>

Extended Green’s relations on Mn(D), a characterization

In this paper, we consider the semiring [Formula: see text] of all [Formula: see text] matrices over a distributive lattice [Formula: see text] and extended Green’s relations [Formula: see text] and [Formula: see text] using [Formula: see text]-ideals. A (left, right) ideal [Formula: see text] of a semiring [Formula: see text] is called a (left, right) [Formula: see text]-ideal if [Formula: see text], where [Formula: see text]. We define [Formula: see text] and [Formula: see text] on a [Formula: see text]-regular semiring [Formula: see text], in which [Formula: see text] is a semilattice, as follows: [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text], where [Formula: see text] is the left [Formula: see text]-ideal generated by [Formula: see text] and [Formula: see text] is the right [Formula: see text]-ideal generated by [Formula: see text]. Here we characterize [Formula: see text] and [Formula: see text] in [Formula: see text] in terms of rows and columns of the matrices.

Subdirect products of an idempotent semiring and a b-lattice of skew-rings

Bandelt and Petrich [Subdirect products of rings and distributive lattices, Proc. Edinburgh Math. Soc. (2) 25(2) (1982) 155–171] characterized a class of additive inverse semirings which are subdirect products of a distributive lattice and a ring. The aim of this paper is to characterize a class of additively regular semirings which are subdirect products of an idempotent semiring and a [Formula: see text]-lattice of skew-rings.

A Hesitant Intuitionistic Fuzzy Set Approach to Study Ideals of Semirings

The classical set theory was extended by the theory of fuzzy set and its several generalizations, for example, intuitionistic fuzzy set, interval valued fuzzy set, cubic set, hesitant fuzzy set, soft set, neutrosophic set, etc. In this paper, the author has combined the concepts of intuitionistic fuzzy set and hesitant fuzzy set to study the ideal theory of semirings. After the introduction and the priliminary of the paper, in Section 3, the author has defined hesitant intuitionistic fuzzy ideals and studied several properities of it using the basic operations intersection, homomorphism and cartesian product. In Section 4, the author has also defined hesitant intuitionistic fuzzy bi-ideals and hesitant intuitionistic fuzzy quasi-ideals of a semiring and used these to find some characterizations of regular semiring. In that section, the author also has discussed some inter-relations between hesitant intuitionistic fuzzy ideals, hesitant intuitionistic fuzzy bi-ideals and hesitant intuitionistic fuzzy quasi-ideals, and obtained some of their related properties.

The congruence 𝒴∗ on quasi completely regular semirings

A semiring [Formula: see text] is said to be a quasi-orthodox semiring if for any [Formula: see text], there exists some positive integer [Formula: see text] such that [Formula: see text]. In this paper, we investigate the congruence generated by [Formula: see text] on quasi completely regular semirings with and without the special conditions like being quasi-orthodox. We further establish that the interval which [Formula: see text] belongs to is [Formula: see text].

Congruences on quasi completely regular semirings

Congruences on quasi completely regular semirings

On semirings all of whose semimodules have injective envelopes

It was shown in [Y. Katsov, Tensor products and injective envelopes of semimodules over additively regular semirings, Algebra Colloq. 4 (1997) 121–131.] that each semimodule over an additively regular semiring has an injective envelope. We show the converse statement is true as well; moreover, the result holds even if each finitely generated semimodule has an injective envelope.

Von Neumann Regular Semiring

The aim of this action is a study and investigate “Von Neumann regular” semirings, some related concepts, e.g. reduced semirings; duo semiring, quasi-duo, and weakly duo semirings; regular, weakly regular and strongly regular semirings, also investigated. Some known results related to those concepts in rings were converted to semirings. Another aim of this paper is characterization Von Neumann Regular condition by the principal right ideal generated by an idempotent element.

Characterizations of ordered $h$-regular semirings by ordered $h$-ideals

The objective of this paper is to study the ordered $h$-regular semirings by the properties of their ordered $h$-ideals. It is proved that each $h$-regular ordered semiring is an ordered $h$-regular semiring but the converse does not follow. Important theorems relating to basic properties of the operator clousre and $h$-regular semirings are given. It is also proved that each regular ordered semiring is an ordered $h$-regular semiring but the converse does not hold. The classifications of the left and the right ordered $h$-regular semirings and the left and the right ordered $h$-weakly regular semirings are also presented.

On characterization of minimal $k$-bi-ideals in $k$-regular and completely $k$-regular semirings

On characterization of minimal $k$-bi-ideals in $k$-regular and completely $k$-regular semirings

Characterizations of ordered k-regular semirings by ordered k-ideals

An ordered semiring [Formula: see text] is called ordered [Formula: see text]-regular if for every element [Formula: see text] of [Formula: see text] there exist [Formula: see text] with [Formula: see text] such that [Formula: see text]. An ordered ideal [Formula: see text] of [Formula: see text] is called an ordered [Formula: see text]-ideal, if [Formula: see text] and [Formula: see text] for some [Formula: see text] then [Formula: see text]. In this work, we characterize ordered [Formula: see text]-regular semirings using their ordered [Formula: see text]-ideals. Moreover, characterizations of left(right) ordered [Formula: see text]-regular semirings and left(right) ordered [Formula: see text]-weakly regular semirings are investigated.

Classes of Regular Semiring

In this paper, it was proved that, if S is a Right regular semiring, then (S, +) is a band under the following cases. 1. If S is multiplicatively subidempotent semiring. 2. If S is an almost idempotent semiring.