Regular Semirings Research Articles

Generalized Soft Intersectional Ideals in Ternary Semirings

In this paper we introduce the notions of soft intersectional ternary subsemirings and soft ntersectional ideals in ternary semirings. We also discuss some basic results associated with these notions. In the last part of the paper we characterize regular and weakly regular ternary semirings by their soft intersectional ideals

On quasi completely regular semirings

We extend the concepts of a completely π-regular semigroup and a GV semigroup to semirings and find a semiring analogue of a structure theorem on GV semigroups. We also show that a semiring S is quasi completely regular if and only if S is an idempotent semiring of quasi skew-rings.

Application of L-fuzzy soft sets to semirings

L-fuzzy soft set is a generalization of fuzzy soft set. In the recent paper L-fuzzy soft sets have emerged as a new tool to characterize semirings. Concepts of L-fuzzy soft subsemirings and L-fuzzy ideals are introduced and some of their properties are studied. It is seen that L-fuzzy soft ideals have ability to characterize regular and weakly regular semirings. Finally fully idempotent semirings are characterized with the help of L-fuzzy soft ideals.

The Clifford semiring congruences on an additive regular semiring

The Clifford semiring congruences on an additive regular semiring

THE ZELEZNIKOW PROBLEM ON A CLASS OF ADDITIVELY IDEMPOTENT SEMIRINGS

AbstractA semiring is a set $S$ with two binary operations $+ $ and $\cdot $ such that both the additive reduct ${S}_{+ } $ and the multiplicative reduct ${S}_{\bullet } $ are semigroups which satisfy the distributive laws. If $R$ is a ring, then, following Chaptal [‘Anneaux dont le demi-groupe multiplicatif est inverse’, C. R. Acad. Sci. Paris Ser. A–B 262 (1966), 274–277], ${R}_{\bullet } $ is a union of groups if and only if ${R}_{\bullet } $ is an inverse semigroup if and only if ${R}_{\bullet } $ is a Clifford semigroup. In Zeleznikow [‘Regular semirings’, Semigroup Forum 23 (1981), 119–136], it is proved that if $R$ is a regular ring then ${R}_{\bullet } $ is orthodox if and only if ${R}_{\bullet } $ is a union of groups if and only if ${R}_{\bullet } $ is an inverse semigroup if and only if ${R}_{\bullet } $ is a Clifford semigroup. The latter result, also known as Zeleznikow’s theorem, does not hold in general even for semirings $S$ with ${S}_{+ } $ a semilattice Zeleznikow [‘Regular semirings’, Semigroup Forum 23 (1981), 119–136]. The Zeleznikow problem on a certain class of semirings involves finding condition(s) such that Zeleznikow’s theorem holds on that class. The main objective of this paper is to solve the Zeleznikow problem for those semirings $S$ for which ${S}_{+ } $ is a semilattice.

On invertible matrices over commutative semirings

In this article, the invertible matrices over commutative semirings are studied. Some properties and equivalent descriptions of the invertible matrices are given and the inverse matrix of an invertible matrix is presented by analogues of the classic adjoint matrix. Also, Cramer's rule over a commutative semiring is established. The main results obtained in this article generalize the corresponding results for matrices over commutative rings, for lattice matrices, for incline matrices, for matrices over zerosumfree semirings and for matrices over additively regular semirings.

Equivalent Identities on Semirings

In this paper mainly we have obtained equivalent conditions on semirings, regular semirings and Idempotent semirings.

On Subtractive Semisimple Semirings

Among other results on subtractive semimodules and semirings, we present various (homological) characterizations of subtractive semisimple semirings. Also, we give complete descriptions of finite subtractive semisimple as well as additively regular (in particular, additively idempotent) subtractive semisimple semirings.

Some abstract algebra from the elementary calculus course

We examine in detail some aspects of the algebra of functions (with respect to pointwise addition and multiplication) as encountered in an elementary calculus course. The set of functions whose domains have finite complements furnishes an example of an additively regular semiring. Related examples can be obtained when ‘finite’ is replaced by various other properties. Consideration of these structures leads to a clarification of the relationship between the set of rational real functions and the quotient field of the polynomial ring over the reals. We give a coherent account of some elementary integration based on the description of a definite integral as a group coset.

On quasi‐ideals of semirings

Several statements on quasi‐ideals of semirings are given in this paper, where these semirings may have an absorbing element O or not. In Section 2 we characterize regular semirings and regular elements of semi‐rings using quasi‐ideals (cf. Thms. 2.1, 2.2 and 2.7). In Section 3 we deal with (O−)minimal and canonical quasi‐ideals. In particular, if the considered semiring S is semiprime or quasi‐reflexive, we present criterions which allow to decide easily whether an (O−)minimal quasi‐ideal of S is canonical (cf. Thms. 3.4 and 3.8). If S is an arbitrary semiring, we prove that for (O−)minimal left and right ideals L and R of S the product 〈RL〉⫅L ⋂ R is either {O} or a canonical quasi‐ideal of S (Thm. 3.9). Moreover, for each canonical quasi‐ideal Q of a semiring S and each element a ∈ S, Qa is either {O} or again a canonical quasi‐ideal of S (Thm. 3.11), and the product 〈Q1Q2〉 of canonical quasi‐ideals Q1, Q2 of S is either {O} or again a canonical quasi‐ideal of S (Thm. 3.12). Corresponding results to those given here for semirings are mostly known as well for rings as for semigroups, but often proved by different methods. All proofs of our paper, however, apply simultaneously to semirings, rings and semigroups (cf. Convention 1.1), and we also formulate our results in a unified way for these three cases. The only exceptions are statements on semirings and semigroups without an absorbing element O, which cannot have corresponding statements on rings since each ring has its zero as an absorbing element.

Representations of weakly regular semirings by sections in a presheaf

(1993). Representations of weakly regular semirings by sections in a presheaf. Communications in Algebra: Vol. 21, No. 8, pp. 2819-2835.

Regular semirings

Regular semirings