THE FIRST ISOMORPHISM THEOREM OF IMPLICATIVE SEMIGROUP WITH APARTNESS (original) (raw)

THE THIRD ISOMORPHISM THEOREM FOR IMPLICATIVE SEMIGROUP WITH APARTNESS

Bulletin of the vInternationalMathematical Virtual Institute (ISSN 2303-4874 (p), ISSN (o) 2303-4955), 2021

Implicative semigroups with apartness have been introduced in 2016 by this author who then analyzed them in several papers. In this paper, a form of the third isomorphism theorem for this type of semigroups is shown, which has no counterpart in the classical semigroup theory.

AN INTRODUCTION TO IMPLICATIVE SEMIGROUPS WITH APARTNESS

The setting of this research is Bishop's constructive mathematics. Following ideas of Chan and Shum, exposed in their famous paper " Homomorphisms of implicative semigroups " , we discuss the structure of implicative semigroups on sets with tight apartness. Moreover, we use anti-orders instead of partial orders. We study concomitant issues induced by existence of apartness and anti-orders giving some specific characterizations of these semigroups. In addition, we introduce the notion of anti-filter in implicative semigroups and give some equivalent conditions that the inhabited real subset of an implicative semigroup is an ordered anti-filter.

A REMARK ON CO-IDEALS IN IMPLICATIVE SEMIGROUPS WITH APARTNESS

Acta Universitatis Apulensis, Mathematics and Informatics, ISSN 1582-5329, 2020

In 2001, Jun and Kim introduced the concept of ideals in implicative semigroups. Implicative semigroups with apartness were introduced and analyzed in 2016-19 by the author in his four published articles. In the last of these (D. A. Romano. On co-ideals in implicative semiroups with apartness. Turk. J. Math. Comput. Sci., 11(2)(2019), 101-106), the author analyzes the concept of co-ideals in such semigroups. In this paper, as a continuation of the aforementioned articles, the author shows that the concept of co-ideals is consistent with the concept of ideals, introduced by Jun and Kim. The author demonstrates this by proving that a strong complement of a co-ideal in an implicative semigroup with apartness is an ideal in terms of Jun and Kim..

A NEW CO-FILTER IN IMPLICATIVE SEMIGROUPS WITH APARTNESS

Acta Universitatis Apulensis, 2020

The concept of implicative semigroups with apartness in Bishop's constructive algebra was introduced and analyzed by the author in several of his papers. In some of them, the concepts of co-ideals and co-filters in such semigroups were introduced and analyzed. This paper continues investigation of implicative semigroups with tight apartness and of their co-filters. In particular, the notion of an implicative co-filters was introduced and analyzed. 2010 Mathematics Subject Classification: 03F65, 20M12, 06F05, 06A06, 06A12.

Semigroups of left quotients—the uniqueness problem

Proceedings of the Edinburgh Mathematical Society, 1992

Let S be a subsemigroup of a semigroup Q. Then Q is a semigroup of left quotients of S if every element of Q can be written as a*b, where a lies in a group -class of Q and a* is the inverse of a in this group; in addition, we insist that every element of S satisfying a weak cancellation condition named square-cancellable lie in a subgroup of Q.J. B. Fountain and M. Petrich gave an example of a semigroup having two non-isomorphic semigroups of left quotients. More positive results are available if we restrict the classes of semigroups from which the semigroups of left quotients may come. For example, a semigroup has at most one bisimple inverse ω-semigroup of left quotients. The crux of the matter is the restrictions to a semigroup S of Green's relations ℛ and ℒ in a semigroup of quotients of S. With this in mind we give necessary and sufficient conditions for two semigroups of left quotients of S to be isomorphic under an isomorphism fixing S pointwise.The above result is then u...

Semilattice Co-Congruence in Gamma\GammaGamma-Semigroups

Turkish Journal of Mathematics and Computer Science, 2020

As a generalization of semigroups, Sen, in 1981, introduced the concept of Γ-semigroups. In the author's paper (D A. Romano. Γ-semigroups with apartness. Bull. Allahabad Math. Soc., 34(1)(2019), 71-83.) it is introduced and analyzed the concept of Γ-semigroups with apartness in Bishop's constructive framework. In this article, as a continuation of previous research, the concept of co-congruences in Γ-semigroups is introduced and analyzed. Additionally, it is investigated (co-ordered) semillatice co-congruence on (co-ordered) Γ-semigroup with apartness.

Γ -SEMIGROUPS WITH APARTNESS

Bulletin of the Allahabad Mathematical Society (Bulletin, Al.M.S.) ISSN No. 0971-0493, 2019

As a generalization of a semigroup, Sen 1981 introduced the con- cept of 􀀀 - semigroups. In this paper we analyze the concept of 􀀀 - semigroups with apartness. The logical setting of this article is the Intuitionistic logic and the principled-philosophical environment is the Bishop's constructive algebra orientation. In this algebraic orientation, the concept of appartnesses in sets is a fundamental concept, just as it is the concept of equality in the classical algebra. In addition, we introduce the concepts of co-ideals in such semigroups and give some properties of the family of such substructures. In addition to introducing the concept of 􀀀 - cocongruences of 􀀀 - semigroup, we also by analyzing the connection between strong extensional homomorphisms of 􀀀 - semigroups and congruences and co-congruences, we prove some assertions in related with co-ideals in such semigroups.

STRONGLY EXTENSIONAL HOMOMORPHISM OF IMPLICATIVE SEMIGROUPS WITH APARTNESS

Sarajevo Journal of Mathematics, 2017

The setting of this research is the Bishop's constructive mathematics. Following the ideas of Chan and Shum, exposed in their famous paper " Homomorphisms of implicative semigroups " ([10]), we discuss the ho-momorphisms between the implicative semigroups S and T with apartness as a continuation of our recently published article [26]. The specificity of this research is the application of Intuitionistic logic instead of Classical. In addition , we concentrate on the structure of the implicative semigroups with appartness and then on their interaction which do not appear in the classical case ([10]). In this paper, the concept of ordered anti-filter has been associated with strongly extensional homomorphisms between such semigroup.

Semigroups with finitely generated universal left congruence

Monatshefte für Mathematik

We consider semigroups such that the universal left congruence ω is finitely generated. Certainly a left noetherian semigroup, that is, one in which all left congruences are finitely generated, satisfies our condition. In the case of a monoid the condition that ω is finitely generated is equivalent to a number of pre-existing notions. In particular, a monoid satisfies the homological finiteness condition of being of type left-FP 1 exactly when ω is finitely generated. Our investigations enable us to classify those semigroups such that ω is finitely generated that lie in certain important classes, such as strong semilattices of semigroups, inverse semigroups, Rees matrix semigroups (over semigroups) and completely regular semigroups. We consider closure properties for the class of semigroups such that ω is finitely generated, including under morphic image, direct product, semi-direct product, free product and 0-direct union. Our work was inspired by the stronger condition, stated for monoids in the work of White, of being pseudo-finite. Where appropriate, we specialise our investigations to pseudo-finite semigroups and monoids. In particular, we answer a question of Dales and White concerning the nature of pseudo-finite monoids. Keywords Monoids • Semigroups • Left congruences • Finitely generated • FP 1 • Pseudo-finite Communicated by J. S. Wilson.

THE THIRD ISOMORPHISM THEOREM FOR (CO-ORDERED) Γ-SEMIGROUPS WITH APARTNESS

Annals of Communications in Mathematics (ACM), ISSN 2582-0818, 2020

The notion of Γ-semigroups has been introduced by M. K. Sen and N. K. Saha. The concept of (co-ordered) Γ-semigroups with apartness in Bishop's constructive algebra was introduced by this author. Many classical notions and processes of semigroups and Γ-semigroups have been extended to (co-ordered) Γ-semigroups with apartness such as ideals, filters and the first theorem of isomorphism of this class of algebraic structures. In this paper, as a continuation of earlier research, the author designs a form of the third iso-morphism theorems for Γ-semigroups and co-ordered Γ-semigroups with apartness which does not have its counterpart in the classical Γ-semigroup theory.