Γ -SEMIGROUPS WITH APARTNESS

Journal of Physics: Conference Series

The theory of constructive semigroups with apartness is a new approach to semigroup theory, and not a new class of semigroups. Of course, our work is partly inspired by classical semigroup theory, but, on the other hand, it is distinguished from it by two significant aspects: first, we use intuitionistic logic rather than classical, secondly, our work is based on the notion of apartness (between elements, elements and sets). Here, the focus is on E. Bishop's approach to constructive mathematics (BISH), [4].

We examine basic notions of special subsets and orders in the context of semigroups with apartness and prove constructive analogues of some classical theorems relating such subsets and orders.

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.

Semigroup Forum, 2016

We examine basic notions of special subsets and orders in the context of semigroups with apartness and prove constructive analogues of some classical theorems relating such subsets and orders.

Fundamenta Informaticae, 2022

This paper has several purposes. We present through a critical review the results from already published papers on the constructive semigroup theory, and contribute to its further development by giving solutions to open problems. We also draw attention to its possible applications in other (constructive) mathematics disciplines, in computer science, social sciences, economics, etc. Another important goal of this paper is to provide a clear, understandable picture of constructive semigroups with apartness in Bishop’s style both to (classical) algebraists and the ones who apply algebraic knowledge.

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..

2021

We give a constructive treatment of some basic concepts and results in semigroup theory. Focusing on semigroups equipped with an apartness relation, we give analogues, from the point of view of apartness, of several classical constructions and results, including transitive closure and congruence closure, free semigroups, periodicity, Rees factors, and Green's relations.

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.

Acta Universitatis Apulensis, Mathematics and Informatics, ISSN 1582-5329, 2021

Implicative semigroups with apartness have been introduced in 2016 by this author who then analyzed them in several papers. Here the author continues his research on such semigroups dealing with co-congruences. In addition, he constructs two quotient structures of these semigroups, one of which has no counterpart in the classical theory of implicative semigroups. Additionally, one form of the First Isomorphism Theorem for this type of semigroups is shown, which can be seen as an extension of corresponding theorem in the classical Semigroup Theory.