ON COEQUALITY RELATION AND ITS COPARTITION ON SET WITH APARTNESS

Bull. Int. Math. Virtual Inst., 2019

This investigation is in the mathematics based on the Intuition-istic logic. A relation ρ is a coequality relation if it is consistent, symmetric and co-transitive. For a coequality relation ρ on a set X with apartness we analyze the family Cop(X) of all classes of the relation. Characteristics of this family allow us to introduce a new concept, 'co-partition' in set with apaerness-a specific family of proper subsets. In addition, a connection between the family of all coequality relations and the family of all co-partitions is given. At the end of this article, some examples and applications in the semigroups with apartness theory are given.

In this paper, basing our consideration on the sets with the apart-ness relation, we analyze characteristics of some special relations to these sets such as co-order and co-quasiorder and coequality relations. In addition, we analyze two special classes of subsets, co-filters and co-ideals, of ordered set under a co-quasiorder relation. This investigation is into the Bishop's Constructive mathematics.

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.

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1988

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.

This investigation is in the Bishop's constructive mathematics. We discuss about co-order, co-quasiorder and coequality relations on set X with appartness. A connection between the family of all co-quasiorder relations and the family of all coequality relations on set X is given. In addition, a connection between the family of all co-quasiorder relations included in the co-order α and the family of all regular coequality relation on X with respect to α is also given.

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.

Setting of this research is Bishop's constructive mathematics, the mathematics developed on Intuitionistic logic. If (X, =, =, θ) is an anti-ordered set, for a coequality q on X we say that it is strongly regular if it is regular and θ • q C ⊆ q C • θ holds. In this case, θ • q C is a quasi-antiorder relation on X such that the relation Θ = π • θ • π −1 on X/q is the maximal anti-order on X/q.

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.