Annals of Communications in Mathematics

AbstractA semigroup is an algebraic structure that consists of a set and a binary operation that is associative, meaning that the order in which the operations are performed does not affect the outcome. For example, addition and multiplication are associative operations. The term “semigroup” was first used in its modern sense by Harold Hilton in his book on finite groups in 1908 [6]. Semigroups have since then been studied extensively in mathematics, and they have numerous applications in different fields, such as computer science, physics, and economics. Nearenness semigroup is a generalization of semigroups. Near set theory, which is a generalization of rough set theory, is based on the determination of universal sets according to the available information about the objects. Nearenness semigroups extend the concept of nearness from set theory to semigroups. ˙Inan and Öztürk applied the notion of near sets defined by J. F. Peters to the semigroups [10]. Our objective in this paper is to establish the definition of ordered semigroups on weak near approximation spaces. In addition, we investigated certain characteristics of these ordered nearness semigroups.

AbstractBCK/BCI-algebra is a class of logical algebras that was defined by K. Iseki and S. Tanaka. BCK-algebras have a lot of generalizations. One of them is d-algebras. Near set theory which is a generalization of rough set theory. This theory is based on the determination of universal sets according to the available information of the objects. Based on the image analysis, the near set theory was created. ¨Ozt¨urk applied the notion of near sets defined by J. F. Peters to the theory of d-algebras. In this paper we introduce upper-nearness d-ideal, upper-near (upper-nearness) d#- ideal, upper-near (upper-nearness) d∗-ideal. We explored what conditions we should put on the ideal for quotient nearness d-algebra to become an nearness d-algebra again. More- over, we introduce quotient nearness d-algebras with the help of upper-nearness d∗-ideals of nearness d-algebras. Finally, we present a theorem involving the canonical homomor- phism and the structure of the kernel for nearness d-algebras. Thus, we aim to make preliminary preparations for proving isomorphism theorems for nearness d-algebras.