Approximation of ideals in semigroups by soft relations

Article type: Research Article

Authors: Kanwal, Rani Sumaira^{; *} | Shabir, Muhammad

Affiliations: Department of Mathematics, Quaid-i-Azam University Islamabad, Pakistan

Correspondence: [*] Corresponding author. Rani Sumaira Kanwal, Department of Mathematics, Quaid-i-Azam University Islamabad, Pakistan. E-mails: [email protected] and [email protected].

Abstract: Soft set, as a parametrized family of subsets of a crisp universal set, has more ability to handle uncertain information. Pawlak introduced the concept of rough set to deal with uncertainty. He used equivalence relation to approximate a set. Many authors generalized the concept and used binary relations to approximate a set. In this paper, we used soft binary relations to approximate a set. We approximate a set by using the aftersets and foresets. In this way we get two sets of soft sets, called the lower approximation and upper approximation with respect to the aftersets and foresets. We applied these concepts on semigroups and approximations of subsemigroups, left (right) ideals, interior ideals and bi-ideals of semigroups are studied. Moreover, some examples are considered to illustrate the paper. In the last we studied the homomorphic images of the lower and upper approximations.

Keywords: Approximation by soft relations, approximation of ideals in semigroups, homomorphism

DOI: 10.3233/JIFS-18834

Journal: Journal of Intelligent & Fuzzy Systems, vol. 35, no. 3, pp. 3895-3908, 2018