The philosophy of soft sets is founded on the fundamental idea of parameterization, while Pawlak's rough sets put more emphasis on the importance of granulation. As a multivalued extension of soft sets, the newly emerging concept called $N$ -soft sets can provide a finer granular structure with higher distinguishable power. This article offers a fresh insight into rough set theory from the perspective of $N$ -soft sets. We reveal a close connection between $N$ -soft sets and rough structures of various types. First, we show how the corresponding structures of Pawlak's rough sets, tolerance rough sets, and multigranulation rough sets can be derived from a given $N$ -soft set. Conversely, we investigate the representation of these distinct rough structures using the corresponding notions derived from suitable $N$ -soft sets. The applicability of these theoretical results is highlighted with a case study using real data regarding hotel rating. The established two-way correspondences between $N$ -soft sets and diverse rough structures are constructive, which can bridge the gap between seemingly disconnected disciplines and hopefully nourish the development of both rough sets and soft sets.

An $N$ -Soft Set Approach to Rough Sets

In this work we define soft covering based rough set and present its related
 properties. Also we present an example in medicine which aims to find the
 patients with high prostate cancer risk. Our datas are 56 patients from
 Selcuk University Meram Medicine Faculty.

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An application of multicriteria group decision making by soft covering based rough sets

The notions of union-soft semigroups, union-soft l-ideals, and union-soft r-ideals are introduced, and related properties are investigated. Characterizations of a union-soft semigroup, a union-soft l-ideal, and a union-soft r-ideal are provided. The concepts of union-soft products and union-soft semiprime soft sets are introduced, and their properties related to union-soft l-ideals and union-soft r-ideals are investigated. Using the notions of union-soft l-ideals and union-soft r-ideals, conditions for an ordered semigroup to be regular are considered. The concepts of concave soft sets and critical soft points are introduced, and their properties are discussed.

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Concave soft sets, critical soft points, and union-soft ideals of ordered semigroups.

In this paper, we dene soft intersection semigroups, soft intersection left (right, two-sided) ideals and bi-ideals of semigroups, give their properties and interrelations and we characterize regular, intra-regular, completely regular, weakly regular and quasi-regular semigroups in terms of these ideals.

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Soft intersection semigroups, ideals and bi-ideals; a new application on semigroup theory I

In this book, the authors define several new types of soft neutrosophic algebraic structures over neutrosophic algebraic structures and we study their generalizations. These soft neutrosophic algebraic structures are basically parameterized collections of neutrosophic sub-algebraic structures of the neutrosophic algebraic structure. An important feature of this book is that the authors introduce the soft neutrosophic group ring, soft neutrosophic semigroup ring with its generalization, and soft mixed neutrosophic N-algebraic structure over neutrosophic group ring, then the neutrosophic semigroup ring and mixed neutrosophic N-algebraic structure respectively.

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Soft Neutrosophic Algebraic Structures and Their Generalization, Vol. 2

Binary relations, in particular, equivalence relations play an important role in both mathematics and information sciences. The concept of soft sets was initiated by Molodtsov as a general mathematical frameworkfordealingwithuncertainty. Thepresentpaperestablishesapossibleconnectionbetweenbinary relations and soft sets. The concept of soft binary relations is introduced and some related properties are investigated. It is shown that any fuzzy relation may be considered as a soft binary relation. Moreover, we discuss the application of soft binary relations in semigroup theory. We consider soft congruence relations over semigroups and show that all soft congruence relations over a semigroup with a fixed parameter set form a lattice. Finally, the notion of soft homomorphisms is presented and isomorphism theorems for soft semigroups are established based on soft congruence relations.

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Soft relations applied to semigroups

In this article we have defined two new subclasses of analytic functions k−Sq[A,B] and k−Kq[A,B] by using q-difference operator in an open unit disk. Furthermore, the necessary and sufficient conditions along with certain other useful properties of these newly defined subclasses have been calculated by using q-difference operator.

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Irfan Muhammad Ali

Papers

Soft relations applied to semigroups

Generalization of k-Uniformly Starlike and Convex Functions Using q-Difference Operator