Soft Maximal and Irreducible Int-Ideals of a Ring
18 Mar 2020-New Mathematics and Natural Computation (World Scientific Publishing Company)-Vol. 16, Iss: 01, pp 37-52
TL;DR: The concepts of soft maximal int-ideal and soft irreducible int-Ideal of a ring are introduced and some properties of these notions have been studied.
Abstract: In this paper, the concepts of soft maximal int-ideal and soft irreducible int-ideal of a ring are introduced and some properties of these notions have been studied. Also we point out some facts of...
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TL;DR: It is shown that the concepts of soft elementwise ring and soft int-ring of a ring are equivalent and the notions of minimal generating set for a soft int -ideal and soft principal int-ideal are presented.
Abstract: In this paper, the notion of soft elementwise ring is introduced and it is shown that the concepts of soft elementwise ring and soft int-ring of a ring are equivalent. The concept of soft direct su...
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TL;DR: The main purpose of this paper is to introduce the basic notions of the theory of soft sets, to present the first results of the the theory, and to discuss some problems of the future.
Abstract: The soft set theory offers a general mathematical tool for dealing with uncertain, fuzzy, not clearly defined objects. The main purpose of this paper is to introduce the basic notions of the theory of soft sets, to present the first results of the theory, and to discuss some problems of the future.
3,759 citations
TL;DR: The basic properties of soft sets are introduced, and compare soft sets to the related concepts of fuzzy sets and rough sets, and a definition of soft groups is given.
1,012 citations
01 Jul 2010
TL;DR: A possible fusion of fuzzy sets and rough sets is proposed to obtain a hybrid model called rough soft sets, based on a Pawlak approximation space, and a concept called soft–rough fuzzy sets is initiated, which extends Dubois and Prade's rough fuzzy sets.
Abstract: Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Dubois and Prade investigated the problem of combining fuzzy sets with rough sets. Soft set theory was proposed by Molodtsov as a general framework for reasoning about vague concepts. The present paper is devoted to a possible fusion of these distinct but closely related soft computing approaches. Based on a Pawlak approximation space, the approximation of a soft set is proposed to obtain a hybrid model called rough soft sets. Alternatively, a soft set instead of an equivalence relation can be used to granulate the universe. This leads to a deviation of Pawlak approximation space called a soft approximation space, in which soft rough approximations and soft rough sets can be introduced accordingly. Furthermore, we also consider approximation of a fuzzy set in a soft approximation space, and initiate a concept called soft---rough fuzzy sets, which extends Dubois and Prade's rough fuzzy sets. Further research will be needed to establish whether the notions put forth in this paper may lead to a fruitful theory.
607 citations
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TL;DR: This paper initiates the study of soft semirings by using the soft set theory, and the notions of soft Semirings, soft subsemirings,soft ideals, idealistic softSemirings and soft semiring homomorphisms are introduced, and several related properties are investigated.
Abstract: Molodtsov introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainty. In this paper, we initiate the study of soft semirings by using the soft set theory. The notions of soft semirings, soft subsemirings, soft ideals, idealistic soft semirings and soft semiring homomorphisms are introduced, and several related properties are investigated.
579 citations
TL;DR: Initial concepts of soft rings are introduced and some operations on soft sets are defined in this paper.
Abstract: Molodtsov (1999) introduced the concept of soft sets in [1]. Then, Maji et al. (2003) defined some operations on soft sets in [2]. Aktas and Ca@?man (2007) defined the notion of soft groups in [3]. Finally, soft semirings are defined by Feng et al. (2008) in [5]. In this paper, we have introduced initial concepts of soft rings.
348 citations