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Journal Article•DOI•

Soft semirings

Feng Feng, Young Bae Jun1, Xianzhong Zhao2•
Gyeongsang National University1, Northwest University (China)2
01 Nov 2008-Computers & Mathematics With Applications-Vol. 56, Iss: 10, pp 2621-2628
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.
Citations
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Journal Article•DOI•
On some new operations in soft set theory

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M. Irfan Ali1, Feng Feng, Xiaoyan Liu, Won Keun Min2, Muhammad Shabir1 •
Quaid-i-Azam University1, Kangwon National University2
01 May 2009-Computers & Mathematics With Applications
TL;DR: This paper points out that several assertions in a previous paper by Maji et al. are not true in general, and gives some new notions such as the restricted intersection, the restricted union, therestricted difference and the extended intersection of two soft sets.
Abstract: Molodtsov introduced the theory of soft sets, which can be seen as a new mathematical approach to vagueness. In this paper, we first point out that several assertions (Proposition 2.3 (iv)-(vi), Proposition 2.4 and Proposition 2.6 (iii), (iv)) in a previous paper by Maji et al. [P.K. Maji, R. Biswas, A.R. Roy, Soft set theory, Comput. Math. Appl. 45 (2003) 555-562] are not true in general, by counterexamples. Furthermore, based on the analysis of several operations on soft sets introduced in the same paper, we give some new notions such as the restricted intersection, the restricted union, the restricted difference and the extended intersection of two soft sets. Moreover, we improve the notion of complement of a soft set, and prove that certain De Morgan's laws hold in soft set theory with respect to these new definitions.

1,223 citations

Cites background from "Soft semirings"

  • ...1 Note that restricted intersection is also known as bi-intersection in [5]....

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Journal Article•DOI•
On soft topological spaces

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Muhammad Shabir1, Munazza Naz1•
Quaid-i-Azam University1
01 Apr 2011-Computers & Mathematics With Applications
TL;DR: It is shown that a soft topological space gives a parametrized family of topological spaces and it is established that the converse does not hold.
Abstract: In the present paper we introduce soft topological spaces which are defined over an initial universe with a fixed set of parameters. The notions of soft open sets, soft closed sets, soft closure, soft interior points, soft neighborhood of a point and soft separation axioms are introduced and their basic properties are investigated. It is shown that a soft topological space gives a parametrized family of topological spaces. Furthermore, with the help of an example it is established that the converse does not hold. The soft subspaces of a soft topological space are defined and inherent concepts as well as the characterization of soft open and soft closed sets in soft subspaces are investigated. Finally, soft T"i-spaces and notions of soft normal and soft regular spaces are discussed in detail. A sufficient condition for a soft topological space to be a soft T"1-space is also presented.

832 citations

Journal Article•DOI•
Soft set theory and uni–int decision making

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Naim Çağman1, Serdar Enginoğlu1•
Gaziosmanpaşa University1
01 Dec 2010-European Journal of Operational Research
TL;DR: An uni-int decision making method which selects a set of optimum elements from the alternatives is constructed which shows that the method can be successfully applied to many problems that contain uncertainties.
Abstract: We firstly redefine the operations of Molodtsov’s soft sets to make them more functional for improving several new results. We also define products of soft sets and uni–int decision function. By using these new definitions we then construct an uni–int decision making method which selects a set of optimum elements from the alternatives. We finally present an example which shows that the method can be successfully applied to many problems that contain uncertainties.

622 citations

Cites methods from "Soft semirings"

  • ...[6] initiated the study of soft semirings by using the soft set theory and investigated several related properties....

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Journal Article•DOI•
Soft sets combined with fuzzy sets and rough sets: a tentative approach

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Feng Feng, Chang-xing Li, Bijan Davvaz1, M. Irfan Ali2•
Yazd University1, Quaid-i-Azam University2
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

Journal Article•DOI•
Soft sets and soft rough sets

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Feng Feng1, Xiaoyan Liu, V. Leoreanu-Fotea, Young Bae Jun2•
Shaanxi Normal University1, Gyeongsang National University2
01 Mar 2011-Information Sciences
TL;DR: It is shown that Pawlak's rough set model can be viewed as a special case of the soft rough sets, and these two notions will coincide provided that the underlying soft set in the soft approximation space is a partition soft set.
Abstract: In this study, we establish an interesting connection between two mathematical approaches to vagueness: rough sets and soft sets. Soft set theory is utilized, for the first time, to generalize Pawlak's rough set model. Based on the novel granulation structures called soft approximation spaces, soft rough approximations and soft rough sets are introduced. Basic properties of soft rough approximations are presented and supported by some illustrative examples. We also define new types of soft sets such as full soft sets, intersection complete soft sets and partition soft sets. The notion of soft rough equal relations is proposed and related properties are examined. We also show that Pawlak's rough set model can be viewed as a special case of the soft rough sets, and these two notions will coincide provided that the underlying soft set in the soft approximation space is a partition soft set. Moreover, an example containing a comparative analysis between rough sets and soft rough sets is given.

494 citations

Cites background from "Soft semirings"

  • ...Soft semirings and several related notions are defined in [7] in order to establish a connection between soft sets and semirings....

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References
More filters
Book•
Fuzzy sets

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Lotfi A. Zadeh
01 Aug 1996
TL;DR: A separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.
Abstract: A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.

52,705 citations

Journal Article•DOI•
Soft set theory—First results

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D. Molodtsov1•
Russian Academy of Sciences1
01 Feb 1999-Computers & Mathematics With Applications
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

"Soft semirings" refers background in this paper

  • ...Molodtsov [7] defined the notion of a soft set in the following way: Let U be an initial universe set and E be a set of parameters....

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  • ...For illustration,Molodtsov considered several examples in [7]....

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  • ...In 1992, Molodtsov [7] introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainties....

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  • ...Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa Soft semirings Feng Feng a,∗, Young Bae Jun b, Xianzhong Zhao c a Department of Applied Mathematics and Applied Physics, Xi’an Institute of Posts and Telecommunications, Xi’an 710061, PR China b Department of Mathematics Education (and RINS), Gyeongsang National University, Chinju 660-701, Republic of Korea c Department of Mathematics, Northwest University, Xi’an 710069, PR China a r t i c l e i n f o Article history: Received 20 February 2008 Received in revised form 4 May 2008 Accepted 23 May 2008 Keywords: Soft sets Soft semirings Soft subsemirings Soft ideals Idealistic soft semirings Soft semiring homomorphisms Fuzzy sets a b s t r a c t Molodtsov introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainty....

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  • ...Molodtsov introduced the concept of soft sets,which can be seen as a newmathematical tool for dealingwith uncertainty....

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Journal Article•DOI•
Soft set theory

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Pabitra Kumar Maji1, Ranjit Biswas1, A. R. Roy1•
Indian Institute of Technology Kharagpur1
01 Feb 2003-Computers & Mathematics With Applications
TL;DR: The authors define equality of two soft sets, subset and super set of a soft set, complement of asoft set, null soft set and absolute soft set with examples and De Morgan's laws and a number of results are verified in soft set theory.
Abstract: In this paper, the authors study the theory of soft sets initiated by Molodtsov. The authors define equality of two soft sets, subset and super set of a soft set, complement of a soft set, null soft set, and absolute soft set with examples. Soft binary operations like AND, OR and also the operations of union, intersection are defined. De Morgan's laws and a number of results are verified in soft set theory.

2,114 citations

"Soft semirings" refers background in this paper

  • ...[10] introduced and investigated several binary operations such as intersection, union, AND-operation, and OR-operation of soft sets....

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  • ...[10] defined and studied several operations on soft sets....

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  • ...Thus a null soft set in [10] is indeed a soft set with an empty support, and we say that a soft set (η, A) is non-null if Supp (η, A) 6= ∅....

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  • ...These examples were also discussed in [11,10]....

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Journal Article•DOI•
Rudiments of rough sets

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Zdziasław Pawlak1, Andrzej Skowron1•
University of Warsaw1
01 Jan 2007-Information Sciences
TL;DR: The basic concepts of rough set theory are presented and some rough set-based research directions and applications are pointed out, indicating that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences.
Abstract: Worldwide, there has been a rapid growth in interest in rough set theory and its applications in recent years. Evidence of this can be found in the increasing number of high-quality articles on rough sets and related topics that have been published in a variety of international journals, symposia, workshops, and international conferences in recent years. In addition, many international workshops and conferences have included special sessions on the theory and applications of rough sets in their programs. Rough set theory has led to many interesting applications and extensions. It seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences, especially in research areas such as machine learning, intelligent systems, inductive reasoning, pattern recognition, mereology, knowledge discovery, decision analysis, and expert systems. In the article, we present the basic concepts of rough set theory and point out some rough set-based research directions and applications.

2,004 citations

"Soft semirings" refers background in this paper

  • ...Alternatively, mathematical theories such as probability theory, fuzzy set theory [1,2], rough set theory [3,4], vague set theory [5] and the interval mathematics [6] were established by researchers to modelling uncertainties appearing in the above fields....

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Journal Article•
Vague sets

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W.-L. Gau1, D.J. Buehrer1•
National Chung Cheng University1
01 Mar 1993-IEEE Transactions on Systems, Man, and Cybernetics
TL;DR: A vague set is a set of objects, each of which has a grade of membership whose value is a continuous subinterval of according to the inequality of the following type:
Abstract: A vague set is a set of objects, each of which has a grade of membership whose value is a continuous subinterval of

1,567 citations

"Soft semirings" refers background in this paper

  • ...Alternatively, mathematical theories such as probability theory, fuzzy set theory [1,2], rough set theory [3,4], vague set theory [5] and the interval mathematics [6] were established by researchers to modelling uncertainties appearing in the above fields....

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Related Papers (5)
Soft set theory—First results

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01 Feb 1999-Computers & Mathematics With Applications
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Soft sets and soft groups

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Soft set theory

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01 Feb 2003-Computers & Mathematics With Applications
Pabitra Kumar Maji, Ranjit Biswas, A. R. Roy
An application of soft sets in a decision making problem

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01 Oct 2002-Computers & Mathematics With Applications
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On some new operations in soft set theory

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01 May 2009-Computers & Mathematics With Applications
M. Irfan Ali, Feng Feng, Xiaoyan Liu, Won Keun Min, Muhammad Shabir