On operations of soft sets Sezgin, Aslıhan; Atagün, Akın Osman
Computers & mathematics with applications (1987),
03/2011, Volume:
61, Issue:
5
Journal Article
Peer reviewed
Open access
Soft set theory, proposed by Molodtsov, has been regarded as an effective mathematical tool to deal with uncertainties. In this paper, first we prove that certain De Morgan’s law hold in soft set ...theory with respect to different operations on soft sets. Then, we discuss the basic properties of operations on soft sets such as intersection, extended intersection, restricted union and restricted difference. Moreover, we illustrate their interconnections between each other. Also we define the notion of restricted symmetric difference of soft sets and investigate its properties. The main purpose of this paper is to extend the theoretical aspect of operations on soft sets.
In this paper, a new uncertainty modelling concept called strait fuzzy set is introduced, which brings new perspectives to both theoretical and practical advances in fuzzy mathematics. This set type ...allows objects/points to be graded with fuzzy membership intervals that are partitions of 0,1 (so that the union of these partitions is 0,1 and their intersection is empty) instead of fuzzy membership degrees represented as exact values in 0,1. Moreover, some basic operations and properties of strait fuzzy sets are studied in detail. The concept of strait fuzzy rough set is put forward and its theoretical aspects are discussed. In addition, two different similarity approaches are proposed for strait fuzzy sets and strait fuzzy rough sets, and applied to measure the similarity rates of vaccines against influenza viruses.
A soft matrix multiplication of matrices in different types was not possible so far. In this study, we generalize the soft matrix products And, And–Not, Or, Or–Not defined in Çağman and Enginoğlu ...(Comput Math Appl 59:3308–3314,
2010
) so as to multiply soft matrices in different types. Furthermore, these generalizations allow us to multiply soft matrices more than two soft matrices. Therefore, we can solve decision making problems with multiple decision makers using a single product. These new operations make the process of solving decision making problems faster, easier and more convenient. Then we construct some effective decision making methods called soft distributive max–min (max–max, min–min, min–max) decision making methods. We also provided Scilab codes to demonstrate our methods.
In this paper, two new uncertainty modeling concepts, namely strait soft set and strait rough set, between the structures of rough sets and soft sets, which will bring new perspectives to both ...theoretical and practical aspects, are presented. A reduction method of alternatives is given using strait soft sets. Another convenience arising from the structure of strait soft sets is that they allow the parameters to be combined. Thus, the fusion of parameters is defined and its use in soft set operations is demonstrated. Strait rough sets naturally contain the characteristics of the rough sets and also allow the parameters to be characterized. The strait soft set and strait rough set are supported by many examples and comparisons. In addition, a new decision-making approach based on the strait soft set and strait rough set is proposed and then followed by real-life applications to illustrate the computational processes.
Soft set theory, proposed by Molodtsov, has been regarded as an effective mathematical tool to deal with uncertainties. In this paper, we introduce and study soft subrings and soft ideals of a ring ...by using Molodtsov’s definition of the soft sets. Moreover, we introduce soft subfields of a field and soft submodule of a left
R
-module. Some related properties about soft substructures of rings, fields and modules are investigated and illustrated by many examples.
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•The row-products of soft matrices are firstly introduced and some related properties are examined in detail.•Two monoids according to the operations And row-product, and Or ...row-product are presented.•It is proposed the novel algorithms based on the row-products of soft matrices to solve the problems of multiple-disjoint decision making.•Finally, Scilab codes of the algorithms which make the process of decision making faster and easier are given.
In this paper, we firstly introduce row-products of the soft matrices and investigate their properties and algebraic structures in detail. We aim to show that these row-products can be used in handling decision making problems. We therefore propose two new methods called a soft max-row decision making method and a multi-soft distributive max–min decision making method employing these operations. These methods are utilized to obtain an optimum choice when the decision makers evaluate the objects of disjoint universe sets according to the parameters during decision making. Also, we argue that the first of them can be employed to solve the decision problems handled in 6,16. The second method that we propose to solve the decision problems involving multi-disjoint universe sets is a generalization of the soft decision method presented in 9. By constructing them, we pioneer the idea that the soft matrices can be used to deal with decision making involving the multi-disjoint universe sets, which it is shortly called a multiple-disjoint decision making. Moreover, we present the outstanding examples to verify the practicality and effectiveness of the emerging methods. Finally, we give Scilab codes for each step of our methods and put forward that these codes make the process of decision making faster and easier.
Soft groups and normalistic soft groups Sezgin, Aslıhan; Atagün, Akın Osman
Computers & mathematics with applications (1987),
07/2011, Volume:
62, Issue:
2
Journal Article
Peer reviewed
Open access
Soft set theory, proposed by Molodtsov, has been regarded as an effective mathematical tool to deal with uncertainties. In this paper, first we correct some of the problematic cases in a previous ...paper by Aktaş and Çag˜man H. Aktaş, N. Çag˜man, Soft sets and soft groups, Inf. Sci. 177 (2007) 2726–2735. Moreover, we introduce the concepts of normalistic soft group and normalistic soft group homomorphism, study their several related properties, and investigate some structures that are preserved under normalistic soft group homomorphisms.
In this paper, we first generalize the products of two fuzzy soft matrices. Through these generalizations, three or more fuzzy soft matrices in the different types can be multiplied. Furthermore, we ...introduce the mean operators and normalized fuzzy weighted mean operators of the fuzzy soft matrices. We discuss the theoretical aspects of these operators. We describe the multicriteria group decision making (MCGDM) problem with different evaluation criterion sets, and then we create two algorithms using the mean operators and generalized products of fuzzy soft matrices to deal with such problems. To show the advantages of the proposed ones, we present the comparison results with some of the preexisting decision making algorithms of fuzzy soft sets. Finally, we create Scilab codes of our algorithms to expedite and facilitate the decision making process.
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•We made a new approach to the classical ring theory via soft set theory, with the concept of soft intersection rings.•We defined ideals, (generalized) bi-ideals, interior ideals and ...quasi-ideals of soft rings.•By defining soft union–intersection product, we obtain the relationship between this new concept and soft intersection ring and its different ideals.•We also characterized regular, regular duo, intra-regular and strongly regular rings by soft intersection rings and ideals.
In this paper, we make a completely new approach to the classical ring theory via soft set theory, with the concept of soft intersection rings, ideals, (generalized) bi-ideals, interior ideals and quasi-ideals. Particularly, we define soft union–intersection product and obtain the relationship between this new concept and soft intersection ring and its different ideals. Moreover, we characterize regular, regular duo, intra-regular and strongly regular rings by soft intersection rings and ideals.
Molodtsov introduced the theory of soft sets, which can be seen as an effective mathematical tool to deal with uncertainties, since it is free from the difficulties that the usual theoretical ...approaches have troubled. In this paper, we apply the definitions proposed by Ali et al. M. I. Ali, F. Feng, X. Liu, W. K. Min and M. Shabir,On some new operations in soft set theory, Comput. Math. Appl. 57 (2009), 1547—1553 to the concept of soft near-rings and substructures of soft near-rings, proposed by Atagün and Sezgin A. O. Atagün and A. Sezgin,Soft Near-rings, submitted and show them with illustrating examples. Moreover, we investigate the properties of idealistic soft near-rings with respect to the near-ring mappings and we show that the structure is preserved under the near-ring epimorphisms. Main purpose of this paper is to extend the study of soft near-rings from a theoretical aspect.
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