On fundamental isomorphism theorems in soft subgroups
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On fundamental isomorphism theorems in soft subgroups 1 N. Çagman ˘
· R. Barzegar2 · S. B. Hosseini2
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Molodsov initiated a novel concept of soft set theory, which is a completely new approach for modeling vagueness and uncertainty, which there is no limited condition to description of objects and is free from the difficulties affecting existing methods. This makes the theory very convenient and easy to apply in practice. After the pioneering work of Molodsov, there has been a great effort to obtain soft set analogues of classical theories. Among other fields, a progressive developments are made in the field of algebraic structure. To extend the soft set in group theory, many researchers introduced the notions of soft subgroup and investigated its applications in group theory and decision making. In this paper, by using the soft sets and their duality, we introduce new concepts on the soft sets, which are called soft quotient subgroup and quotient dual soft subgroup. We then derive their algebraic properties and, in sequel, investigate the fundamental isomorphism theorems in soft subgroups analogous to the group theory. Keywords Soft sets · Soft subgroups · Normal soft subgroups · Homomorphism of soft subgroups · Quotient-terms
1 Introduction Most of our traditional tools for formal modeling, reasoning and computing are crisp, deterministic and precise in character. But many complicated problems in economics, engineering, environment, social science, medical science, etc., involve data which are not always all crisp. We cannot always use the classical methods because of various types of uncertainties present in these problems. As a new mathematical tool for dealing with uncertainties, incomplete information and the study of intelligent systems characterized by insufficient control systems, for the reason, in 1965 the notion of fuzzy sets was introduced by Zadeh (1965) and in 1982 the concept of rough set theory proposed by Pawlak (1982, 1985); Pawlak and Skowron (2007). But each Communicated by A. Di Nola.
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N. Ça˘gman [email protected] R. Barzegar [email protected] S. B. Hosseini [email protected]
1
Department of Mathematics, Tokat Gaziosmanpasa University, Tokat, Turkey
2
Islamic Azad University, Sari Branch, Sari, Iran
of these theories has its inherent difficulties as pointed out by Molodtsov (1999). Molodtsov (1999) initiated a novel concept of soft set theory, which is a completely new approach for modeling vagueness and uncertainty, which there is no limited condition to description of objects and is free from the difficulties affecting existing methods. This makes the theory very convenient and easy to apply in practice. Soft set theory has potential applications in many different fields; some of them are the algebraic structure of soft sets (Acar et al. 2010; Akta¸s and Ça˘gman 2007; Atagün and Sezgin 2011; Feng et al. 2008; Jun 2008; Jun et al. 2009; Jun and Park 2008; Jun et al. 2010; Wen 2008; Yin and
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