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International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(2): 125-140

Published online June 25, 2024

https://doi.org/10.5391/IJFIS.2024.24.2.125

© The Korean Institute of Intelligent Systems

Near-Generalized Approximations in Groups Based on a Set-Valued Mapping

Dian Winda Setyawati1, Subiono1, and Bijan Davvaz2

1Department of Mathematics, Institut Teknologi Sepuluh Nopember, Kampus ITS, Sukolilo-Surabaya, Indonesia
2Department of Mathematical Sciences, Yazd University, Yazd, Iran

Correspondence to :
Subiono (subiono2008@matematika.its.ac.id)

Received: June 5, 2023; Revised: August 24, 2023; Accepted: June 5, 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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A normal subgroup of a group can partition a group into equivalence classes. Therefore, approximations can be constructed within a group. The near approximations in a group are extensions of the approximations in a group. A set-valued mapping T from group G to the set of all non-empty subsets of group G′ can establish generalized approximations in group G based on the set-valued mapping T. In this study, we introduce the notion of near-generalized approximations in a group based on set-valued mapping, an extension of the concept of generalized approximations in a group based on set-valued mapping and near approximations in a group. We then present some properties of nearby subgroups in a group based on set-valued mapping. Furthermore, we compare these types of near-generalized and generalized approximations in a group based on set-valued mapping.

Keywords: Normal subgroup, Group, Approximations, Near approximations, Generalized approximations, Near-generalized approximations

Conflict of Interest

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No potential conflict of interest relevant to this article was reported.

Biographies

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DianWinda Setyawati is a doctoral student in the Department of Mathematics at Institut Teknologi Sepuluh Nopember, Indonesia. Since 2003, she has worked as a lecturer in the Department of Mathematics at Institut Teknologi Sepuluh Nopember, Indonesia. Her research areas include group theory, ring theory, and rough set theory.

Subiono is a professor in the Department of Mathematics at Institut Teknologi Sepuluh Nopember, Indonesia. He has worked as a lecturer in the Department of Mathematics at Institut Teknologi Sepuluh Nopember, Indonesia, since 1984. His research areas are min-maxpluss algebra and applied algebra.

Bijan Davvaz received his B.Sc. degree in Applied Mathematics from Shiraz University in 1988 and his M.Sc. degree in Pure Mathematics from Tehran University in 1990. In 1998, he received his Ph.D. in mathematics at Tarbiat Modares University. He is a member of the Editorial Boards of 24 mathematical journals. He has also served as the Head of the Department of Mathematics (1998–2002), Chairman of the Faculty of Science (2004–2006), and Vice-President for Research (2006–2008) at Yazd University, Iran. His areas of interest include algebra (groups, rings, and modules), algebraic hyperstructures, rough sets, and fuzzy logic. He has published 12 books on mathematics. He is currently a distinguished professor of Mathematics at Yazd University.

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(2): 125-140

Published online June 25, 2024 https://doi.org/10.5391/IJFIS.2024.24.2.125

Copyright © The Korean Institute of Intelligent Systems.

Near-Generalized Approximations in Groups Based on a Set-Valued Mapping

Dian Winda Setyawati1, Subiono1, and Bijan Davvaz2

1Department of Mathematics, Institut Teknologi Sepuluh Nopember, Kampus ITS, Sukolilo-Surabaya, Indonesia
2Department of Mathematical Sciences, Yazd University, Yazd, Iran

Correspondence to:Subiono (subiono2008@matematika.its.ac.id)

Received: June 5, 2023; Revised: August 24, 2023; Accepted: June 5, 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A normal subgroup of a group can partition a group into equivalence classes. Therefore, approximations can be constructed within a group. The near approximations in a group are extensions of the approximations in a group. A set-valued mapping T from group G to the set of all non-empty subsets of group G′ can establish generalized approximations in group G based on the set-valued mapping T. In this study, we introduce the notion of near-generalized approximations in a group based on set-valued mapping, an extension of the concept of generalized approximations in a group based on set-valued mapping and near approximations in a group. We then present some properties of nearby subgroups in a group based on set-valued mapping. Furthermore, we compare these types of near-generalized and generalized approximations in a group based on set-valued mapping.

Keywords: Normal subgroup, Group, Approximations, Near approximations, Generalized approximations, Near-generalized approximations

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