Bipolar-Valued Fuzzy Subgroups, Normal Subgroups, and Homomorphisms on Dib’s Fuzzy Space

Authors

  • Fadi Al-Zu’bi Department of Mathematical Sciences, Faculty of Science & Technology,6 Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia
  • Abd Ghafur Ahmad Department of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia
  • Abd Ulazeez Alkouri Department of Mathematics, Faculty of Science, Ajloun National University, P.O. 43, Ajloun- 26810, Jordan
  • Maslina Darus Department of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia
  • Sadeq Damrah Department of Mathematics and Physics, College of Engineering, Australian University West Mishref, Safat 13015, Kuwait.

DOI:

https://doi.org/10.29020/nybg.ejpam.v18i4.6761

Keywords:

Fuzzy group, Fuzzy space, Bipolar valued fuzzy space, Bipolar valued fuzzy subgroup, Bipolar valued fuzzy homomorphisms, Bipolar valued fuzzy normal subgroup, Dib fuzzy group theory

Abstract

Fuzzy group theory has evolved beyond single-valued memberships to account for dual polarity and uncertainty. Building on Dib’s fuzzy space and bipolar-valued fuzzy sets, we develop a unified algebraic theory of bipolar-valued fuzzy (BVF) subgroups, including BVF normal subgroups and BVF homomorphisms, via a BVF binary operation (BVFBO) on a BVF-space. We establish necessary and sufficient subgroup criteria, characterize normality through coset symmetry in BVF-space, and prove homomorphism properties that align BVF structures with their classical counterparts through correspondence theorems. The framework clarifies when associativity holds between subgroup elements and ambient BVF-group elements and provides constructive examples. This generalization resolves limitations tied to the absence of a bipolar fuzzy universal set and supports applications in polarity-sensitive decision systems and network analysis.

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Published

2025-11-05

Issue

Vol. 18 No. 4: (October 2025)

Section

Algebra

License

Copyright (c) 2025 Fadi Al-Zu’bi, Abd Ghafur Ahmad, Abd Ulazeez Alkouri, Maslina Darus , Sadeq Damrah

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How to Cite

Bipolar-Valued Fuzzy Subgroups, Normal Subgroups, and Homomorphisms on Dib’s Fuzzy Space. (2025). European Journal of Pure and Applied Mathematics, 18(4), 6761. https://doi.org/10.29020/nybg.ejpam.v18i4.6761