Composition Series and Jordan-Holder Theorem under Fuzzy Multigroups (European Journal of Pure and Applied Mathematics, 2026)
The theory of fuzzy multigroups is an algebraic structure derivable from the application of fuzzy multisets to groups. Various concepts in group theory have been considered in fuzzy multigroup theory. However, simple group, maximal normal subgroup, normal series, composition series, and the Jordan-Hölder Theorem are open problems in fuzzy multigroup theory. Hence, this paper defines simple fuzzy multigroups, maximal normal fuzzy submultigroups, normal series for fuzzy multigroups, and composition series for fuzzy multigroups with illustrations. In addition, the Jordan-Hölder Theorem in fuzzy multigroup theory is established. We show that each finite fuzzy multigroup of a finite group possesses a composition series, and also prove that two composition series for a finite fuzzy multigroup of a finite group are comparable.
Keywords: fuzzy multigroup; fuzzy multiset; order of fuzzy multigroup; simple fuzzy multigroup; maximal normal fuzzy submultigroup; normal series; composition series; Jordan-Holder Theorem