Poonam Kumar Sharma

This paper presents a systematic investigation of intuitionistic fuzzy algebraic structures associated with group algebras. We introduce the concept of an intuitionistic fuzzy group subalgebra (IFGSA) of an intuitionistic fuzzy group algebra (IFGA) constructed from the group algebra K[G], where G is a finite group and K is a field. Structural properties of IFGSAs are examined, with particular emphasis on their behavior under intuitionistic fuzzy group algebra homomorphisms (IFGA-homomorphisms). The image and inverse image of IFGSAs are studied, and it is proved that the intersection of an arbitrary family of IFGSAs is again an IFGSA. Furthermore, the notion of an intuitionistic fuzzy augmentation ideal (IFAI) in an IFGA is introduced and analyzed. It is shown that the intersection of an arbitrary family of IFAIs remains an IFAI. The image and inverse image of IFAIs under IFGA-homomorphisms are also investigated. Finally, isomorphism theorems for IFGAs are established, extending classical group algebra results to the intuitionistic fuzzy framework.

This paper introduces the concept of an intuitionistic fuzzy group algebra associated with a finite group G and an intuitionistic fuzzy group A on G. We establish its structural properties, showing that it simultaneously behaves as an intuitionistic fuzzy algebra and an intuitionistic fuzzy G-module. Extending author's earlier results on the semi-simplicity of intuitionistic fuzzy G-modules, we explore links to complete reducibility and injectivity. Further, we study intersections, (α,β)-cuts, and homomorphic images of such algebras, and define intuitionistic fuzzy group algebra homomorphisms. Finally, we prove that the class of all intuitionistic fuzzy group algebras forms a category.

This article is a continuation of the author’s earlier work [11]. Here we extend the study of group actions on intuitionistic fuzzy structures by focusing on intuitionistic fuzzy primary and semiprimary ideals of Γ-rings. We introduce and characterize intuitionistic fuzzy G-primary and G-semiprimary ideals, establishing their relationships with G-invariant intuitionistic fuzzy ideals. Several structural properties and related results are investigated to deepen the understanding of these generalized ideals under group actions. Furthermore, we explore the behavior of intuitionistic fuzzy G-primary and G-semiprimary ideals under G-homomorphisms, specifically analyzing their image and pre-image. These results contribute to the ongoing development of intuitionistic fuzzy algebraic structures in the context of Γ-rings.

In this paper, we discuss the commutativity of an intuitionistic L-fuzzy subgroup of a group. Some necessary and sufficient conditions for an intuitionistic L-fuzzy subgroup to be commutative are derived. The relationship between commutativity and normality of intuitionistic L-fuzzy subgroups is briefly studied. It is also proved that any commutative intuitionistic L-fuzzy subgroup of a finite group admits a decomposition as a direct product of intuitionistic L-fuzzy subgroups of Sylow subgroups.

In this paper, we introduce and explore some novel concepts within the frame work of intuitionistic fuzzy module theory. First, we define the notion of an intuitionistic fuzzy chained module as a generalisation of chained modules, establishing it foundational properties. We then characterise intuitionistic fuzzy chained modules, in terms of its level-cut submodules. In addition to this, we describe an intuitionistic fuzzy chained module in terms of its intuitionistic fuzzy cyclic submodules. Finally, we demonstrate that under specific conditions, the intuitionistic fuzzy multiplication module is an intuitionistic fuzzy chained module.

As a generalization of the concepts of an intuitionistic fuzzy prime ideal and a prime intuitionistic fuzzy ideal, the concepts of an intuitionistic fuzzy 2-absorbing ideal and a 2-absorbing intuitionistic fuzzy ideal of a lattice are introduced. Some results on such intuitionistic fuzzy ideals are proved. It is shown that the radical of an intuitionistic fuzzy ideal of L is a 2-absorbing intuitionistic fuzzy ideal if and only if it is a 2-absorbing primary intuitionistic fuzzy ideal of L. We also introduce and study these concepts in the product of lattices.