Construction and properties of homomorphisms on intuitionistic fuzzy subgroups

Abstract

This paper introduces and develops the theory of intuitionistic fuzzy group homomorphisms, extending classical group homomorphic structures into the intuitionistic fuzzy algebraic framework. Based on the recently defined concept of intuitionistic fuzzy functions by Milles et al. (2020), we first formalize the notions of surjective and injective intuitionistic fuzzy functions and illustrate them with concrete examples. We then define intuitionistic fuzzy group homomorphisms between intuitionistic fuzzy subgroups and establish a series of fundamental theorems that generalize well-known classical results. Our principal contributions show that the composition of two intuitionistic fuzzy homomorphisms remains an intuitionistic fuzzy homomorphism, and that both intuitionistic fuzzy subgroups and intuitionistic fuzzy normal subgroups are preserved under direct and inverse images of such homomorphisms. Furthermore, we introduce the kernel of an intuitionistic fuzzy group homomorphism and prove that it forms an intuitionistic fuzzy subgroup; a key theorem establishes the relationship between this kernel and the injectivity of the homomorphism. These results systematically transfer classical homomorphism theory into the intuitionistic fuzzy setting, enriching the algebraic study of intuitionistic fuzzy structures. The work provides a foundation for future research on intuitionistic fuzzy homomorphisms across broader algebraic systems such as rings, modules, and vector spaces.

Keywords

• intuitionistic fuzzy sets
• intuitionistic fuzzy subgroups
• intuitionistic fuzzy normal subgroups
• intuitionistic fuzzy functions
• intuitionistic fuzzy homomorphisms

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