TRIANGULAR NORMS AND LATTICE-VALUED MODULE THEORY: ON ⊛-SUBMODULES AND ⊛-MODULE HOMOMORPHISMS
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE, vol.16, no.1, pp.119-129, 2026 (Peer-Reviewed Journal)
- Publication Type: Article / Article
- Volume: 16 Issue: 1
- Publication Date: 2026
- Journal Name: BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE
- Journal Indexes: Academic Search Ultimate (EBSCO), MathSciNet, zbMATH
- Page Numbers: pp.119-129
- Open Archive Collection: AVESIS Open Access Collection
- Recep Tayyip Erdoğan University Affiliated: Yes
Abstract
Thispaperinvestigates lattice-valued relations and functions within the framework of triangular norms (t-norms) and their applications to module theory. Building upon Demirci’s foundational work on fuzzy functions defined via fuzzy equivalence relations, we introduce and develop the concept of ⊛-module homomorphisms. Our approach extends classical module homomorphism theory to the lattice-valued setting by leveraging the structural properties of t-norms and ⊛-equivalence relations. We establish necessary preliminary concepts including t-norms, ⊛-equivalence relations, and ⊛-functions, with special attention to infinitely ∨-distributive t-norms. We then characterize ⊛-submodules and ⊛-congruence relations, proving key theorems regarding their behavior. The core contribution is the formal definition and detailed analysis of ⊛-module homomorphisms. We prove that kernels and images of ⊛-module homomorphisms are themselves ⊛-submodules, and establish fundamental homomorphism theorems in the lattice-valued context.