DOI: 10.12688/f1000research.183797.2 ISSN: 2046-1402

Fuzzy Congruences on Heyting Algebras: Characterizations via Fuzzy Ideals and Filters.

DEREBEW NIGUSSIE DERSO
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Background Heyting algebras serve as algebraic models for intuitionistic logic, with classical congruence relations playing a key role in their structural analysis. This paper extends classical congruence theory to the fuzzy setting, motivated by the need to handle gradations of equivalence and logical truth. Methods Building on the foundational work of Assaye et al. (2019) on classical Heyting algebra congruences, we introduce fuzzy congruence relations via : fuzzy implicatively and multiplicatively closed subsets. The construction generalizes standard techniques by incorporating membership degrees. Results We establish fuzzy versions of the First Isomorphism Theorem and correspondence theorems linking prime fuzzy congruences, ideals, and filters. Furthermore, we characterize fuzzy congruences in terms of fuzzy kernels and cokernels, providing a complete algebraic description. Conclusions The systematic study of fuzzy congruences on Heyting algebras yields a robust framework that unifies fuzzy logic and universal algebra. These results pave the way for further investigations into fuzzy quotient algebras and their applications to many-valued reasoning.

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