Fuzzy ideals and fuzzy congruences of co-residuated lattices

In order to expand the theory of fuzzy logic algebra, the fuzzy ideals and fuzzy congruences of co-residuated lattices and their interrelationships were studied. Firstly, by using Heyting algebra as the valuation domain, the concepts of fuzzy ideals and fuzzy congruences of co-residuated lattices were introduced, and their mutual induction methods were studied to prove a one-to-one correspondence between them. Secondly, the equivalent descriptions of fuzzy ideals and fuzzy congruence relationships were studied using cut set and strong cut set methods. The research shows that fuzzy ideals and the fuzzy congruences are two equivalent concepts and thus will play the same role in structure and classification problems. The research conclusion enriches the relevant theory of fuzzy logic algebraic algebras and can provide certain theoretical reference for further study of other algebraic systems.