Fuzzy normed BCK-algebras and BCI-algebras

Abstract

In this paper, we introduce and study the notion of fuzzy normed BCK-algebras and fuzzy normed BCI-algebras as a natural extension of clas sical normed algebraic structures into the fuzzy setting. A fuzzy norm on a BCK/BCI-algebra is defined as a mapping from the algebra and a positive real parameter into the unit interval satisfying suitable axioms analogous to those of fuzzy normed linear spaces. Several examples are presented to illustrate the validity of the axioms. Fundamental properties of fuzzy normed BCK/BCI algebras are established, including monotonicity, chained triangle inequalities, and order-related behaviors. It is shown that every BCK/BCI-algebra admits a fuzzy norm, and the behavior of fuzzy norms under algebra homomorphisms is investigated. Necessary and sufficient conditions are obtained for the transfer of fuzzy norms via injective, surjective, and bijective homomorphisms. A charac terization theorem is proved showing that the main fuzzy norm inequality can be reduced to a simpler condition. These results generalize known concepts in fuzzy algebra and provide a new analytical framework for studying BCK/BCI algebras.