Subalgebras and Ideals in Fuzzy Normed BCK-algebras and BCI-Algebras

We study the interplay between fuzzy BCK/BCI-norms and algebraic substructures in BCK- and BCI-algebras. We show that level subsets induced by a fuzzy BCK/BCI-norm need not form ideals or subalgebras in general, and introduce strong fuzzy normed BCK/BCI-algebras via a strengthened fuzzy triangle inequality. Under this framework, we prove that a level subset is an ideal if and only if the underlying fuzzy norm is strong, and we establish corresponding results for subalgebras, highlighting essential differences between the BCK and BCI cases. We also examine when fuzzy sets, fuzzy subalgebras, and fuzzy ideals can induce fuzzy normed structures, showing that additional injectivity and order conditions are necessary. Several examples illustrate the sharpness of the results. The results have natural interpretations in approximate reasoning, knowledge representation, and the algebraic study of logical systems under uncertainty.