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The notion of normal pseudo-BCI-algebras is studied and some characterizations of it are given. Extensions of pseudo-BCI-algebras are also considered.

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ABSTRACT. Characterizations of fuzzy filters in a BE-algebra are established. Conditions for a fuzzy set to be a fuzzy filter are given. For a fuzzy set µ the least fuzzy filter containing µ is constructed. The homomorphic properties of fuzzy filters of a BE-algebra are provided. Finally, characterizations of Noetherian BE-algebras and Artinian BE-algebras via fuzzy filters are obtained.

The notions of an ideal and a fuzzy ideal in BN-algebras are introduced. The properties and characterizations of them are investigated. The concepts of normal ideals and normal congruences of a BN-algebra are also studied, the properties of them are displayed, and a one-to-one correspondence between them is presented. Conditions for a fuzzy set to be a fuzzy ideal are given. The relationships between ideals and fuzzy ideals of a BN-algebra are established. The homomorphic properties of fuzzy ideals of a BN-algebra are provided. Finally, characterizations of Noetherian BN-algebras and Artinian BN-algebras via fuzzy ideals are obtained.