We introduce the concept of a translation ideal in pseudo-BCH algebras and investigate its properties. We prove that for any pseudo-BCH algebra 𝔛 there is a one-to-one order-preserving correspondence between the closed translation ideals of 𝔛 and the relative congruences on 𝔛.
In this paper we introduce the notion of BF-algebras, which is a generalization of B-algebras. We also introduce the notions of an ideal and a normal ideal in BF-algebras. We investigate the properties and characterizations of them.
Some connections between BM-algebras and its related topics are studied. It is proved that the class of medial BH-algebras coincides with the class of BM-algebras. Moreover, the congruence lattice of a BM-algebra is investigated.
Characterizations of fuzzy ideals of a pseudo-BCK algebra are established. Conditions for a fuzzy set to be a fuzzy ideal are given. Given a fuzzy set μ, the least fuzzy ideal containing μ is constructed. The homomorphic properties of fuzzy ideals of a pseudo-BCK algebra are provided. Finally, characterizations of Noetherian pseudo-BCK algebras and Artinian pseudo-BCK algebras in terms of fuzzy ideals are given.