In this paper, we introduce the concept of a new algebraic structure: independent UP-algebras (in short, IUP-algebras), which is independent of UP-algebras. We also introduce the concepts of IUP-subalgebras, IUP-filters, IUP-ideals, and strong IUP-ideals of IUP-algebras and investigate their properties and relationships. Finally, we discuss the concept of homomorphisms between IUP-algebras and also study the direct and inverse images of four special subsets.

In this paper, we introduce the concept of (inf, sup)-hesitant fuzzy subalgebras, which is a general concept of interval-valued fuzzy subalgebras, in BCK/BCI-algebras and investigate its properties. We characterize (inf, sup)-hesitant fuzzy subalgebras in terms of sets, fuzzy sets, hesitant fuzzy sets, interval-valued fuzzy sets, Pythagorean fuzzy sets, negative fuzzy sets and bipolar fuzzy sets. Furthermore, characterizations of subalgebras, fuzzy subalgebras, anti-fuzzy subalgebras, negative fuzzy subalgebras, Pythagorean fuzzy subalgebras and bipolar fuzzy subalgebras of BCK/BCI-algebras are given in terms of (inf, sup)-hesitant fuzzy subalgebras and interval-valued fuzzy subalgebras.