Generalizations of (∈,∈∨q)-fuzzy subalgebras in BCK/BCI-algebras

More general form of the notion of quasi-coincidence of a fuzzy point with a fuzzy set is considered, and generalizations of results in the papers [Y. B. Jun, On (α,β)(α,β)-fuzzy subalgebras of BCK/BCI-algebras, Bull. Korean Math. Soc. 42 (4) (2005) 703–711; Y. B. Jun, Fuzzy subalgebras of type (α,β)(α,β) in BCK/BCI-algebras, Kyungpook Math. J. 47 (2007) 403–410] are discussed. The notions of (∈,qk)-fuzzy subalgebras and (∈,∈∨qk)-fuzzy subalgebras in a BCK/BCI-algebra XX are introduced, and several properties are investigated. Characterizations of (∈,∈∨qk)-fuzzy subalgebra in a BCK/BCI-algebra XX are discussed.