Fuzzy Subalgebras of Type (α, β) in BCK/BCI-Algebras

Abstract

Using the belongs to relation (2) and quasi-coincidence with relation (q) between fuzzy points and fuzzy sets, the concept of (, )-fuzzy subalgebras where  and  areany two of {2, q, 2 _q, 2 ^q} with  6=2 ^q was already introduced, and related properties were investigated (see [3]). In this paper, we give a condition for an (2,2_ q)- fuzzy subalgebra to be an (2,2)-fuzzy subalgebra. We provide characterizations of an (2,2_ q)-fuzzy subalgebra. We show that a proper(2,2)-fuzzy subalgebra A of X with additional conditions can be expressed as the union of two proper non-equivalent (2,2)- fuzzy subalgebras of X. We also prove that if A is a proper (2,2_ q)-fuzzy subalgebra of a CK/BCI-algebra X such that #{A (x) | A (x) < 0.5}  2, then there exist two proper non-equivalent (2,2_ q)-fuzzy subalgebras ofX such that A can be expressed as the unionof them.