Some Concepts of BE-algebras

Abstract

BE-algebras are important tools for certain investigations in algebraic logic since they can be considered as fragments of any propositional logic containing a logical connective implication and the constant 1 which is considered as the logical value “true.” The notion of BE-algebras was introduced and extensively studied by H.S. Kim and Y.H. Kim in (Sci Math Jpn, Online e-2006, 1299–1302, [153]). These classes of BE-algebras were introduced as a generalization of the class of BCK-algebras of K. I$\acute{s}$eki and S. Tanaka (Math Jpn 23(1):1–26, 1979, [124]). I. Chajda and J. Kuhr (Miskolc Math Notes 8(1):11–21, 2007, [38]) studied the algebraic structure derived from a BCK-algebra. They viewed commutative BCK-algebras as semilattices whose sections have antitone involutions, and it is known that bounded commutative BCK-algebras are equivalent to MV-algebras. In (Rezaei and Saeid, Afrika Matematika 22(2):115–127, 2011, [202]), A. Rezaei and A.B. Saeid introduced the concept of fuzzy subalgebras of BE-algebras and studied its nature. They stated and proved the Foster’s results on homomorphic images and inverse images in fuzzy topological BE-algebras.