On linear fuzzifications of groupoids with special emphasis on BCK-algebras

In this paper, for a given groupoid (X, *) and a relation R on X we consider a related structure LF_RX = {< x, y, z > ∈ X³ | xRy, yRz}, and we discuss three different types on LF_RX, and we construct a linear fuzzy real BCK-algebra (LF_≤X, $\ominus$, ε(0)) and a new BCK-algebra (LF_≤X, $\oast$, ε(0)) from a BCK-algebra (X, *, 0).