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 LFRX = { ∈ X3 | xRy, yRz}, and we discuss three different types on LFRX, and we construct a linear fuzzy real BCK-algebra LF≤X, $\ominus$, e0 and a new BCK-algebra LF≤X, $\oast$, e0 from a BCK-algebra X, *, 0.