The term endomorphism is derived from the Greek adverb endon (“inside”) and morphosis (“to form” or “to shape”). In an algebra, an endomorphism of a group, module, ring, vector space, etc., is a homomorphism from the algebra to itself (with surjectivity not required). In 2001, Sergio Celani (Int J Math Math Sci, 29(1):55–61, 2002)  gave a representation theorem for Hilbert algebras by means of ordered sets and characterized the homomorphisms of Hilbert algebras in terms of applications defined between the sets of all irreducible deductive systems of the associated algebras. In , Chul Kon Bae (J Korea Soc Math Edu, 24(1):7–10, 1985) investigated some properties on homomorphisms in BCK-algebras. In his paper, he mainly studied the properties of the compositions of homomorphisms of BCK-algebras. In , Z. Chen, Y. Huang and E.H. Roh (Comm Korean Math Soc, 10(3):499–518, 1995) considered the centralizer C(S) of a given set with respect to the semigroup End(X) of all endomorphisms of an implicative BCK-algebras X with the condition (S). They obtained a series of interesting results those indicated the embedding of X into the centralizer C(S).