Abstract
In this paper, we generalize the notion of an implicativity discussed in B C K -algebras, and apply it to some groupoids and B C K -algebras. We obtain some relations among those axioms in the theory of groupoids.
Highlights
As a generalization of BCK-algebras, the notion of d-algebras was introduced by Neggers and
We generalized the notion of an implicativity discussed mainly in BCK-algebras by using the notion of a word, and obtained several properties in groupoids and BCK-algebras
By using the notion of Bin( X )-product 2, we generalized the notion of the implicativity in different directions, and obtained the notion of a weakly (i-)implicativity. We applied these notions to BCK-algebras and several groupoids, and investigated some relations among them
Summary
As a generalization of BCK-algebras, the notion of d-algebras was introduced by Neggers and. They discussed some relations between d-algebras and BCK-algebras as well as several other relations between d-algebras and oriented digraphs. The notion of an implicativity has a very important role in the study of BCK-algebras. We generalize the notion of the implicativity, which is a useful tool for investigation of BCK-algebras by using the notion of a word in general algebraic structures, the most simple mathematical structure, i.e., in the theory of a groupoid. We generalized the notion of the implicativity by using Bin( X )-product “2”, and obtain the notion of a weakly i-implicativity, and obtain several properties in BCK-algebras and other algebraic structures
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