On quotient binary algebras.

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On quotient binary algebras.

[section]1. Introduction

By an algebra X = (X, *, 0), we mean a non-empty set X together with a binary operation * and a some distinguished 0.

Y. L. Liu et al. [4] studied an algebraic structure called a BCI-algebra which is an algebra (X, *, 0) with a binary operation * such that for all x, y, z [member of] X, satisfies the following properties:

(1) ((x * y) * (x * z)) * (z * y) = 0;

(2) (x * (x * y)) * y = 0;

(3) x * x = 0;

(4) x * y = 0 and y * x = 0 imply x = y.

In 2003, E. H. Roh et al. [3] introduced an algebra (X; *, [less than or equal to], 0) with a binary operation * and a nullary operation 0. Moreover, a binary relation [less than or equal to] on X is called difference algebra if it satisfies for all x, y, z [member of] X :

(D1) (X, [less than or equal to]) is a poset;

(D2) x [less than or equal to] y implies x * z [less than or equal to] y * z;

(D3) (x * y) * z [less than or equal to] (x * z) * y;

(D4) 0 [less than or equal to] x * x;

(D5) x [less than or equal to] y ...

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