Abstract. We introduce a new notion, called a Q-algebra, which is a generalization of the idea of BCH /BCI /BCK -algebras and we generalize some theorems discussed in BCIalgebras. Moreover, we introduce the notion of "quadratic" Q-algebra, and show that every quadratic Q-algebra (X; * ,e), e ∈ X, has a product of the form x * y = x − y + e, where x, y ∈ X when X is a field with |X| ≥ 3.2000 Mathematics Subject Classification. 06F35, 03G25.
Introduction. Imai and Iséki introduced two classes of abstract algebras:BCKalgebras and BCI -algebras (see [4,5]). It is known that the class of BCK -algebras is a proper subclass of the class of BCI -algebras. In [2,3] Hu and Li introduced a wide class of abstract algebras: BCH -algebras. They have shown that the class of BCI -algebras is a proper subclass of the class of BCH -algebras. Neggers and Kim (see [8]) introduced the notion of d-algebras, that is, (I) x * x = e; (IX) e * x = e; (VI) x * y = e and y * x = e imply x = y, which is another useful generalization of BCK -algebras, after which they investigated several relations between d-algebras and BCK -algebras, as well as other relations between d-algebras and oriented digraphs. At the same time, Jun, Roh, and Kim [6] introduced a new notion, called a BH -algebra, that is, (I) x * x = e; (II) x * e = x; (VI) x * y = e and y * x = e imply x = y, which is a generalization of BCH /BCI /BCK -algebras, and they showed that there is a maximal ideal in bounded BH -algebras. We introduce a new notion, called a Q-algebra, which is a generalization of BCH /BCI /BCK -algebras and generalize some theorems from the theory of BCIalgebras. Moreover, we introduce the notion of "quadratic" Q-algebra, and obtain the result that every quadratic Q-algebra (X; * ,e), e ∈ X, is of the form x * y = x − y + e, where x, y ∈ X and X is a field with |X| ≥ 3, that is, the product is linear in a special way.
