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 BCI -algebras. 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 .