fq-Derivations of b-algebras

Abstract

Let X = (X,∗,0) be a B-algebra and f a self-map on X. We study some properties of X for the self-map d f q is an outside and inside fq-derivation of X, respectively, as follows:

(∀x, y ∈ X)(d^f_q(x ∗ y) = f(x) ∗ d^f_q (y)),

(∀x, y ∈ X)(d^f_q (x ∗ y) = d^f_q(x) ∗ f(y)).

In addition, we define and study some properties of (right-left) and (left-right) fq-derivation of X, respectively, as follows:

(∀x, y ∈ X)(d^f_q(x ∗ y) = (f(x) ∗ d^f_q(y))∧(d^f_q(x) ∗ f(y))),

(∀x, y ∈ X)(d^f_q(x ∗ y) = (d^f_q(x) ∗ f(y))∧(f(x) ∗ d^f_q(y))).