On Derivations of Subtraction Algebras

Abstract

The aim of this paper is to introduce the notion of derivations of subtraction algebras. We define a derivation of a subtraction algebra X as a function d on X satisfying d(x − y) = (d(x) − y) ∧ (x − d(y)) for all x, y ∈ X. Then it is characterized as a function d satisfying d(x − y) = d(x) − y for all x, y ∈ X. Also we define a simple derivation as a function da on X satisfying da(x) = x−a for all x ∈ X. Then every simple derivation is a derivation and every derivation can be partially a simple derivation on intervals. For any derivation d of a subtraction algebra X, Ker(d) and Im(d) are ideals of X, and X/Ker(d) ∼= Im(d) and X/Im(d) ∼= Ker(d). Finally, we show that every subtraction algebra X is embedded in Im(d) × Ker(d) for any derivation d of X.