On Derivations of Subtraction Algebras

Authors : Yong Ho YON, Kyung Ho KİM

Pages : 157-168

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Publication Date : 2012-02-01

Article Type : Research Paper

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 dinsert ignore into journalissuearticles values(x − y); = insert ignore into journalissuearticles values(dinsert ignore into journalissuearticles values(x); − y); ∧ insert ignore into journalissuearticles values(x − dinsert ignore into journalissuearticles values(y);); for all x, y ∈ X. Then it is characterized as a function d satisfying dinsert ignore into journalissuearticles values(x − y); = dinsert ignore into journalissuearticles values(x); − y for all x, y ∈ X. Also we define a simple derivation as a function da on X satisfying dainsert ignore into journalissuearticles values(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, Kerinsert ignore into journalissuearticles values(d); and Iminsert ignore into journalissuearticles values(d); are ideals of X, and X/Kerinsert ignore into journalissuearticles values(d); ∼= Iminsert ignore into journalissuearticles values(d); and X/Iminsert ignore into journalissuearticles values(d); ∼= Kerinsert ignore into journalissuearticles values(d);. Finally, we show that every subtraction algebra X is embedded in Iminsert ignore into journalissuearticles values(d); × Kerinsert ignore into journalissuearticles values(d); for any derivation d of X.
Keywords : Subtraction algebra, Derivation, Simple derivation, Non expansive map, Dual closure operator, Boolean algebra, 2000 AMS Classification 03 G 25