On Derivations of Subtraction Algebras

Yıl 2012, Cilt: 41 Sayı: 2, 157 - 168, 01.02.2012

Yong Ho Yon Kyung Ho Kim

Öz

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.

Anahtar Kelimeler

Subtraction algebra, Derivation, Simple derivation, Non-expansive map, Dual closure operator, Boolean algebra, 2000 AMS Classification: 03 G 25

Kaynakça

• Abbott, J. C. Sets, Lattices and Boolean Algebras (Allyn and Bacon, Boston, 1969).
• Schein, B. M. Difference semigroups, Comm. Algebra 20, 2153–2169, 1992.
• Xin, X. L., Li, T. Y. and Lu, J. H. On derivations of lattices, Inform. Sci. 178, 307–316, 2008.
• Zelinka, B. Subtraction semigroups, Math. Bohem. 120, 445–447, 1995.