Endomorphic GE-derivations

Abstract

Using the binary operation "hooked left arrow" on a GE-algebra X given by (x, y) = (y * x) * x and the GEendomorphism Omega : X -> X, the notion of Omega (l , r)-endomorphic (resp., Omega (r , l)-endomorphic) GE-derivation is introduced, and several properties are investigated. Also, examples that illustrate these are provided. Conditions under which Omega (l , r)-endomorphic GE-derivations or Omega (l , r)-endomorphic GE-derivations to satisfy certain equalities and inequalities are studied. We explored the conditions under which f becomes order preserving when f is an Omega (l , r)-endomorphic GE-derivation or an Omega (r , l)-endomorphic GE-derivation on X. The f-kernel and Omega-kernel of f formed by the Omega (r , l)-endomorphic GE-derivation or Omega (l , r)-endomorphic GE-derivation turns out to be GE-subalgebras. It is observed that the Omega-kernel off is a GE-filter of X. The condition under which the f-kernel off formed by the Omega (r , l)-endomorphic GE-derivation or Omega (l , r)-endomorphic GE-derivation becomes a GE-filter is explored.