Endomorphic GE-derivations

Abstract

Using the binary operation '$ {\Lsh} $' on a GE-algebra $ {X} $ given by $ {\Lsh}({x}, {y}) = ({y}{*} {x}){*} {x} $ and the GE-endomorphism $ {\Omega} : {X} \rightarrow {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 of $ {f} $ is a GE-filter of $ {X} $. The condition under which the $ {f} $-kernel of $ {f} $ formed by the $ {\Omega}_{(r, l)} $-endomorphic GE-derivation or $ {\Omega}_{(l, r)} $-endomorphic GE-derivation becomes a GE-filter is explored.

Keywords

• $ {\omega}_{(l, r)} $-endomorphic ge-derivation
• $ {\omega}_{(r, l)} $-endomorphic ge-derivation
• endomorphism
• ge-filter
• kernel