Endomorphic GE-derivations

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.