Endomorphic GE-derivations

Using the binary operation " $\Lsh$" on a GE-algebra $X$ given by $\Lsh (x,y)=(y\ast x)\ast x$ and the GE-endomorphism $\mathrm{\Omega }:X\to X$, the notion of ${\mathrm{\Omega }}_{(l,r)}$-endomorphic (resp., ${\mathrm{\Omega }}_{(r,l)}$-endomorphic) GE-derivation is introduced, and several properties are investigated. Also, examples that illustrate these are provided. Conditions under which ${\mathrm{\Omega }}_{(l,r)}$-endomorphic GE-derivations or ${\mathrm{\Omega }}_{(l,r)}$-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 ${\mathrm{\Omega }}_{(l,r)}$-endomorphic GE-derivation or an ${\mathrm{\Omega }}_{(r,l)}$-endomorphic GE-derivation on $X$. The $f$-kernel and $\mathrm{\Omega }$-kernel of $f$ formed by the ${\mathrm{\Omega }}_{(r,l)}$-endomorphic GE-derivation or ${\mathrm{\Omega }}_{(l,r)}$-endomorphic GE-derivation turns out to be GE-subalgebras. It is observed that the $\mathrm{\Omega }$-kernel of $f$ is a GE-filter of $X$. The condition under which the $f$-kernel of $f$ formed by the ${\mathrm{\Omega }}_{(r,l)}$-endomorphic GE-derivation or ${\mathrm{\Omega }}_{(l,r)}$-endomorphic GE-derivation becomes a GE-filter is explored.