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)} $-endomorphic GE-derivation becomes a GE-filter is explored.
Citation: Young Bae Jun, Ravikumar Bandaru, Amal S. Alali. Endomorphic GE-derivations[J]. AIMS Mathematics, 2025, 10(1): 1792-1813. doi: 10.3934/math.2025082
Using the binary operation "$ {\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.
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Young Bae Jun, Ravikumar Bandaru, Amal S. Alali. Endomorphic GE-derivations[J]. AIMS Mathematics, 2025, 10(1): 1792-1813. doi: 10.3934/math.2025082
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