Quasigroups, Units and Belousov’s Problem # 18

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Submitted: 2024-12-10; Published: 2024-12-10

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Abstract

We investigate {left, right, middle} units of quasigroups and families of identities which might imply their existence. A prominent role is played by the newly introduced notion of derivative operation, generalizing Belousov's notions of left/right derivative operations for quasigroups. Partial solutions of the Belousov's Problem # 18 and its generalizations are obtained. Several related problems remain open.