Units in generalized derivatives of quasigroups

TL;DR

This paper investigates the existence of units in generalized derivatives of quasigroups, identifying numerous cases and providing proofs or counterexamples for each, thereby advancing the algebraic understanding of quasigroup derivatives.

Contribution

It systematically studies units in generalized derivatives of quasigroups, covering 1944 cases with proofs or counterexamples, expanding the theoretical framework.

Findings

Existence of units varies across different cases.

Many cases have been conclusively proved or disproved.

The research deepens understanding of algebraic structures in quasigroup derivatives.

Abstract

We proceed the research of generalized quasigroup derivatives started in early papers of the last co-author ([20, p. 212], [13]). For any quasigroup there exist 648 generalized derivatives. Here we study the problem about existence of units (left, right, middle) in quasigroups that are a generalized derivative of a quasigroup. There exist 1944 various cases. For every case we find a proof or a counterexample, often using Prover or Mace [15, 14].