Units in generalized derivatives of quasigroups

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].

Identities and generalized derivatives of quasigroups

“…By the letter T we denote the set of all quasigroup translations of a fixed quasigroup (Q, •) and their inverses relatively one fixed element, say, relatively element a. (12) , A (13) , A (23) , A (123) , A (132) }, α, β, γ ∈ T , and in every case one of the translations α, β, γ is an identity permutation, is called an isostrophic (generalized) derivative of quasigroup (Q, •) with respect to element a [14].…”

“…This paper is a prolongation of research about units in generalized derivatives of quasigroups started in [14], [16], [20].…”

“…In order to make reading of this paper more comfortable, we repeat some concepts and definitions from [14].…”