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Elementary properties of \(n\)-ary groupoids with division. (Russian) Zbl 0648.20070
An n-ary groupoid Q(A) is called division groupoid (written \({\mathcal D}\)- groupoid) if the equation \(A(a_ 1^{s-1},x,a^ n_{s+1})=b\) has a solution for all \(a_ 1,...,a_ n,b\in Q\) and every \(s\in \{1,2,...,n\}\). A \({\mathcal D}\)-groupoid Q(A) is called regular if \(A(a_ 1^{s-1},x,a^ n_{s+1})=A(a_ 1^{s-1},y,a^ n_{s+1})\) implies \(A(b_ 1^{s-1},x,b^ n_{s+1})=A(b_ 1^{s-1},y,b^ n_{s+1})\) for all \(b_ 1,...,b_ n\in Q\) and every \(s\in \{1,...,n\}.\)
The authors investigate the questions when: (1) a homomorphic image of a \({\mathcal D}\)-groupoid is a quasigroup, (2) a homomorphic image of a regular \({\mathcal D}\)-groupoid is a loop, (3) a mapping of a regular \({\mathcal D}\)- groupoid Q(A) in a regular \({\mathcal D}\)-groupoid Q’(A’) is a homotopy or epitopy, (4) an epitopic image of a \({\mathcal D}\)-groupoid is a regular \({\mathcal D}\)-groupoid.
Reviewer: E.Brozikov√°
MSC:
20N15 \(n\)-ary systems \((n\ge 3)\)
20N05 Loops, quasigroups
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