Elementary properties of \(n\)-ary groupoids with division. (Russian) Zbl 0648.20070

Shakaleo, Manuel Pedro; Prodan, N. I.

Mat. Issled. 102, 106-110 (1988).

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.Broziková

MSC:

20N15	\(n\)-ary systems \((n\ge 3)\)
20N05	Loops, quasigroups

Keywords:

n-ary groupoid; division groupoid; quasigroup; loop; regular \({\mathcal D}\)- groupoid; homotopy; epitopy