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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.Broziková
##### MSC:
 20N15 $$n$$-ary systems $$(n\ge 3)$$ 20N05 Loops, quasigroups
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