## Medial quasigroups of type $$(n,k)$$.(English)Zbl 1236.20066

The author demonstrates how the apparatus of groupoid terms might be employed for studying properties of parallelism in the ($$n,k$$)-quasigroups. She considers a particular type of block designs admitting a high degree of regularity. The author pays attention to Steiner systems of type $$S(2,k,n)$$ arising from idempotent medial quasigroups of a particular kind. She proves that each of them corresponds either to an affine space ($$n>k^2$$, $$k>2$$, in this case necessarily $$n=km$$) or to a Desarguesian affine plane ($$n=k^2$$, $$k>3$$).

### MSC:

 20N05 Loops, quasigroups 05B25 Combinatorial aspects of finite geometries
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### References:

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