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Keywords:
Quasigroup; idempotent groupoid term; mediality; incidence structure; parallelism; affine space; desarguesian affine plane
Summary:
Our aim is to demonstrate how the apparatus of groupoid terms (on two variables) might be employed for studying properties of parallelism in the so called $(n,k)$-quasigroups. We show that an incidence structure associated with a medial quasigroup of type $(n,k)$, $n>k\ge 3$, is either an affine space of dimension at least three, or a desarguesian plane. Conversely, if we start either with an affine space of order $k>2$ and dimension $m$, or with a desarguesian affine plane of order $k>2$ then there is a medial quasigroup of type $(k^m,k)$, $m>2$ such that the incidence structure naturally associated to a quasigroup is isomorphic with the starting one (the simplest case $k=2$ can be examined separately but is of little interest). The proofs are mostly based on properties of groupoid term functions, applied to idempotent medial quasigroups (idempotency means that $x\cdot x=x$ holds, and mediality means that the identity $(xy)(uv)=(xu)(yv)$ is satisfied).
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