Geodesic
Medial quasigroups of type $(n,k)$
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 49 (2010) no. 2, pp. 107-122 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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).
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).
Classification : 05B25, 20N05
Keywords: Quasigroup; idempotent groupoid term; mediality; incidence structure; parallelism; affine space; desarguesian affine plane 
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     author = {Van\v{z}urov\'a, Alena},
     title = {Medial quasigroups of type $(n,k)$},
     journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
     pages = {107--122},
     year = {2010},
     volume = {49},
     number = {2},
     mrnumber = {2796951},
     zbl = {1236.20066},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AUPO_2010_49_2_a8/}
}
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Vanžurová, Alena. Medial quasigroups of type $(n,k)$. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 49 (2010) no. 2, pp. 107-122. http://geodesic.mathdoc.fr/item/AUPO_2010_49_2_a8/