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Medial quasigroups of type ( n , k )

Alena Vanžurová

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2010)

  • Volume: 49, Issue: 2, page 107-122
  • ISSN: 0231-9721

Abstract

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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 ≥ 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 · x = x holds, and mediality means that the identity ( x y ) ( u v ) = ( x u ) ( y v ) is satisfied).

How to cite

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Vanžurová, Alena. "Medial quasigroups of type $(n,k)$." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 49.2 (2010): 107-122. <http://eudml.org/doc/116518>.

@article{Vanžurová2010,
abstract = {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).},
author = {Vanžurová, Alena},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {Quasigroup; idempotent groupoid term; mediality; incidence structure; parallelism; affine space; desarguesian affine plane; medial quasigroups; idempotent groupoid terms; mediality; incidence structures; parallelism; affine spaces; Desarguesian affine plane},
language = {eng},
number = {2},
pages = {107-122},
publisher = {Palacký University Olomouc},
title = {Medial quasigroups of type $(n,k)$},
url = {http://eudml.org/doc/116518},
volume = {49},
year = {2010},
}

TY - JOUR
AU - Vanžurová, Alena
TI - Medial quasigroups of type $(n,k)$
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2010
PB - Palacký University Olomouc
VL - 49
IS - 2
SP - 107
EP - 122
AB - 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).
LA - eng
KW - Quasigroup; idempotent groupoid term; mediality; incidence structure; parallelism; affine space; desarguesian affine plane; medial quasigroups; idempotent groupoid terms; mediality; incidence structures; parallelism; affine spaces; Desarguesian affine plane
UR - http://eudml.org/doc/116518
ER -

References

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