A geometric approach to universal quasigroup identities

Václav J. Havel

Displaying similar documents to “A geometric approach to universal quasigroup identities”

Quasigroups arisen by right nuclear extension

Péter T. Nagy, Izabella Stuhl (2012)

Commentationes Mathematicae Universitatis Carolinae

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The aim of this paper is to prove that a quasigroup $Q$ with right unit is isomorphic to an $f$ -extension of a right nuclear normal subgroup $G$ by the factor quasigroup $Q / G$ if and only if there exists a normalized left transversal $Σ \subset Q$ to $G$ in $Q$ such that the right translations by elements of $Σ$ commute with all right translations by elements of the subgroup $G$ . Moreover, a loop $Q$ is isomorphic to an $f$ -extension of a right nuclear normal subgroup $G$ by a loop if and only if $G$ is middle-nuclear, and...

Right division in Moufang loops

Maria de Lourdes M. Giuliani, Kenneth Walter Johnson (2010)

Commentationes Mathematicae Universitatis Carolinae

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If $(G, \cdot)$ is a group, and the operation $(*)$ is defined by $x * y = x \cdot y^{- 1}$ then by direct verification $(G, *)$ is a quasigroup which satisfies the identity $(x * y) * (z * y) = x * z$ . Conversely, if one starts with a quasigroup satisfying the latter identity the group $(G, \cdot)$ can be constructed, so that in effect $(G, \cdot)$ is determined by its right division operation. Here the analogous situation is examined for a Moufang loop. Subtleties arise which are not present in the group case since there is a choice of defining identities and the identities...

On multiplication groups of relatively free quasigroups isotopic to Abelian groups

Aleš Drápal (2005)

Czechoslovak Mathematical Journal

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If $Q$ is a quasigroup that is free in the class of all quasigroups which are isotopic to an Abelian group, then its multiplication group $M l t Q$ is a Frobenius group. Conversely, if $M l t Q$ is a Frobenius group, $Q$ a quasigroup, then $Q$ has to be isotopic to an Abelian group. If $Q$ is, in addition, finite, then it must be a central quasigroup (a $T$ -quasigroup).

Medial quasigroups of prime square order

David Stanovský (2016)

Commentationes Mathematicae Universitatis Carolinae

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We prove that, for any prime $p$ , there are precisely $2 p^{4} - p^{3} - p^{2} - 3 p - 1$ medial quasigroups of order $p^{2}$ , up to isomorphism.

Bicrossproduct Hopf quasigroups

Jennifer Klim, Shahn Majid (2010)

Commentationes Mathematicae Universitatis Carolinae

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We recall the notion of Hopf quasigroups introduced previously by the authors. We construct a bicrossproduct Hopf quasigroup $k M ▹ ◂ k (G)$ from every group $X$ with a finite subgroup $G \subset X$ and IP quasigroup transversal $M \subset X$ subject to certain conditions. We identify the octonions quasigroup $G_{𝕆}$ as transversal in an order 128 group $X$ with subgroup $ℤ_{2}^{3}$ and hence obtain a Hopf quasigroup $k G_{𝕆} > ◂ k (ℤ_{2}^{3})$ as a particular case of our construction.