Displaying similar documents to “ m -medial n -quasigroups”

Ternary quasigroups and the modular group

Jonathan D. H. Smith (2008)

Commentationes Mathematicae Universitatis Carolinae

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For a positive integer n , the usual definitions of n -quasigroups are rather complicated: either by combinatorial conditions that effectively amount to Latin n -cubes, or by 2 n identities on n + 1 different n -ary operations. In this paper, a more symmetrical approach to the specification of n -quasigroups is considered. In particular, ternary quasigroups arise from actions of the modular group.

Medial quasigroups of type ( n , k )

Alena Vanžurová (2010)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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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...

A geometric approach to universal quasigroup identities

Václav J. Havel (1993)

Archivum Mathematicum

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In the present paper we construct the accompanying identity I ^ of a given quasigroup identity I . After that we deduce the main result: I is isotopically invariant (i.e., for every guasigroup Q it holds that if I is satisfied in Q then I is satisfied in every quasigroup isotopic to Q ) if and only if it is equivalent to I ^ (i.e., for every quasigroup Q it holds that in Q either I , I ^ are both satisfied or both not).