107 research outputs found
Medial quasigroups of prime square order
We prove that, for any prime , there are precisely
medial quasigroups of order , up to isomorphism
Distributive and trimedial quasigroups of order 243
We enumerate three classes of non-medial quasigroups of order up to
isomorphism. There are non-medial trimedial quasigroups of order
(extending the work of Kepka, B\'en\'eteau and Lacaze), non-medial
distributive quasigroups of order (extending the work of Kepka and
N\v{e}mec), and non-medial distributive Mendelsohn quasigroups of order
(extending the work of Donovan, Griggs, McCourt, Opr\v{s}al and
Stanovsk\'y).
The enumeration technique is based on affine representations over commutative
Moufang loops, on properties of automorphism groups of commutative Moufang
loops, and on computer calculations with the \texttt{LOOPS} package in
\texttt{GAP}
Connected quandles and transitive groups
We establish a canonical correspondence between connected quandles and
certain configurations in transitive groups, called quandle envelopes. This
correspondence allows us to efficiently enumerate connected quandles of small
orders, and present new proofs concerning connected quandles of order p and 2p.
We also present a new characterization of connected quandles that are affine
Distributive groupoids are symmetric-by-medial: An elementary proof
summary:We present an elementary proof (purely in equational logic) that distributive groupoids are symmetric-by-medial
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