20 research outputs found
On left distributive left idempotent groupoids
summary:We study the groupoids satisfying both the left distributivity and the left idempotency laws. We show that they possess a canonical congruence admitting an idempotent groupoid as factor. This congruence gives a construction of left idempotent left distributive groupoids from left distributive idempotent groupoids and right constant groupoids
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}
Skew left braces and 2-reductive solutions of the Yang-Baxter equation
We study 2-reductive non-involutive non-degenerate set-theoretic solutions of
the Yang-Baxter equation. We give a combinatorial construction of any such
solution of any (even infinite) size. We also prove that solutions associated
to a skew left brace are 2-reductive if and only if the skew left brace is
nilpotent of class 2. Moreover, all such skew left braces are actually bi-skew
left braces. We focus on these structures and we give several equivalent
properties characterizing solutions associated to bi-skew left braces.Comment: 21 page
Cocyclic braces and indecomposable cocyclic solutions of the Yang-Baxter equation
We study indecomposable involutive set-theoretic solutions of the Yang-Baxter
equation with cyclic permutation groups (cocyclic solutions). In particular, we
show that there is no one-to-one correspondence between indecomposable cocyclic
solutions and cocyclic braces which contradicts recent results in
\cite{Rump21}