Arithmetical forms of quasigroups
Automorphic loops and metabelian groups
Given a uniquely 2-divisible group , we study a commutative loop which arises as a result of a construction in “Engelsche elemente noetherscher gruppen” (1957) by R. Baer. We investigate some general properties and applications of “” and determine a necessary and sufficient condition on in order for to be Moufang. In “A class of loops categorically isomorphic to Bruck loops of odd order” (2014) by M. Greer, it is conjectured that is metabelian if and only if is an automorphic loop. We...
Autotopies des quasigroupes et des systèmes associatifs
Axiomatization of quasigroups
Quasigroups were originally described combinatorially, in terms of existence and uniqueness conditions on the solutions to certain equations. Evans introduced a universal-algebraic characterization, as algebras with three binary operations satisfying four identities. Now, quasigroups are redefined as heterogeneous algebras, satisfying just two conditions respectively known as hypercommutativity and hypercancellativity.
Axioms for trimedial quasigroups
We give new equations that axiomatize the variety of trimedial quasigroups. We also improve a standard characterization by showing that right semimedial, left F-quasigroups are trimedial.