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Antiflexible Latin directed triple systems

Andrew R. Kozlik (2015)

Commentationes Mathematicae Universitatis Carolinae

It is well known that given a Steiner triple system one can define a quasigroup operation · upon its base set by assigning x · x = x for all x and x · y = z , where z is the third point in the block containing the pair { x , y } . The same can be done for Mendelsohn triple systems, where ( x , y ) is considered to be ordered. But this is not necessarily the case for directed triple systems. However there do exist directed triple systems, which induce a quasigroup under this operation and these are called Latin directed triple systems....

Automorphic loops and metabelian groups

Mark Greer, Lee Raney (2020)

Commentationes Mathematicae Universitatis Carolinae

Given a uniquely 2-divisible group G , we study a commutative loop ( G , ∘ ) 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 G in order for ( G , ∘ ) to be Moufang. In “A class of loops categorically isomorphic to Bruck loops of odd order” (2014) by M. Greer, it is conjectured that G is metabelian if and only if ( G , ∘ ) is an automorphic loop. We...

Axiomatization of quasigroups

Jonathan D.H. Smith (2006)

Discussiones Mathematicae - General Algebra and Applications

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

Michael K. Kinyon, Jon D. Phillips (2004)

Commentationes Mathematicae Universitatis Carolinae

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.

Bol loop actions.

Sbitneva, Larissa (2000)

Commentationes Mathematicae Universitatis Carolinae

Bol loop actions

Larissa V. Sbitneva (2000)

Commentationes Mathematicae Universitatis Carolinae

The notions of left Bol and Bol-Bruck actions are introduced. A purely algebraic analogue of a Nono family (Lie triple family), the so called Sabinin-Nono family, is given. It is shown that any Sabinin-Nono family is a left Bol-Bruck action. Finally it is proved that any local Nono family is a local left Bol-Bruck action. On general matters see [L.V. Sabinin 91, 99].

Bol loops with a large left nucleus

Orin Chein, Edgar G. Goodaire (2008)

Commentationes Mathematicae Universitatis Carolinae

Possession of a unique nonidentity commutator/associator is a property which dominates the theory of loops whose loop rings, while not associative, nevertheless satisfy an ``interesting'' identity. Indeed, until now, with the exception of some ad hoc examples, the only known class of Bol loops whose loop rings satisfy the right Bol identity have this property. In this paper, we identify another class of loops whose loop rings are ``strongly right alternative'' and present various constructions of...

Bol-loops of order 3 · 2 n

Daniel Wagner, Stefan Wopperer (2007)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

In this article we construct proper Bol-loops of order 3 · 2 n using a generalisation of the semidirect product of groups defined by Birkenmeier and Xiao. Moreover we classify the obtained loops up to isomorphism.

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