Displaying similar documents to “Rectangular loops.”

On Mikheev's construction of enveloping groups

J. I. Hall (2010)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Mikheev, starting from a Moufang loop, constructed a groupoid and reported that this groupoid is in fact a group which, in an appropriate sense, is universal with respect to enveloping the Moufang loop. Later Grishkov and Zavarnitsine gave a complete proof of Mikheev's results. Here we give a direct and self-contained proof that Mikheev's groupoid is a group, in the process extending the result from Moufang loops to Bol loops.

A scoop from groups: equational foundations for loops

Phillips, J. D., Petr Vojtěchovský (2008)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Groups are usually axiomatized as algebras with an associative binary operation, a two-sided neutral element, and with two-sided inverses. We show in this note that the same simplicity of axioms can be achieved for some of the most important varieties of loops. In particular, we investigate loops of Bol-Moufang type in the underlying variety of magmas with two-sided inverses, and obtain ``group-like'' equational bases for Moufang, Bol and C-loops. We also discuss the case when the inverses...

The free commutative automorphic 2 -generated loop of nilpotency class 3

Dylene Agda Souza de Barros, Alexander Grishkov, Petr Vojtěchovský (2012)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

A loop is automorphic if all its inner mappings are automorphisms. We construct the free commutative automorphic 2 -generated loop of nilpotency class 3 . It has dimension 8 over the integers.

Commutators and associators in Catalan loops

Jan M. Raasch (2010)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Various commutators and associators may be defined in one-sided loops. In this paper, we approximate and compare these objects in the left and right loop reducts of a Catalan loop. To within a certain order of approximation, they turn out to be quite symmetrical. Using the general analysis of commutators and associators, we investigate the structure of a specific Catalan loop which is non-commutative, but associative, that appears in the original number-theoretic application of Catalan...

Identities and the group of isostrophisms

Aleš Drápal, Viktor Alekseevich Shcherbakov (2012)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

In this paper we reexamine the concept of isostrophy. We connect it to the notion of term equivalence, and describe the action of dihedral groups that are associated with loops by means of isostrophy. We also use it to prove and present in a new way some well known facts on m -inverse loops and middle Bol loops.