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Inverse property, flexible loops

J. D. Phillips — 2000

Bollettino dell'Unione Matematica Italiana

Uno dei metodi migliori per scoprire le proprietà di un cappio chiuso è studiarne il gruppo di moltiplicazione [3], [4]. In questo breve saggio descriviamo i gruppi di moltiplicazione di una classe importante di cappi, e cioè di quella dei cappi flessibili che posseggono la proprietà inversa.

The commingling of commutativity and associativity in Bol loops

Jon D. Phillips — 2016

Commentationes Mathematicae Universitatis Carolinae

Commutative Moufang loops were amongst the first (nonassociative) loops to be investigated; a great deal is known about their structure. More generally, the interplay of commutativity and associativity in (not necessarily commutative) Moufang loops is well known, e.g., the many associator identities and inner mapping identities involving commutant elements, especially those involving the exponent three. Here, we investigate all of this in the variety of Bol loops.

On Moufang A-loops

Jon D. Phillips — 2000

Commentationes Mathematicae Universitatis Carolinae

In a series of papers from the 1940’s and 1950’s, R.H. Bruck and L.J. Paige developed a provocative line of research detailing the similarities between two important classes of loops: the diassociative A-loops and the Moufang loops ([1]). Though they did not publish any classification theorems, in 1958, Bruck’s colleague, J.M. Osborn, managed to show that diassociative, commutative A-loops are Moufang ([5]). In [2] we relaunched this now over 50 year old program by examining conditions under which...

F-quasigroups isotopic to groups

Tomáš KepkaMichael K. KinyonJon D. Phillips — 2010

Commentationes Mathematicae Universitatis Carolinae

In Kepka T., Kinyon M.K., Phillips J.D., , , we showed that every loop isotopic to an F-quasigroup is a Moufang loop. Here we characterize, via two simple identities, the class of F-quasigroups which are isotopic to groups. We call these quasigroups FG-quasigroups. We show that FG-quasigroups are linear over groups. We then use this fact to describe their structure. This gives us, for instance, a complete description of the simple FG-quasigroups. Finally, we show an equivalence of equational classes...

F-quasigroups and generalized modules

Tomáš KepkaMichael K. KinyonJon D. Phillips — 2008

Commentationes Mathematicae Universitatis Carolinae

In Kepka T., Kinyon M.K., Phillips J.D., , J. Algebra (2007), 435–461, we showed that every F-quasigroup is linear over a special kind of Moufang loop called an NK-loop. Here we extend this relationship by showing an equivalence between the class of (pointed) F-quasigroups and the class corresponding to a certain notion of generalized module (with noncommutative, nonassociative addition) for an associative ring.

A scoop from groups: equational foundations for loops

Phillips, J. D.Petr Vojtěchovský — 2008

Commentationes Mathematicae Universitatis Carolinae

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 are only...

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