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On ideals in regular ternary semigroups

Tapan K. Dutta, Sukhendu Kar, Bimal K. Maity (2008)

Discussiones Mathematicae - General Algebra and Applications

In this paper we study some interesting properties of regular ternary semigroups, completely regular ternary semigroups, intra-regular ternary semigroups and characterize them by using various ideals of ternary semigroups.

On left distributive left idempotent groupoids

Přemysl Jedlička (2005)

Commentationes Mathematicae Universitatis Carolinae

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.

On Lie semiheaps and ternary principal bundles

Andrew James Bruce (2024)

Archivum Mathematicum

We introduce the notion of a Lie semiheap as a smooth manifold equipped with a para-associative ternary product. For a particular class of Lie semiheaps we establish the existence of left-invariant vector fields. Furthermore, we show how such manifolds are related to Lie groups and establish the analogue of principal bundles in this ternary setting. In particular, we generalise the well-known ‘heapification’ functor to the ambience of Lie groups and principal bundles.

On loops that are abelian groups over the nucleus and Buchsteiner loops

Piroska Csörgö (2008)

Commentationes Mathematicae Universitatis Carolinae

We give sufficient and in some cases necessary conditions for the conjugacy closedness of Q / Z ( Q ) provided the commutativity of Q / N . We show that if for some loop Q , Q / N and Inn Q are abelian groups, then Q / Z ( Q ) is a CC loop, consequently Q has nilpotency class at most three. We give additionally some reasonable conditions which imply the nilpotency of the multiplication group of class at most three. We describe the structure of Buchsteiner loops with abelian inner mapping groups.

On loops whose inner permutations commute

Piroska Csörgö, Tomáš Kepka (2004)

Commentationes Mathematicae Universitatis Carolinae

Multiplication groups of (finite) loops with commuting inner permutations are investigated. Special attention is paid to the normal closure of the abelian permutation group.

On Mikheev's construction of enveloping groups

J. I. Hall (2010)

Commentationes Mathematicae Universitatis Carolinae

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.

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