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Subloops of sedenions

Benard M. Kivunge, Jonathan D. H Smith (2004)

Commentationes Mathematicae Universitatis Carolinae

This note investigates sedenion multiplication from the standpoint of loop theory. New two-sided loops are obtained within the version of the sedenions introduced by the second author. Conditions are given for the satisfaction of standard loop-theoretical identities within these loops.

Symmetries in hexagonal quasigroups

Vladimír Volenec, Mea Bombardelli (2007)

Archivum Mathematicum

Hexagonal quasigroup is idempotent, medial and semisymmetric quasigroup. In this article we define and study symmetries about a point, segment and ordered triple of points in hexagonal quasigroups. The main results are the theorems on composition of two and three symmetries.

Ternary quasigroups and the modular group

Jonathan D. H. Smith (2008)

Commentationes Mathematicae Universitatis Carolinae

For a positive integer n , the usual definitions of n -quasigroups are rather complicated: either by combinatorial conditions that effectively amount to Latin n -cubes, or by 2 n identities on n + 1 different n -ary operations. In this paper, a more symmetrical approach to the specification of n -quasigroups is considered. In particular, ternary quasigroups arise from actions of the modular group.

The Cayley graph and the growth of Steiner loops

P. Plaumann, L. Sabinina, I. Stuhl (2014)

Commentationes Mathematicae Universitatis Carolinae

We study properties of Steiner loops which are of fundamental importance to develop a combinatorial theory of loops along the lines given by Combinatorial Group Theory. In a summary we describe our findings.

The centre of a Steiner loop and the maxi-Pasch problem

Andrew R. Kozlik (2020)

Commentationes Mathematicae Universitatis Carolinae

A binary operation “ · ” which satisfies the identities x · e = x , x · x = e , ( x · y ) · x = y and x · y = y · x is called a Steiner loop. This paper revisits the proof of the necessary and sufficient conditions for the existence of a Steiner loop of order n with centre of order m and discusses the connection of this problem to the question of the maximum number of Pasch configurations which can occur in a Steiner triple system (STS) of a given order. An STS which attains this maximum for a given order is said to be maxi-Pasch. We show that...

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.

The endocenter and its applications to quasigroup representation theory

Jon D. Phillips, Jonathan D. H. Smith (1991)

Commentationes Mathematicae Universitatis Carolinae

A construction is given, in a variety of groups, of a ``functorial center'' called the endocenter. The endocenter facilitates the identification of universal multiplication groups of groups in the variety, addressing the problem of determining when combinatorial multiplication groups are universal.

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

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.

The hyperbolic triangle centroid

Abraham A. Ungar (2004)

Commentationes Mathematicae Universitatis Carolinae

Some gyrocommutative gyrogroups, also known as Bruck loops or K-loops, admit scalar multiplication, turning themselves into gyrovector spaces. The latter, in turn, form the setting for hyperbolic geometry just as vector spaces form the setting for Euclidean geometry. In classical mechanics the centroid of a triangle in velocity space is the velocity of the center of momentum of three massive objects with equal masses located at the triangle vertices. Employing gyrovector space techniques we find...

The operation A B A in operator algebras

Marcell Gaál (2020)

Commentationes Mathematicae Universitatis Carolinae

The binary operation a b a , called Jordan triple product, and its variants (such as e.g. the sequential product a b a or the inverted Jordan triple product a b - 1 a ) appear in several branches of operator theory and matrix analysis. In this paper we briefly survey some analytic and algebraic properties of these operations, and investigate their intimate connection to Thompson type isometries in different operator algebras.

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