Strictly quadratic functional equations on quasigroups. I.
Structure of triabelian quasigroups
Structure of weakly abelian quasigroups
Subloops of sedenions
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
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
For a positive integer , the usual definitions of -quasigroups are rather complicated: either by combinatorial conditions that effectively amount to Latin -cubes, or by identities on different -ary operations. In this paper, a more symmetrical approach to the specification of -quasigroups is considered. In particular, ternary quasigroups arise from actions of the modular group.
The category of totally symmetric quasigroups is binding
The Cayley graph and the growth of Steiner loops
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
A binary operation “” which satisfies the identities , , and 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 with centre of order 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
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 construction of orthogonal k-skeins and latin k-cubes.
The endocenter and its applications to quasigroup representation theory
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 equational theory of paramedial cancellation groupoids
The free commutative automorphic -generated loop of nilpotency class
A loop is automorphic if all its inner mappings are automorphisms. We construct the free commutative automorphic -generated loop of nilpotency class . It has dimension over the integers.
The holomorphic automorphism group of the complex disk.
The hyperbolic square and Möbius transformations.
The hyperbolic triangle centroid
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 inner mappings of extra loops
The operation in operator algebras
The binary operation , called Jordan triple product, and its variants (such as e.g. the sequential product or the inverted Jordan triple product ) 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.
The structure of hemigroups.