Ternary structures and partial semigroups
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 congruence lattice of implication algebras.
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 free one-generated left distributive algebra: basics and a simplified proof of the division algorithm
The left distributive law is the law a· (b· c) = (a·b) · (a· c). Left distributive algebras have been classically used in the study of knots and braids, and more recently free left distributive algebras have been studied in connection with large cardinal axioms in set theory. We provide a survey of results on the free left distributive algebra on one generator, A, and a new, simplified proof of the existence of a normal form for terms in A. Topics included are: the confluence of A, the linearity...
The group ring of Richard Thompson’s Group has no minimal non-zero ideals
We use a total order on Thompson’s group to show that the group ring has no minimal non-zero ideals.
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 hyperrings of order 3.
The ideal structure of simple ternary algebras
The inner mappings of extra loops
The lattice of solid -subgroups of a retractable group
The Levitzki Radical for Omega-groups