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Termal groupoids

Jaroslav Ježek (2002)

Czechoslovak Mathematical Journal

We investigate the factor of the groupoid of terms through the largest congruence with a given set among its blocks. The set is supposed to be closed for overterms.

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.

Ternary semigroups of morphisms of objects in categories

Antoni Chronowski, Miroslav Novotný (1995)

Archivum Mathematicum

In this paper the notion of a ternary semigroup of morphisms of objects in a category is introduced. The connection between an isomorphism of categories and an isomorphism of ternary semigroups of morphisms of suitable objects in these categories is considered. Finally, the results obtained for general categories are applied to the categories 𝐑𝐄𝐋 n + 1 and 𝐀𝐋𝐆 n which were studied in [5].

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 free one-generated left distributive algebra: basics and a simplified proof of the division algorithm

Richard Laver, Sheila Miller (2013)

Open Mathematics

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...

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