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Classification of quasigroups according to directions of translations I

Fedir Sokhatsky, Alla Lutsenko (2020)

Commentationes Mathematicae Universitatis Carolinae

It is proved that every translation in a quasigroup has two independent parameters. One of them is a bijection of the carrier set. The second parameter is called a direction here. Properties of directions in a quasigroup are considered in the first part of the work. In particular, totally symmetric, semisymmetric, commutative, left and right symmetric and also asymmetric quasigroups are characterized within these concepts. The sets of translations of the same direction are under consideration in...

Classification of quasigroups according to directions of translations II

Fedir Sokhatsky, Alla Lutsenko (2021)

Commentationes Mathematicae Universitatis Carolinae

In each quasigroup Q there are defined six types of translations: the left, right and middle translations and their inverses. Two translations may coincide as permutations of Q , and yet be different when considered upon the web of the quasigroup. We shall call each of the translation types a direction and will associate it with one of the elements ι , l , r , s , l s and r s , i.e., the elements of a symmetric group S 3 . Properties of the directions are considered in part 1 of “Classification of quasigroups according...

Classification results in quasigroup and loop theory via a combination of automated reasoning tools

Volker Sorge, Simon Colton, Roy McCasland, Andreas Meier (2008)

Commentationes Mathematicae Universitatis Carolinae

We present some novel classification results in quasigroup and loop theory. For quasigroups up to size 5 and loops up to size 7, we describe a unique property which determines the isomorphism (and in the case of loops, the isotopism) class for any example. These invariant properties were generated using a variety of automated techniques --- including machine learning and computer algebra --- which we present here. Moreover, each result has been automatically verified, again using a variety of techniques...

Clifford algebras, Möbius transformations, Vahlen matrices, and B -loops

Jimmie Lawson (2010)

Commentationes Mathematicae Universitatis Carolinae

In this paper we show that well-known relationships connecting the Clifford algebra on negative euclidean space, Vahlen matrices, and Möbius transformations extend to connections with the Möbius loop or gyrogroup on the open unit ball B in n -dimensional euclidean space ℝ n . One notable achievement is a compact, convenient formula for the Möbius loop operation a * b = ( a + b ) ( 1 - a b ) - 1 , where the operations on the right are those arising from the Clifford algebra (a formula comparable to ( w + z ) ( 1 + w ¯ z ) - 1 for the Möbius loop multiplication...

Closure conditions of commutativity

Václav J. Havel, Josef Klouda (1994)

Archivum Mathematicum

There are investigated some closure conditions of Thomsen type in 3-webs which gurantee that at least one of coordinatizing quasigroups of a given 3-web is commutative.

Combinatorial aspects of code loops

Petr Vojtěchovský (2000)

Commentationes Mathematicae Universitatis Carolinae

The existence and uniqueness (up to equivalence defined below) of code loops was first established by R. Griess in [3]. Nevertheless, the explicit construction of code loops remained open until T. Hsu introduced the notion of symplectic cubic spaces and their Frattini extensions, and pointed out how the construction of code loops followed from the (purely combinatorial) result of O. Chein and E. Goodaire contained in [2]. Within this paper, we focus on their combinatorial construction and prove...

Commutation of operations and its relationship with Menger and Mann superpositions

Fedir M. Sokhatsky (2004)

Discussiones Mathematicae - General Algebra and Applications

The article considers a problem from Trokhimenko paper [13] concerning the study of abstract properties of commutations of operations and their connection with the Menger and Mann superpositions. Namely, abstract characterizations of some classes of operation algebras, whose signature consists of arbitrary families of commutations of operations, Menger and Mann superpositions and their various connections are found. Some unsolved problems are given at the end of the article.

Commutative subloop-free loops

Martin Beaudry, Louis Marchand (2011)

Commentationes Mathematicae Universitatis Carolinae

We describe, in a constructive way, a family of commutative loops of odd order, n ≥ 7 , which have no nontrivial subloops and whose multiplication group is isomorphic to the alternating group 𝒜 n .

Commutators and associators in Catalan loops

Jan M. Raasch (2010)

Commentationes Mathematicae Universitatis Carolinae

Various commutators and associators may be defined in one-sided loops. In this paper, we approximate and compare these objects in the left and right loop reducts of a Catalan loop. To within a certain order of approximation, they turn out to be quite symmetrical. Using the general analysis of commutators and associators, we investigate the structure of a specific Catalan loop which is non-commutative, but associative, that appears in the original number-theoretic application of Catalan loops.

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