Quasigroups arisen by right nuclear extension

Péter T. Nagy; Izabella Stuhl

Displaying similar documents to “Quasigroups arisen by right nuclear extension”

A geometric approach to universal quasigroup identities

Václav J. Havel (1993)

Archivum Mathematicum

Similarity:

In the present paper we construct the accompanying identity $\hat{I}$ of a given quasigroup identity $I$ . After that we deduce the main result: $I$ is isotopically invariant (i.e., for every guasigroup $Q$ it holds that if $I$ is satisfied in $Q$ then $I$ is satisfied in every quasigroup isotopic to $Q$ ) if and only if it is equivalent to $\hat{I}$ (i.e., for every quasigroup $Q$ it holds that in $Q$ either $I, \hat{I}$ are both satisfied or both not).

Medial quasigroups of prime square order

David Stanovský (2016)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We prove that, for any prime $p$ , there are precisely $2 p^{4} - p^{3} - p^{2} - 3 p - 1$ medial quasigroups of order $p^{2}$ , up to isomorphism.

Classification of quasigroups according to directions of translations I

Fedir Sokhatsky, Alla Lutsenko (2020)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

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...

Solution of Belousov's problem

Maks A. Akivis, Vladislav V. Goldberg (2001)

Discussiones Mathematicae - General Algebra and Applications

Similarity:

The authors prove that a local n-quasigroup defined by the equation $x_{n + 1} = F (x ₁, . . ., x ₙ) = (f ₁ (x ₁) + . . . + f ₙ (x ₙ)) / (x ₁ + . . . + x ₙ)$ , where $f_{i} (x_{i})$ , i,j = 1,...,n, are arbitrary functions, is irreducible if and only if any two functions $f_{i} (x_{i})$ and $f_{j} (x_{j})$ , i ≠ j, are not both linear homogeneous, or these functions are linear homogeneous but $f_{i} (x_{i}) / x_{i} \neq f_{j} (x_{j}) / x_{j}$ . This gives a solution of Belousov’s problem to construct examples of irreducible n-quasigroups for any n ≥ 3.

Extending the structural homomorphism of LCC loops

Piroska Csörgö (2005)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

A loop $Q$ is said to be left conjugacy closed if the set $A = {L_{x} / x \in Q}$ is closed under conjugation. Let $Q$ be an LCC loop, let $ℒ$ and $ℛ$ be the left and right multiplication groups of $Q$ respectively, and let $I (Q)$ be its inner mapping group, $M (Q)$ its multiplication group. By Drápal’s theorem [3, Theorem 2.8] there exists a homomorphism $Λ : ℒ \to I (Q)$ determined by $L_{x} \to R_{x}^{- 1} L_{x}$ . In this short note we examine different possible extensions of this $Λ$ and the uniqueness of these extensions.

Symmetric linear operator identities in quasigroups

Reza Akhtar (2017)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Let $G$ be a quasigroup. Associativity of the operation on $G$ can be expressed by the symbolic identity $R_{x} L_{y} = L_{y} R_{x}$ of left and right multiplication maps; likewise, commutativity can be expressed by the identity $L_{x} = R_{x}$ . In this article, we investigate symmetric linear identities: these are identities in left and right multiplication symbols in which every indeterminate appears exactly once on each side, and whose sides are mirror images of each other. We determine precisely which identities imply associativity...