Medial quasigroups of prime square order

David Stanovský

Displaying similar documents to “Medial quasigroups of prime square order”

Classification of quasigroups according to directions of translations I

Fedir Sokhatsky, Alla Lutsenko (2020)

Commentationes Mathematicae Universitatis Carolinae

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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

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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.

Quasigroup covers of division groupoids

Jaroslav J. Ježek, Tomáš Kepka, Petr Němec (2023)

Commentationes Mathematicae Universitatis Carolinae

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Let $G$ be a division groupoid that is not a quasigroup. For each regular cardinal $α > | G |$ we construct a quasigroup $Q$ on $G \times α$ that is a quasigroup cover of $G$ (i.e., $G$ is a homomorphic image of $Q$ and $G$ is not an image of any quasigroup that is a proper factor of $Q$ ). We also show how to easily obtain quasigroup covers from free quasigroups.

Symmetric linear operator identities in quasigroups

Reza Akhtar (2017)

Commentationes Mathematicae Universitatis Carolinae

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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...

Classification of quasigroups according to directions of translations II

Fedir Sokhatsky, Alla Lutsenko (2021)

Commentationes Mathematicae Universitatis Carolinae

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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...

Nonassociative triples in involutory loops and in loops of small order

Aleš Drápal, Jan Hora (2020)

Commentationes Mathematicae Universitatis Carolinae

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A loop of order $n$ possesses at least $3 n^{2} - 3 n + 1$ associative triples. However, no loop of order $n > 1$ that achieves this bound seems to be known. If the loop is involutory, then it possesses at least $3 n^{2} - 2 n$ associative triples. Involutory loops with $3 n^{2} - 2 n$ associative triples can be obtained by prolongation of certain maximally nonassociative quasigroups whenever $n - 1$ is a prime greater than or equal to $13$ or $n - 1 = p^{2 k}$ , $p$ an odd prime. For orders $n \leq 9$ the minimum number of associative triples is reported for both general...

Linear operator identities in quasigroups

Reza Akhtar (2022)

Commentationes Mathematicae Universitatis Carolinae

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We study identities of the form $L_{x_{0}} ϕ_{1} \dots ϕ_{n} R_{x_{n + 1}} = R_{x_{n + 1}} ϕ_{σ (1)} \dots ϕ_{σ (n)} L_{x_{0}}$ in quasigroups, where $n \geq 1$ , $σ$ is a permutation of ${1, ..., n}$ , and for each $i$ , $ϕ_{i}$ is either $L_{x_{i}}$ or $R_{x_{i}}$ . We prove that in a quasigroup, every such identity implies commutativity. Moreover, if $σ$ is chosen randomly and uniformly, it also satisfies associativity with probability approaching $1$ as $n \to \infty$ .

Antiflexible Latin directed triple systems

Andrew R. Kozlik (2015)

Commentationes Mathematicae Universitatis Carolinae

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It is well known that given a Steiner triple system one can define a quasigroup operation $\cdot$ upon its base set by assigning $x \cdot x = x$ for all $x$ and $x \cdot y = z$ , where $z$ is the third point in the block containing the pair ${x, y}$ . The same can be done for Mendelsohn triple systems, where $(x, y)$ is considered to be ordered. But this is not necessarily the case for directed triple systems. However there do exist directed triple systems, which induce a quasigroup under this operation and these are called Latin directed...

Quasigroups arisen by right nuclear extension

Péter T. Nagy, Izabella Stuhl (2012)

Commentationes Mathematicae Universitatis Carolinae

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The aim of this paper is to prove that a quasigroup $Q$ with right unit is isomorphic to an $f$ -extension of a right nuclear normal subgroup $G$ by the factor quasigroup $Q / G$ if and only if there exists a normalized left transversal $Σ \subset Q$ to $G$ in $Q$ such that the right translations by elements of $Σ$ commute with all right translations by elements of the subgroup $G$ . Moreover, a loop $Q$ is isomorphic to an $f$ -extension of a right nuclear normal subgroup $G$ by a loop if and only if $G$ is middle-nuclear, and...