Classification of quasigroups according to directions of translations I

Fedir Sokhatsky; Alla Lutsenko

Displaying similar documents to “Classification of quasigroups according to directions of translations I”

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...

Medial quasigroups of prime square order

David Stanovský (2016)

Commentationes Mathematicae Universitatis Carolinae

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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.

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.

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...

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 $n \to \infty$ .

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...

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...

Construction of Mendelsohn designs by using quasigroups of $(2, q)$ -varieties

Lidija Goračinova-Ilieva, Smile Markovski (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $q$ be a positive integer. An algebra is said to have the property $(2, q)$ if all of its subalgebras generated by two distinct elements have exactly $q$ elements. A variety $𝒱$ of algebras is a variety with the property $(2, q)$ if every member of $𝒱$ has the property $(2, q)$ . Such varieties exist only in the case of $q$ prime power. By taking the universes of the subalgebras of any finite algebra of a variety with the property $(2, q)$ , $2 < q$ , blocks of Steiner system of type $(2, q)$ are obtained. The stated correspondence...

The centre of a Steiner loop and the maxi-Pasch problem

Andrew R. Kozlik (2020)

Commentationes Mathematicae Universitatis Carolinae

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A binary operation “ $\cdot$ ” which satisfies the identities $x \cdot e = x$ , $x \cdot x = e$ , $(x \cdot y) \cdot x = y$ and $x \cdot y = y \cdot x$ 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 . We show that...

Automorphic loops and metabelian groups

Mark Greer, Lee Raney (2020)

Commentationes Mathematicae Universitatis Carolinae

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Given a uniquely 2-divisible group $G$ , we study a commutative loop $(G, \circ)$ which arises as a result of a construction in “Engelsche elemente noetherscher gruppen” (1957) by R. Baer. We investigate some general properties and applications of “ $\circ$ ” and determine a necessary and sufficient condition on $G$ in order for $(G, \circ)$ to be Moufang. In “A class of loops categorically isomorphic to Bruck loops of odd order” (2014) by M. Greer, it is conjectured that $G$ is metabelian if and only if $(G, \circ)$ is an automorphic...