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

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 ϕ n R x n + 1 = R x n + 1 ϕ σ ( 1 ) ϕ σ ( n ) L x 0 in quasigroups, where n 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 .

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 · upon its base set by assigning x · x = x for all x and x · 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...

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 × 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.

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...

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 9 the minimum number of associative triples is reported for both general...

Purity of level m stratifications

Marc-Hubert Nicole, Adrian Vasiu, Torsten Wedhorn (2010)

Annales scientifiques de l'École Normale Supérieure

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Let k be a field of characteristic p &gt; 0 . Let D m be a BT m over k (i.e., an m -truncated Barsotti–Tate group over k ). Let S be a k -scheme and let X be a BT m over S . Let S D m ( X ) be the subscheme of S which describes the locus where X is locally for the fppf topology isomorphic to D m . If p 5 , we show that S D m ( X ) is pure in S , i.e. the immersion S D m ( X ) S is affine. For p { 2 , 3 } , we prove purity if D m satisfies a certain technical property depending only on its p -torsion D m [ p ] . For p 5 , we apply the developed techniques to show that...

Sum-product theorems and incidence geometry

Mei-Chu Chang, Jozsef Solymosi (2007)

Journal of the European Mathematical Society

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In this paper we prove the following theorems in incidence geometry. 1. There is δ > 0 such that for any P 1 , , P 4 , and Q 1 , , Q n 2 , if there are n ( 1 + δ ) / 2 many distinct lines between P i and Q j for all i , j , then P 1 , , P 4 are collinear. If the number of the distinct lines is < c n 1 / 2 then the cross ratio of the four points is algebraic. 2. Given c > 0 , there is δ > 0 such that for any P 1 , P 2 , P 3 2 noncollinear, and Q 1 , , Q n 2 , if there are c n 1 / 2 many distinct lines between P i and Q j for all i , j , then for any P 2 { P 1 , P 2 , P 3 } , we have δ n distinct lines between P and Q j . 3. Given...