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On the structure of finite paramedial quasigroups

V. A. Shcherbacov, D. I. Pushkashu (2010)

Commentationes Mathematicae Universitatis Carolinae

Information on the structure of finite paramedial quasigroups, including a classification of finite simple paramedial quasigroups, is given. The problem ``Classify the finite simple paramedial quasigroups'' was posed by J. Ježek and T. Kepka at the conference LOOPS'03, Prague 2003.

On the uniqueness of loops M ( G , 2 )

Petr Vojtěchovský (2003)

Commentationes Mathematicae Universitatis Carolinae

Let G be a finite group and C 2 the cyclic group of order 2. Consider the 8 multiplicative operations ( x , y ) ( x i y j ) k , where i , j , k { - 1 , 1 } . Define a new multiplication on G × C 2 by assigning one of the above 8 multiplications to each quarter ( G × { i } ) × ( G × { j } ) , for i , j C 2 . If the resulting quasigroup is a Bol loop, it is Moufang. When G is nonabelian then exactly four assignments yield Moufang loops that are not associative; all (anti)isomorphic, known as loops M ( G , 2 ) .

On universal quasigroup identities

Elena Brožíková (1992)

Mathematica Bohemica

The paper deals with quasigroup identities under isotopies. The terminology is taken from [2], [3] and [4]. Stimulated by geometric illustrations, V. D. Belousov in [2] has presented two important identity properties and posed a question for which identities these properties are necessary and sufficient for the identity to be invariant under isotopies. Inspired by V. D. Belousov, G. Monoszová investigated in [6] one special kind of identities for which both Belousov’s properties give necessary and...

On varieties of left distributive left idempotent groupoids

David Stanovský (2004)

Discussiones Mathematicae - General Algebra and Applications

We describe a part of the lattice of subvarieties of left distributive left idempotent groupoids (i.e. those satisfying the identities x(yz) ≈ (xy)(xz) and (xx)y ≈ xy) modulo the lattice of subvarieties of left distributive idempotent groupoids. A free groupoid in a subvariety of LDLI groupoids satisfying an identity xⁿ ≈ x decomposes as the direct product of its largest idempotent factor and a cycle. Some properties of subdirectly ireducible LDLI groupoids are found.

On π-Groupoids

Zoran Stojaković, Janez Ušan (1979)

Publications de l'Institut Mathématique

Overlapping latin subsquares and full products

Joshua M. Browning, Petr Vojtěchovský, Ian M. Wanless (2010)

Commentationes Mathematicae Universitatis Carolinae

We derive necessary and sufficient conditions for there to exist a latin square of order n containing two subsquares of order a and b that intersect in a subsquare of order c . We also solve the case of two disjoint subsquares. We use these results to show that: (a) A latin square of order n cannot have more than n m n h / m h subsquares of order m , where h = ( m + 1 ) / 2 . Indeed, the number of subsquares of order m is bounded by a polynomial of degree at most 2 m + 2 in n . (b) For all n 5 there exists a loop of order n in which every...

P -hypergoupes

Thomas N. Vougiouklis, L. Konguetsof (1987)

Acta Universitatis Carolinae. Mathematica et Physica

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