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A class of Bol loops with a subgroup of index two

Petr Vojtěchovský (2004)

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 . We describe all situations in which the resulting quasigroup is a Bol loop. This paper also corrects an error in P. Vojtěchovsk’y: On the uniqueness of loops M ( G , 2 ) .

A class of commutative loops with metacyclic inner mapping groups

Aleš Drápal (2008)

Commentationes Mathematicae Universitatis Carolinae

We investigate loops defined upon the product ℤ m × ℤ k by the formula ( a , i ) ( b , j ) = ( ( a + b ) / ( 1 + t f i ( 0 ) f j ( 0 ) ) , i + j ) , where f ( x ) = ( s x + 1 ) / ( t x + 1 ) , for appropriate parameters s , t ∈ ℤ m * . Each such loop is coupled to a 2-cocycle (in the group-theoretical sense) and this connection makes it possible to prove that the loop possesses a metacyclic inner mapping group. If s = 1 , then the loop is an A-loop. Questions of isotopism and isomorphism are considered in detail.

A class of quasigroups solving a problem of ergodic theory

Jonathan D. H. Smith (2000)

Commentationes Mathematicae Universitatis Carolinae

A pointed quasigroup is said to be semicentral if it is principally isotopic to a group via a permutation on one side and a group automorphism on the other. Convex combinations of permutation matrices given by the one-sided multiplications in a semicentral quasigroup then yield doubly stochastic transition matrices of finite Markov chains in which the entropic behaviour at any time is independent of the initial state.

A class of torsion-free abelian groups characterized by the ranks of their socles

Ulrich F. Albrecht, Anthony Giovannitti, H. Pat Goeters (2002)

Czechoslovak Mathematical Journal

Butler groups formed by factoring a completely decomposable group by a rank one group have been studied extensively. We call such groups, bracket groups. We study bracket modules over integral domains. In particular, we are interested in when any bracket R -module is R tensor a bracket group.

A classification of rational languages by semilattice-ordered monoids

Libor Polák (2004)

Archivum Mathematicum

We prove here an Eilenberg type theorem: the so-called conjunctive varieties of rational languages correspond to the pseudovarieties of finite semilattice-ordered monoids. Taking complements of members of a conjunctive variety of languages we get a so-called disjunctive variety. We present here a non-trivial example of such a variety together with an equational characterization of the corresponding pseudovariety.

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