Relations Among Some Classes Of Quasigroups.
Relatives of K-loops: Theory and examples
A K-loop or Bruck loop is a Bol loop with the automorphic inverse property. An overview of the most important theorems on K-loops and some of their relatives, especially Kikkawa loops, is given. First, left power alternative loops are discussed, then Kikkawa loops are considered. In particular, their nuclei are determined. Then the attention is paid to general K-loops and some special classes of K-loops such as 2-divisible ones. To construct examples, the method of derivation is introduced. This...
Remarks on projectivities.
Right division in Moufang loops
If is a group, and the operation is defined by then by direct verification is a quasigroup which satisfies the identity . Conversely, if one starts with a quasigroup satisfying the latter identity the group can be constructed, so that in effect is determined by its right division operation. Here the analogous situation is examined for a Moufang loop. Subtleties arise which are not present in the group case since there is a choice of defining identities and the identities produced by...
Rot-Quasigroups
Rot-quasigroups as isotopes of Abelian groups
Schreier loops
We study systematically the natural generalization of Schreier's extension theory to obtain proper loops and show that this construction gives a rich family of examples of loops in all traditional common, important loop classes.
Secure two-way on-line communication by using quasigroup enciphering with almost public key.
Selfdistributive groupoids of small orders
After enumerating isomorphism types of at most five-element left distributive groupoids, we prove that a distributive groupoid with less than 81 elements is necessarily medial.
Selfdistributive groupoids. Part A1. Non-indempotent left distributive groupoids
Selfdistributive groupoids. Part D1: Left distributive semigroups
Selfdistributive grupoids, Part A2: Non-idempotent left distributive quasigroups
Self-invariant 1-factorizations of complete graphs and finite Bol loops of exponent 2.
Self-orthogonal cyclic n-quasigroups.
Self-orthogonal cyclic n-quasigroups. (Short Communication).
Semigroup representations of medial groupoids
Semisymmetrization and Mendelsohn quasigroups
The semisymmetrization of an arbitrary quasigroup builds a semisymmetric quasigroup structure on the cube of the underlying set of the quasigroup. It serves to reduce homotopies to homomorphisms. An alternative semisymmetrization on the square of the underlying set was recently introduced by A. Krapež and Z. Petrić. Their construction in fact yields a Mendelsohn quasigroup, which is idempotent as well as semisymmetric. We describe it as the Mendelsohnization of the original quasigroup. For quasigroups...
Separable Quasigroups.
Simple paramedial groupoids
Simple quasigroups whose inner permutations commute
Simple quasigroups with commuting inner permutations are medial.