r- ортогональные квазигруппы. I
r- ортогональные квазигруппы. II
Realizations of Loops and Groups defined by short identities
In a recent paper, those quasigroup identities involving at most three variables and of “length” six which force the quasigroup to be a loop or group have been enumerated by computer. We separate these identities into subsets according to what classes of loops they define and also provide humanly-comprehensible proofs for most of the computer-generated results.
Rechtsdistributive Multiplikationen auf homogenen Räumen.
Rectangular groupoids
Recursive inseparability of the sets of identically valid and finitely refutable formulas of some elementary theories of varieties.
Recursively differentiable quasigroups and complete recursive codes
Criteria of recursive differentiability of quasigroups are given. Complete recursive codes which attains the Joshibound are constructed using recursively differentiable -ary quasigroups.
Reflection loops of spaces with congruence and hyperbolic incidence structure
In an absolute space with congruence there are line reflections and point reflections. With the help of point reflections one can define in a natural way an addition + of points which is only associative if the product of three point reflection is a point reflection again. In general, for example for the case that is a linear space with hyperbolic incidence structure, the addition is not associative. is a K-loop or a Bruck loop.
Regular mappings of groupoids
Regular permutation sets and loops
Two suitable composition laws are defined in a regular permutation set in order to find new characterizations of some important classes of loops.
Regulated buildups of 3-configurations
We deal with two types of buildups of 3-configurations: a generating buildup over a given edge set and a regulated one (according to maximal relative degrees of vertices over a penetrable set of vertices). Then we take account to minimal generating edge sets, i.e., to edge bases. We also deduce the fundamental relation between the numbers of all vertices, of all edges from edge basis and of all terminal elements. The topic is parallel to a certain part of Belousov' “Configurations in algebraic...
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