F-quasigroups and generalized modules

Units in generalized derivatives of quasigroups

We proceed the research of generalized quasigroup derivatives started in early papers of the last co-author ([20, p. 212], [13]). For any quasigroup there exist 648 generalized derivatives. Here we study the problem about existence of units (left, right, middle) in quasigroups that are a generalized derivative of a quasigroup. There exist 1944 various cases. For every case we find a proof or a counterexample, often using Prover or Mace [15, 14].

F-quasigroups isotopic to groups

In math.GR/0510298, we showed that every loop isotopic to an F-quasigroup is a Moufang loop. Here we characterize, via two simple identities, the class of F-quasigroups which are isotopic to groups. We call these quasigroups FG-quasigroups. We show that FG-quasigroups are linear over groups. We then use this fact to describe their structure. This gives us, for instance, a complete description of the simple FG-quasigroups. Finally, we show an equivalence of equational classes between pointed FG-quasigroups and central generalized modules over a particular ring.

Isomorphism Invariants for Linear Quasigroups

For a unital ring $S$, an $S$-linear quasigroup is a unital $S$-module, with automorphisms $ρ$ and $λ$ giving a (nonassociative) multiplication $x\cdot y=x^ρ+y^λ$. If $S$ is the field of complex numbers, then ordinary characters provide a complete linear isomorphism invariant for finite-dimensional $S$-linear quasigroups. Over other rings, it is an open problem to determine tractably computable isomorphism invariants. The paper investigates this isomorphism problem for $\mathbb{Z}$-linear quasigroups. We consider the extent to which ordinary characters classify $\mathbb{Z}$-linear quasigroups …

图淬群： racks 和右拟群的广义 Cayley 图

Graph quandles: Generalized Cayley graphs of racks and right quasigroups

本文研究了基于右拟群作用于图的广义 Cayley 图，介绍了基于标记的图不变量，并证明了所有右拟群可由无边图或完全图实现，解决了 Bardakov 的两个问题。

On the structure of some modules over generalized soluble groups

Let R be a ring and G a group. An R-module A is said to be artinian-by-(finite rank) if TorR(A) is artinian and A/TorR(A) has finite R-rank. The authors study ZG-modules A such that A/CA(H) is artinian-by-(finite rank) (as a Z-module) for every proper subgroup H.

广义交叉模的覆盖与提升

Coverings and liftings of generalized crossed modules

本文定义广义cat$^1$-群并构造其到广义交叉模的函子，引入覆盖与提升概念，旨在证明给定广义交叉模的覆盖范畴与提升范畴等价。