Sylow Theory for Quasigroups

This paper is intended as a first step toward a general Sylow theory for quasigroups and Latin squares. A subset of a quasigroup lies in a nonoverlapping orbit if its respective translates under the elements of the left multiplication group remain disjoint. In the group case, each nonoverlapping orbit contains a subgroup, and Sylow's Theorem guarantees nonoverlapping orbits on subsets whose order is a prime‐power divisor of the group order. For the general quasigroup case, the paper investigates the relationship between non‐overlapping orbits and structural properties of a quasigroup. Divisors of the order of a finite quasigroup are classified by the behavior of nonoverlapping orbits. In a dual direction, Sylow properties of a subquasigroup P of a finite left quasigroup Q may be defined directly in terms of the homogeneous space , and also in terms of the behavior of the isomorphism type within the so‐called Burnside order, a labeled order structure on the full set of all isomorphism types of irreducible permutation representations.     

P3BD‐closed sets

A concept called P3BD‐closed set is introduced to describe a set of positive integers which is both PBD‐closed and 3BD‐closed. The theory of P3BD‐closure is developed and a few examples of P3BD‐closed sets are exhibited. In the process, the existence spectrum of OLIQ ^?s (overlarge sets of idempotent quasigroups with their own idempotent orthogonal mates) is almost determined. A pair of orthogonal OLIQ ^?s is shown to asymptotically exist. The existence of OLIQ s (overlarge sets of idempotent quasigroups with their own orthogonal mates not necessarily specifying idempotency) is also established with only 10 possible exceptions remained. Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 19:407‐421, 2011     

Lexicographic product decompositions of half linearly ordered loops

In this paper we prove for an hl-loop Q an assertion analogous to the result of Jakubík concerning lexicographic products of half linearly ordered groups. We found conditions under which any two lexicographic product decompositions of an hl-loop Q with a finite number of lexicographic factors have isomorphic refinements.     

Hopf Quasimodules and Yetter-Drinfeld Modules over Hopf Quasigroups

We introduce the notions of a four-angle Hopf quasimodule and an adjoint quasiaction over a Hopf quasigroup H in a symmetric monoidal category C.If H possesses an adjoint quasiaction,we show that symmetric Yetter-Drinfeld categories are trivial,and hence we obtain a braided monoidal category equivalence between the category of right Yetter-Drinfeld modules over H and the category of four-angle Hopf modules over H under some suitable conditions.     

The spectrum for large sets of idempotent quasigroups

In this article we construct a large set of idempotent quasigroups of order 14. Combined with the results in Chang, JCMCC; and Teirlinck and Lindner, Eur J Combin 9 (1988), 83–89, this shows that the spectrum for large sets of idempotent quasigroups of order n [briefly, LIQ(n)] is the set all integers n≥ 3 with the exception of n=6. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 79–82, 2000     

幂等对称拟群的超大集

一个v阶拟群(X,。)等价于一个v(v-1)×3的部分正交表(每行由三个不同元素构成).若对任意a,b∈X,都有a。b=b。a,则称(X,。)为对称的.设y为v+1元集,y的由三个不同元素构成的(v+1)v(v-1)个有序三元组作成集合T(v+1).若T(v+1)可分拆成v+1个分别作用在Y＼{y}上的两两不交的部分正交表By,则称{(y＼{y},By);y∈y}为幂等对称拟群的超大集.本文证明了存在v阶幂等对称拟群的超大集当且发v≡1(mod 2),v≥3.     

Hopf拟群的代数形变

利用2-余循环对原有Hopf拟群的结构进行代数形变,进而构造出新的Hopf拟群,并讨论了新构造的Hopf拟群上的余拟三角结构与原有Hopf拟群上的余拟三角结构之间的关系,推广了Hopf代数中的相应结论.     

基于正交拟群的流密码算法

利用正交拟群的特点, 克服Edon80的弱点, 设计了称为Double40的二进制加法同步流密码算法. 它基于一对8阶相互正交的自正交拟群, 使得Johansson和Hell的密钥恢复攻击对Double40无法奏效.     

一类基于拟群的Hash函数

基于Hash函数是用于信息安全领域中的加密算法,因此利用剩余类环和有限域理论给出一种基于拟群运算的具有良好抗碰撞性的Hash函数,并对其安全性作出分析.     

A coalgebraic approach to quasigroup permutation representations

The paper identifies the class of all permutation representations of a given finite quasigroup as a covariety of coalgebras. Each permutation representation decomposes as a sum of homomorphic images of homogeneous spaces. For a group, permutation representations in the present sense specialise to the classical concept. Burnside's Lemma, with a new proof, is extended from groups to quasigroups. Received March 13, 2002; accepted in final form September 18, 2002. RID="h1" ID="h1"This paper was written while the author was a guest of the Institute of Mathematics and Information Sciences at Warsaw University of Technology, on Faculty Professional Development Assignment from Iowa State University.