Burnside Orders,Burnside Algebras and Partition Lattices

The permutation representation theory of groups has been extended, through quasigroups, to one-sided left (or right) quasigroups. The current paper establishes a link with the theory of ordered sets, introducing the concept of a Burnside order that generalizes the poset of conjugacy classes of subgroups of a finite group. Use of the Burnside order leads to a simplification in the proof of key properties of the Burnside algebra of a left quasigroup. The Burnside order for a projection left quasigroup structure on a finite set is defined by the lattice of set partitions of that set, and it is shown that the general direct and restricted tensor product operations for permutation representations of the projection left quasigroup structure both coincide with the operation of intersection on partitions. In particular, the mark matrix of the Burnside algebra of a projection left quasigroup, a permutation-theoretic concept, emerges as dual to the zeta function of a partition lattice, an order-theoretic concept.     

Separation in topological algebras

One of the basic results from the theory of topological groups is that aT ₀ topological group is already completely regular. It is also known [1] thatT ₀ quasigroups are regular. Taylor [8] showed that a fragment of these results holds in any congruence permutable variety, namely that in such a variety aT ₀ topological algebra is already Hausdorff. Gumm [2] extended this result to 3-permutable varieties and showed that, more generally,T ₀ topological algebras in congruencen-permutable varieties areT ₁. In this paper we show thatT ₀ topological algebras inn-permutable varieties satisfy a separation condition strictly stronger thanT ₁. We also give some counterexamples that show that some of these separation results are the best possible.Presented by S. Burris.This research was part of the author's thesis at The University of Colorado, directed by Walter Taylor. The author wishes to thank professor Taylor for his many contributions.     

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.     

The Clone of a Topological Space: an inspiring book by Walter Taylor

This article is a survey of our results inspired by Walter Taylor’s book, The Clone of a Topological Space. This paper is dedicated to Walter Taylor. Received June 28, 2005; accepted in final form January 3, 2006.     

On quasigroups with few associative triples

There are numerous application of quasigroups in cryptology. It turns out that quasigroups with the relatively small number of associative triples can be utilized in designs of hash functions. In this paper we provide both a new lower bound and a new upper bound on the minimum number of associative triples over quasigroups of a given order.     

Algebraic properties of quantum quasigroups

Quantum quasigroups provide a self-dual framework for the unification of quasigroups and Hopf algebras. This paper furthers the transfer program, investigating extensions to quantum quasigroups of various algebraic features of quasigroups and Hopf algebras. Part of the difficulty of the transfer program is the fact that there is no standard model-theoretic procedure for accommodating the coalgebraic aspects of quantum quasigroups. The linear quantum quasigroups, which live in categories of modules under the direct sum, are a notable exception. They form one of the central themes of the paper.From the theory of Hopf algebras, we transfer the study of grouplike and setlike elements, which form separate concepts in quantum quasigroups. From quasigroups, we transfer the study of conjugate quasigroups, which reflect the triality symmetry of the language of quasigroups. In particular, we construct conjugates of cocommutative Hopf algebras. Semisymmetry, Mendelsohn, and distributivity properties are formulated for quantum quasigroups. We classify distributive linear quantum quasigroups that furnish solutions to the quantum Yang-Baxter equation. The transfer of semisymmetry is designed to prepare for a quantization of web geometry.     

On the congruence modularity conjecture

A new condition of compatibility with projections, applicable to some Maltsev filters, is defined and shown to hold, among others, for the filter of congruence-modular varieties. As a consequence, it is shown that there exist no simple counterexamples (in a specified sense) to the modularity conjecture. This paper is dedicated to Walter Taylor. Received November 5, 2005; accepted in final form April 3, 2006.     

Loops, their cores and symmetric spaces

This paper is devoted to the relations among affine symmetric spaces, smooth Bol and Moufang loops, smooth left distributive quasigroups and differentiable 3-nets. The results are used to prove the analyticity of smooth Moufang loops and left distributive quasigroups with involutive left translations as well as to show the Lie nature of transformation groups naturally related to some classes of smooth binary systems and 3-nets. In the last section we establish power series expansion for local loops with weak associativity conditions and apply the methods of the previous sections in order to describe geodesic loops having euclidean lines either as their geodesic lines or as geodesic lines of their core. The first author was partly supported by the Deutsche Forschungsgemeinschaft and by OTKA Grant no. T020545.     

Uniform non-amenability of free Burnside groups

The aim of this paper is to show that free Burnside groups of sufficiently large odd exponent are non-amenable in a certain strong sense. More precisely their left regular representations are isolated from the trivial representation uniformly on finite generating sets. It follows that free Burnside groups are of uniform exponential growth. This answers a question of de la Harpe. This work has been partially supported by the NSF grant DMS-0605093 and by the RFBR Grant # 05-01-00892. Received: 25 April 2006     

Algebraic lattices and Boolean algebras

In this paper we establish several equivalent conditions for an algebraic lattice to be a finite Boolean algebra. This paper is dedicated to Walter Taylor. Received February 11, 2005; accepted in final form October 9, 2005.     

Nonexpansive algebras

A nonexpansive algebra is a pseudometric algebra in which the operations are all nonexpansive. We study such algebras, particularly in the case of algebras in permutable and n-permutable varieties, leading to new characterizations of such varieties. Free nonexpansive algebras are also investigated. This paper is dedicated to Walter Taylor. Received September 27, 2005; accepted in final form February 4, 2006. The author would like to thank the referee for the many helpful suggestions.     

Effective Codescent Morphisms in Some Varieties of Universal Algebras

The paper gives a sufficient condition formulated in a syntactical form for all codescent morphisms of a variety of universal algebras satisfying the amalgamation property to be effective. This result is further used in proving that all codescent morphisms of (left/right-)quasigroups, loops, and magmas are effective.     

Hyperidentities and solid varieties

Let V be a variety of type τ. A type τ hyperidentity of V is an identity of V which also holds in an additional stronger sense: for every substitution of terms of the variety (of appropriate arity) for the operation symbols in the identity, the resulting equation holds as an identity of the variety. Such identities were first introduced by Walter Taylor in [27] in 1981. A variety is called solid if all its identities also hold as hyperidentities. For example, the semigroup variety of rectangular bands is a solid variety. For any fixed type τ, the collection of all solid varieties of type τ forms a complete lattice which is a sublattice of the lattice L(τ) of all varieties of type τ. In this paper we give an overview of the study of hyperidentities and solid varieties, particularly for varieties of semigroups, culminating in the construction of an infinite collection of solid varieties of arbitrary type. This paper is dedicated to Walter Taylor. Received July 16, 2005; accepted in final form January 3, 2006. This paper is an expanded version of a talk presented at the Conference on Algebras, Lattices and Varieties in Honour of Walter Taylor, in Boulder Colorado, August 2004. The author’s research is supported by NSERC of Canada.     

n-closure systems and n-closure operators

It is very well known and permeating the whole of mathematics that a closure operator on a given set gives rise to a closure system, whose constituent sets form a complete lattice under inclusion, and vice-versa. Recent work of Wille on triadic concept analysis and subsequent work by the author on polyadic concept analysis led to the introduction of complete trilattices and complete n-lattices, respectively, that generalize complete lattices and capture the order-theoretic structure of the collection of concepts associated with polyadic formal contexts. In the present paper, polyadic closure operators and polyadic closure systems are introduced and they are shown to be in a relationship similar to the one that exists between ordinary (dyadic) closure operators and ordinary (dyadic) closure systems. Finally, the algebraic case is given some special consideration. This paper is dedicated to Walter Taylor. Received March 10, 2005; accepted in final form March 7, 2006.     

Irresponsibility indicates an inability to be strong

In this paper we provide sufficient conditions on a given finite algebra to prevent it from being strongly dualizable. We then use these conditions to prove that certain unary algebras are not strongly dualizable. This paper is dedicated to Walter Taylor. Received November 16, 2005; accepted in final form May 12, 2006.     

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.     

Sylow Theory for Quasigroups II: Sectional Action

The first paper in this series initiated a study of Sylow theory for quasigroups and Latin squares based on orbits of the left multiplication group. The current paper is based on so‐called pseudo‐orbits, which are formed by the images of a subset under the set of left translations. The two approaches agree for groups, but differ in the general case. Subsets are described as sectional if the pseudo‐orbit that they generate actually partitions the quasigroup. Sectional subsets are especially well behaved in the newly identified class of conflatable quasigroups, which provides a unified treatment of Moufang, Bol, and conjugacy closure properties. Relationships between sectional and Lagrangean properties of subquasigroups are established. Structural implications of sectional properties in loops are investigated, and divisors of the order of a finite quasigroup are classified according to the behavior of sectional subsets and pseudo‐orbits. An upper bound is given on the size of a pseudo‐orbit. Various interactions of the Sylow theory with design theory are discussed. In particular, it is shown how Sylow theory yields readily computable isomorphism invariants with the resolving power to distinguish each of the 80 Steiner triple systems of order 15.     

On profinite completions and canonical extensions

We show that if a variety V of monotone lattice expansions is finitely generated, then profinite completions agree with canonical extensions on V. The converse holds for varieties of finite type. This paper is dedicated to Walter Taylor. Received May 14, 2005; accepted in final form September 8, 2005.     

带可分解性质的幂等拟群大集

具有幂单正交侣的幂等拟群称为可分解的. 具有幂等正交侣的幂等拟群称为几乎可分解的. 若v 元集合上的所有分量互不相同的3-向量能够分拆成互不相交(幂等3-向量除外) 的v-2 个v 阶幂等拟群, 则称之为v 阶幂等拟群大集. 本文使用t-平衡设计(t=2; 3) 的方法给出了可分解幂等拟群大集、几乎可分解幂等拟群大集及可分解对称幂等拟群大集(即可分解高尔夫设计) 的构造方法, 给出了其存在性的若干结果.