Permutation representations of left quasigroups

The concept of a permutation representation has recently been extended from groups to quasigroups. Following a suggestion of Walter Taylor, the concept is now further extended to left quasigroups. The paper surveys the current state of the theory, giving new proofs where necessary to cover the general case of left quasigroups. Both the Burnside Lemma and the Burnside algebra appear in this new context. This paper is dedicated to Walter Taylor. Received August 9, 2005; accepted in final form March 7, 2006.     

Quasigroups satisfying balanced but not Belousov equations are group isotopes

Summary V. D. Belousov (1925–88) made numerous contributions to the study of quasigroups. In particular, his lengthy 1966 paper Balanced identities in quasigroups [4] contains what has been described as a very significant and remarkable theorem [11, pp. 68–69]. Remarkable though it was, this theorem provided only a partial answer to the question as to which balanced equations on quasigroups gave rise to group isotopes. Although not specifically addressed in the paper [12], a characterization of the balanced equations in question may be derived from a generalization of Belousov's Theorem due to E. Falconer. The first author explicitly solved the problem in 1979; however his characterization was of a technical nature and depended on machinery developed over three papers [13].In 1985 Belousov found a characterization which is not only elegant but also lends itself to a simple proof [5]. The purpose of this paper is to provide sufficient background for the non specialist to understand and enjoy what we too would describe as a remarkable theorem.     

Right distributive quasigroups on algebraic varieties

In this paper we develop a structure theory of algebraic right distributive quasigroups which correspond to closed and connected conjugacy classes generating algebraic Fischer groups (in the sense of [6]) such that the mappingx x ^–1 ax, fora , is an automorphism of (as variety). We also give examples of algebraic Fischer groups where this does not happen. It becomes clear that the class of algebraic right distributive quasigroups has nice properties concerning subquasigroups, normal subquasigroups and direct product.We give a complete classification of one- and two-dimensional as well as of minimal algebraic right distributive quasigroups.     

The geometric mean and B-loop quasigroups of the convex cone of positive definite matrices

In this paper we consider the convex cone of positive definite matrices as algebraic system equipped with geometric mean and B-loop from the standard matrix polar decomposition. Some algebraic structures of these quasigroups are investigated in the context of matrix theory. In particular, their autotopism groups are completely determined: they are isomorphic to the group of positive real numbers.Received: 28 April 2004     

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.     

The Order of Automorphisms of Quasigroups

We prove quadratic upper bounds on the order of any autotopism of a quasigroup or Latin square, and hence also on the order of any automorphism of a Steiner triple system or 1‐factorization of a complete graph. A corollary is that a permutation σ chosen uniformly at random from the symmetric group will almost surely not be an automorphism of a Steiner triple system of order n, a quasigroup of order n or a 1‐factorization of the complete graph . Nor will σ be one component of an autotopism for any Latin square of order n. For groups of order n it is known that automorphisms must have order less than n, but we show that quasigroups of order n can have automorphisms of order greater than n. The smallest such quasigroup has order 7034. We also show that quasigroups of prime order can possess autotopisms that consist of three permutations with different cycle structures. Our results answer three questions originally posed by D.  Stones.     

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.     

Belousov equations on ternary quasigroups

Summary The study of Belousov equations in binary quasigroups was initiated by V. D. Belousov. Krape and Taylor showed that every finite set of Belousov equations was equivalent to a single Belousov equation which was in some sense no longer than any single member of the set. This led to the concept of an irreducible Belousov equation, that is one which is not equivalent to an equation with fewer variables. Krape and Taylor determined the structure of the irreducible equations by establishing a correspondence between them and specific polynomials overZ ₂.In this paper it is shown that the structure of the ternary equations is richer than the binary counterpart, although the main result is similar to the binary case in as far as a system of ternary Belousov equations is equivalent to a single Belousov equation which is no longer than any member of the system or the system is equivalent to a pair of equations each with three variables.     

Bartholdi zeta and L-functions of weighted digraphs, their coverings and products

Since a zeta function of a regular graph was introduced by Ihara [Y. Ihara, On discrete subgroups of the two by two projective linear group over p-adic fields, J. Math. Soc. Japan 19 (1966) 219-235], many kinds of zeta functions and L-functions of a graph or a digraph have been defined and investigated. Most of the works concerning zeta and L-functions of a graph contain the following: (1) defining a zeta function, (2) defining an L-function associated with a (regular) graph covering, (3) providing their determinant expressions, and (4) computing the zeta function of a graph covering and obtaining its decomposition formula as a product of L-functions. As a continuation of those works, we introduce a zeta function of a weighted digraph and an L-function associated with a weighted digraph bundle. A graph bundle is a notion containing a cartesian product of graphs and a (regular or irregular) graph covering. Also we provide determinant expressions of the zeta function and the L-function. Moreover, we compute the zeta function of a weighted digraph bundle and obtain its decomposition formula as a product of the L-functions.     

Construction ofn-cyclic quasigroups and applications

Aequationes mathematicae -     

Some (2,n) combinatorial Stein groupoids

A Stein groupoid (quasigroup) is a groupoid (quasigroup) satisfying the identityx(xy)=yx. We show that, for certain two variable identities, the variety of Stein groupoids defined by any one of these identities has the properties that every groupoid in the variety is a quasigroup and that the free groupoid generated by two elements is of finite (small) order which we determine. These results provide characterizations of some Stein quasigroups of small order and we give some further characterizations involving other identities.     

Cotorsion pairs, model category structures, and representation theory

We make a general study of Quillen model structures on abelian categories. We show that they are closely related to cotorsion pairs, which were introduced by Salce [Sal79] and have been much studied recently by Enochs and coauthors [EJ00]. This gives a method of constructing model structures on abelian categories, which we illustrate by building two model structures on the category of modules over a (possibly noncommutative) Gorenstein ring. The homotopy category of these model structures is a generalization of the stable module category much used in modular representation theory. This stable module category has also been studied by Benson [Ben97]. Received: 14 December 2000; in final form: 17 December 2001 / Published online: 5 September 2002     

Decomposing thick subcategories of the stable module category

Let be the stable category of finitely generated modular representations of a finite group G over a field k. We prove a Krull-Remak-Schmidt theorem for thick subcategories of . It is shown that every thick tensor-ideal of (i.e. a thick subcategory which is a tensor ideal) has a (usually infinite) unique decomposition into indecomposable thick tensor-ideals. This decomposition follows from a decomposition of the corresponding idempotent kG-module into indecomposable modules. If is the thick tensor-ideal corresponding to a closed homogeneous subvariety W of the maximal ideal spectrum of the cohomology ring , then the decomposition of reflects the decomposition of W into connected components. Received: 27 April 1998 / In revised form: 16 July 1998     

Semidirect Product of Loops and Fibrations

Starting from two loops (H, +) and (K, ·), a new loop L can be defined by means of a suitable map Θ : K → Sym H (cf. [3]). Such a loop is called semidirect product of H and K with respect to Θ and denoted by H ×_Θ K =: L. Here we consider the class of those semidirect products in which Θ : K → Aut(H, +) is a homomorphism, this situation being quite akin to the group case. Some relevant algebraic properties of the loop L (Bol condition, Moufang etc.) can be inherited from H and K. In the case that K is a group we investigate the possibility of describing L by a partition (or fibration). In this way we propose a generalization of [8] for the non associative case. Received: September 20, 2007. Revised: November 8, 2007.     

The varieties of loops of Bol-Moufang type

A loop identity is of Bol-Moufang type if two of its three variables occur once on each side, the third variable occurs twice on each side, and the order in which the variables appear on both sides is the same, viz. ((xy)x)z = x(y(xz)). Loop varieties defined by one identity of Bol-Moufang type include groups, Bol loops, Moufang loops and C-loops. We show that there are exactly 14 such varieties, and determine all inclusions between them, providing all necessary counterexamples, too. This extends and completes the programme of Fenyves [Fe69]. Received October 23, 2003; accepted in final form April 12, 2005.     

Bol loops with non-normal nuclei

The purpose of this paper is to expand the collection of Bol loops that have nuclei that are not normal. In doing so, we will show that for each of the Bol loops given by Daniel and Karl Robinson in [2] there are uncountably many more. Clearly this will require us to give examples of infinite Bol loops with non-normal nuclei, which until now have apparently been absent from the literature.Received: 20 August 2001