Tomáš Kepka, Michael K. Kinyon, J. D. Phillips
We solve a problem of Belousov which has been open since 1967: to characterize the loop isotopes of F-quasigroups. We show that every F-quasigroup has a Moufang loop isotope which is a central product of its nucleus and Moufang center. We then use the loop to reveal the structure of the associated F-quasigroup.
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We prolong Kunen research about existence of units (left, right, two-sided) in quasigroups with classical Bol-Moufang type identities. These identities were listed in Fenvesh article.
A quasigroup 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, the order in which the variables appear on both sides is the same, and the only binary operation used is the multiplication, viz. $((xy)x)z=x(y(xz))$. Many well-known varieties of quasigroups are of Bol-Moufang type. We show that there are exactly 26 such varieties, determine all inclusions between them, and provide all necessary counterexamples. We also determine which of these varieties consist of loops or one-sided loops, and fully describe the va…
We enumerate three classes of non-medial quasigroups of order $243=3^5$ up to isomorphism. There are $17004$ non-medial trimedial quasigroups of order $243$ (extending the work of Kepka, Bénéteau and Lacaze), $92$ non-medial distributive quasigroups of order $243$ (extending the work of Kepka and Němec), and $6$ non-medial distributive Mendelsohn quasigroups of order $243$ (extending the work of Donovan, Griggs, McCourt, Opršal and Stanovský). The enumeration technique is based on affine representations over commutative Moufang loops, on properties of automorphism groups of commutative Moufang…
In Search of Homology for Quasigroups of Bol-Moufang Type
本文通过同型仿射拟群扩张为 Bol-Moufang 型拟群建立（上）同调理论，定义第二、第三边界运算并计算多个例子的第二同调群，同时探讨其与小范畴函子系数同调的关系。
Power graphs of both groups and semigroups have been widely studied. While the power graph of a quasigroup can be defined analogously to that of a group, power graphs of quasigroups and loops have thus far been little studied. In this paper we begin transferring results on the power graphs of groups to the context of loops by addressing a question posed by Peter Cameron: if two Moufang loops have isomorphic undirected power graphs, must they have isomorphic directed power graphs? It is known that two groups with isomorphic undirected power graphs must have isomorphic directed power graphs. We …
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.
In math.GR/0510298, we showed that every F-quasigroup is linear over a special kind of Moufang loop called an NK-loop. Here we extend this relationship by showing an equivalence between the equational class of (pointed) F-quasigroups and the equational class corresponding to a certain notion of generalized module (with noncommutative, nonassociative addition) for an associative ring.
Haar-Type Measures on Topological Quasigroups and Kunen's Theorem
本文研究拓扑拟群上的Haar型测度，探讨Moufang型恒等式对模系数cocycle的影响，并提出Kunen定理的测度论解释。
It is proved that any left F-quasigroup is isomorphic to the direct product of a left F-quasigroup with a unique idempotent element and isotope of a special form of a left distributive quasigroup. The similar theorems are proved for right F-quasigroups, left and right SM- and E-quasigroups. Information on simple quasigroups from these quasigroup classes is given, for example, finite simple F-quasigroup is a simple group or a simple medial quasigroup. It is proved that any left F-quasigroup is isotopic to the direct product of a group and a left S-loop. Some properties of loop isotopes of F-qua…
This is a companion to our lectures GAP and loops, to be delivered at the Workshops Loops 2007, Prague, Czech Republic. In the lectures we introduce the GAP package LOOPS, describe its capabilities, and explain in detail how to use it. In this paper we first outline the philosophy behind the package and its main features, and then we focus on three particular computational problems: construction of loop isomorphisms, classification of small Frattini Moufang loops of order 64, and the search for loops of nilpotency class higher than two with an abelian inner mapping group. In particular, this i…
The decomposition theorem for torsion abelian groups holds analogously for torsion commutative diassociative loops. With this theorem in mind, we investigate commutative diassociative loops satisfying the additional condition (trivially satisfied in the abelian group case) that all $n$th powers are central, for a fixed $n$. For $n=2$, we get precisely commutative $C$ loops. For $n=3$, a prominent variety is that of commutative Moufang loops. Many analogies between commutative C and Moufang loops have been noted in the literature, often obtained by interchanging the role of the primes 2 and 3. …
Using groups with triality we obtain some general multiplication formulas in Moufang loops, construct Moufang extensions of abelian groups, and describe the structure of minimal extensions for finite simple Moufang loops over abelian groups.
All known Moufang sets arise, in some way or another, from an algebraic structure which can be called `division' in some way. In this PhD dissertation, I made an attempt to develop a theory of local Moufang sets, which generalize Moufang sets in a way to allow constructing using the corresponding local algebraic structures. In the first of the two major parts of the dissertation, I develop the theory of local Moufang sets, while in the second part, some examples are constructed. The most general example constructed arises from a local Jordan pair (which corresponds to a local Jordan algebra), …
For most (and possibly all) non-associative finite simple Moufang loops, three generators of order 3 can be chosen so that each two of them generate a group isomorphic to $(3, 3 | 3, p)$. The subgroup structure of $(3, 3 | 3, p)$ depends on the solvability of a certain quadratic congruence, and it is described here in terms of generators.
We investigate the relation between the structure of a Moufang loop and its inner mapping group. Moufang loops of odd order with commuting inner mappings have nilpotency class at most two. $6$-divisible Moufang loops with commuting inner mappings have nilpotency class at most two. There is a Moufang loop of order $2^{14}$ with commuting inner mappings and of nilpotency class three.
We give an interpretation of the construction of torsors from preceding work (Bertram, Kinyon: Associative Geometries. I, J. Lie Theory 20) in terms of classical projective geometry. For the Desarguesian case, this leads to a reformulation of certain results from lot.cit., whereas for the Moufang case the result is new. But even in the Desarguesian case it sheds new light on the relation between the lattice structure and the algebraic structures of a projective space.
Buildings have been introduced by J. Tits in order to study semi-simple algebraic groups from a geometrical point of view. One of the most important results in the theory of buildings is the classification of irreducible spherical buildings of rank at least 3. About 25 years ago, M. Ronan and J. Tits defined the class of twin buildings, which generalize spherical buildings in a natural way. The motivation of their definition is provided by the theory of Kac-Moody groups. A 2-spherical twin building is uniquely determined by its local structure in almost all cases: The so-called foundation is t…
Approximate Subloops in Moufang Loops
本文研究Moufang环中的近子环结构，建立内在乘积集恒等式和覆盖结果，提出有限核降维原理，并在交换Moufang环中推导出Freiman型定理及多项式覆盖定理。
Let $L$ be a Moufang loop which is centrally nilpotent of class 2. We first show that the nuclearly-derived subloop (normal associator subloop) $L^*$ of $L$ has exponent dividing 6. It follows that $L_p$ (the subloop of $L$ of elements of $p$-power order) is associative for $p>3$. Next, a loop $L$ is said to be a {\it small Frattini Moufang loop}, or SFML, if $L$ has a central subgroup $Z$ of order $p$ such that $C\isom L/Z$ is an elementary abelian $p$-group. $C$ is thus given the structure of what we call a {\it coded vector space}, or CVS. (In the associative/group case, CVS's are either or…
It is well known that if a group G factorizes as G = NH where H\leq G and N is normal in G then the group structure of G is determined by the subgroups H and N, the intersection of N with H and how H acts on N with a homomorphism f : H -> Aut(N). Here we generalize the idea by creating extensions using the semi-automorphism group of N. We show that if G=NH is a Moufang loop, N is a normal subloop, and H=<u> is a finite cyclic group of order coprime to three then the binary operation of G depends only on the binary operation of N, the intersection of N with H, and how u permutes the elements of…
Modular Cocycles and Haar-Type Measures on Topological Loops
本文研究拓扑循环上的Haar型测度与模cocycle，揭示非结合性对测度扭曲的影响及循环恒等式对模数据的结构限制。
Let $Δ$ be a spherical building each of whose irreducible components is infinite, has rank at least 2 and satisfies the Moufang condition. We show that $Δ$ can be given the structure of a topological building that is compact and totally disconnected precisely when $Δ$ is the building at infinity of a locally finite affine building.
Topological spaces related with topological quasigroups
本文研究拓扑拟群相关的拓扑空间结构，探讨正则性、完备性、分离性、嵌入性、同胚同构及紧化与完备性关系，并证明相关定理。
Automorphic loops are loops in which all inner mappings are automorphisms. This variety of loops includes, for instance, groups and commutative Moufang loops. We study uniquely 2-divisible automorphic loops, particularly automorphic loops of odd order, from the point of view of the associated Bruck loops (motivated by Glauberman's work on uniquely 2-divisible Moufang loops) and the associated Lie rings (motivated by a construction of Wright). We prove that every automorphic loop $Q$ of odd order is solvable, contains an element of order $p$ for every prime $p$ dividing $|Q|$, and $|S|$ divides…
On group and loop spheres
本文将经典合成代数推广到一般环上可能退化的二次型，用三元代数结构定义群球面与 Moufang 球面空间，并建立三元 Moufang 环与 Albert-Cayley-Dickson 构造及广义三元八元数代数的联系。
Let $G$ be a finite group and $H$ a subgroup of $G$. Each left transversal (with identity) of $H$ in $G$ has a left loop (left quasigroup with identity) structure induced by the binary operation of $G$. We say two left transversals are isomorphic if they are isomorphic with respect to the induced left loop structures. In this paper, we develop a method to calculate the number of isomorphism classes of transversals of $H$ in $G$. Also with the help of this we calculate the number of non-isomorphic left loops of a given order.
The Moufang loop named for Richard Parker is a central extension of the extended binary Golay code. It the prototypical example of a general class of nonassociative structures known today as code loops, which have been studied from a number of different algebraic and combinatorial perspectives. This expository article aims to highlight an experimental approach to computing in code loops, by a combination of a small amount of precomputed information and making use of the rich identities that code loops' twisted cocycles satisfy. As a byproduct we demonstrate that one can reconstruct the multipl…
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