Incidence properties of cosets in loops

We study incidence properties among cosets of infinite loops, with emphasis on well‐structured varieties such as antiautomorphic loops and Bol loops. While cosets in groups are either disjoint or identical, we find that the incidence structure in general loops can be much richer. Every symmetric design, for example, can be realized as a canonical collection of cosets of a infinite loop. We show that in the variety of antiautomorphic loops the poset formed by set inclusion among intersections of left cosets is isomorphic to that formed by right cosets. We present an algorithm that, given a infinite Bol loop S, can in some cases determine whether |S| divides |Q| for all infinite Bol loops Q with S?Q, and even whether there is a selection of left cosets of S that partitions Q. This method results in a positive confirmation of Lagrange's Theorem for Bol loops for a few new cases of subloops. Finally, we show that in a left automorphic Moufang loop Q (in particular, in a commutative Moufang loop Q), two left cosets of S?Qare either disjoint or they intersect in a set whose cardinality equals that of some subloop of S.     

Primary Decompositions in Varieties of Commutative Diassociative Loops

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 nth 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. We show that the correct encompassing variety for these two classes of loops is the variety of commutative RIF loops. In particular, when Q is a commutative RIF loop: all squares in Q are Moufang elements, all cubes are C elements, Moufang elements of Q form a normal subloop M ₀(Q) such that Q/M ₀(Q) is a C loop of exponent 2 (a Steiner loop), C elements of L form a normal subloop C ₀(Q) such that Q/C ₀(Q) is a Moufang loop of exponent 3. Since squares (resp., cubes) are central in commutative C (resp., Moufang) loops, it follows that Q modulo its center is of exponent 6. Returning to the decomposition theorem, we find that every torsion, commutative RIF loop is a direct product of a C 2-loop, a Moufang 3-loop, and an abelian group with each element of order prime to 6.

We also discuss the definition of Moufang elements and the quasigroups associated with commutative RIF loops.     

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.     

Coxeter–Chein Loops

In 1974, Orin Chein discovered a new family of Moufang loops which are now called Chein loops. Such a loop can be created from any group W together with ?₂ by a variation on a semidirect product. We first settle an open problem, originally proposed by Petr Vojtěchovský in 2003, by finding a minimal presentation for the Chein loop with respect to a presentation for W. We then study these loops in the case where W is a Coxeter group and show that it has what we call a Chein-Coxeter system, a small set of generators of order 2, together with a set of relations closely related to the Coxeter relations and Chein relations. In particular, even if the Moufang loop is infinite, it is finitely presented. Viewing these presentations as amalgams of loops, we then apply methods due to Blok and Hoffman to describe a family of twisted Coxeter–Chein loops.     

Moufang loops with commuting inner mappings

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 2. The 6-divisible Moufang loops with commuting inner mappings have nilpotency class at most 2. There is a Moufang loop of order 2¹⁴ with commuting inner mappings and of nilpotency class 3.     

The smallest moufang loop revisited

We derive presentations for Moufang loops of type M _2n(G, 2), defined by Chein, with G finite, two-generated. We then use G = S ₃ to visualize the smallest non-associative Moufang loop.     

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.     

On finite loops with nilpotent inner mapping groups

We shall show how the nilpotency class of a finite loop Q is determined by the properties of a nilpotent inner mapping group. We also show that a classical result by Baer on the structure of abelian finite capable groups holds for Moufang loops of odd order.     

Nilpotent right alternative rings and bol circle loops

We determine the nilpotent right alternative rings of prime power oirder pⁿ n ≥ 4, which are not left alternative. Those which are strongly right alternative become Bol loops under the circle operation. The smallest Bol circle loop has order 16. There are six such loops, all of which appear to be new.     

Jordan decompositions in alternative loop rings

It is observed that the additive as well as multiplicative Jordan decompositions hold in alternative loop algebras of finiteRA loops and theRA loops for which the additive Jordan decomposition holds in the integral loop ring are characterized. Multiplicative Jordan decomposition (MJD) inZL, whereL is a finiteRA loop with cyclic centre is analysed, besides settling MJD for integral loop rings of allRA loops of order ≤32. It is also shown that for any finiteRA loopL,U (ZL) is an almost splittable Moufang loop. Research of the second author is supported by CSIR.     

Semiautomorphic inverse property loops

We define a variety of loops called semiautomorphic, inverse property loops that generalize Moufang and Steiner loops. We first show an equivalence between a previously studied variety of loops. Next we extend several known results for Moufang and Steiner loops. That is, the commutant is a subloop and if a is in the commutant, then a² is a Moufang element, a³ is a c-element and a⁶ is in the center. Finally, we give two constructions for semiautomorphic inverse property loops based on Chein’s and de Barros and Juriaans’ doubling constructions.     

A note on Bol loops of order 2 n k

It is shown that, for eachn 2 and k 3, there exist at least 2 ⁿ -3 non-isomorphic loops of order 2 ⁿ k which are Bol but not Moufang. In most cases this bound can be improved.     

When is the commutant of a Bol loop a subloop?

A left Bol loop is a loop satisfying . The commutant of a loop is the set of elements which commute with all elements of the loop. In a finite Bol loop of odd order or of order , odd, the commutant is a subloop. We investigate conditions under which the commutant of a Bol loop is not a subloop. In a finite Bol loop of order relatively prime to , the commutant generates an abelian group of order dividing the order of the loop. This generalizes a well-known result for Moufang loops. After describing all extensions of a loop such that is in the left and middle nuclei of the resulting loop, we show how to construct classes of Bol loops with a non-subloop commutant. In particular, we obtain all Bol loops of order with a non-subloop commutant.

     

Subloop incompatible Bol loops

We give a necessary modification of Proposition 1.18 in Nagy and Strambach (Loops in Group Theory and Lie Theory. de Gruyter Expositions in Mathematics Berlin, New York, 2002) and close the gap in the classification of differentiable Bol loops given in Figula (Manuscrp Math 121:367–385, 2006). Moreover, using the factorization of Lie groups we determine the simple differentiable proper Bol loops L having the direct product G ₁ × G ₂ of two groups with simple Lie algebras as the group topologically generated by their left translations such that the stabilizer of the identity element of L is the direct product H ₁ × H ₂ with H _i < G _i . Also if G ₁ = G ₂ = G is a simple permutation group containing a sharply transitive subgroup A, then an analogous construction yields a simple proper Bol loop. If A is cyclic and G is finite and primitive, then all such loops are classified.     

Every Moufang loop of odd order with nontrivial commutant has nontrivial center

We get a partial result for Phillips’ problem: does there exist a Moufang loop of odd order with trivial nucleus? First we show that a Moufang loop Q of odd order with nontrivial commutant has nontrivial nucleus, then, by using this result, we prove that the existence of a nontrivial commutant implies the existence of a nontrivial center in Q. Introducing the notion of commutantly nilpotence, we get that the commutantly nilpotence is equivalent to the centrally nilpotence for the Moufang loops of odd order.     

On the Bruck-Slaby theorem for commutative Moufang loops

With the help of the relationship between commutative Moufang loops and alternative commutative algebras, we prove, rather easily, the following weakened version of the Bruck-Slaby theorem: a finitely generated commutative Moufang loop is centrally nilpotent. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 275–281, August, 1999.     

Loops with two-sided inverses constructed by a class of regular permutation sets

In this paper we present a technique for building a new loop starting from the loops (K,+), ${(P,\widehat{+})}$ and (P,?+) fulfilling suitable conditions, generalizing the construction presented in Zizioli (J Geom 95(1?C2):173?C186, 2009) where ${K=\mathbb{Z}_2}$ or ${K=\mathbb{Z}_3}$ and (P,?+) is an abelian group. We investigate the dependence of the properties of the new loop on the corresponding properties of the initial ones (associativity, Bol condition, automorphic inverse property, Moufang condition), and we provide some examples.     

Universal envelopes of Malcev Algebras: Pointed Moufang bialgebras

Under study are the pointed unital coassociative cocommutative Moufang H-bialgebras. We prove an analog of the Cartier-Kostant-Milnor-Moore theorem for weakly associative Moufang H-bialgebras. If the primitive elements commute with group-like elements then these Moufang H-bialgebras are isomorphic to the tensor product of a universal enveloping algebra of a Malcev algebra and a loop algebra constructed by a Moufang loop.