Bol Loops Research Articles

HOLOMORPH OF GENERALIZED BOL LOOPS II

The notion of the holomorph of a generalized Bol loop (GBL) is characterized afresh. The holomorph of a right inverse property loop (RIPL) is shown to be a GBL if and only if the loop is a GBL and some bijections of the loop are right (middle) regular. The holomorph of a RIPL is shown to be a GBL if and only if the loop is a GBL and some elements of the loop are right (middle) nuclear. Necessary and sufficient conditions for t holomorph of a RIPL to be a Bol loop are deduced. Some algebraic properties and commutative diagrams are established for a RIPL whose holomorph is a GBL.

Multidimensional smooth loops with universal elasticity

Let be a universal (isotopically invariant) identity that is derived from the elasticity identity . One of the authors has previously shown that a) each local loop of dimension with identity (briefly, a loop ) is a smooth middle Bol loop of dimension ; b) smooth two-dimensional loops are Lie groups; c) up to isotopy, there exist only two three-dimensional loops : the loops and . In this paper, the loops and are extended to the multidimensional case. The fact that each smooth loop of dimension corresponds to a unique multidimensional three-web on a manifold of dimension is key to our work. In addition, the class of loops under investigation is characterized by the fact that the torsion tensor of the corresponding web has rank 1 (that is, the algebra generated by this tensor has a one-dimensional derived algebra). This enables us to express the differential equations of the problem in an invariant form. The system of equations thus obtained was found to be amenable to integration in the most general case, and the equations of the required loops have been obtained in local coordinates. Bibliography: 17 titles.

Holomorph of generalized Bol loops II

The notion of the holomorph of a generalized Bol loop (GBL) is characterized afresh. The holomorph of a right inverse property loop (RIPL) is shown to be a GBL if and only if the loop is a GBL and some bijections of the loop are right (middle) regular. The holomorph of a RIPL is shown to be a GBL if and only if the loop is a GBL and some elements of the loop are right (middle) nuclear. Necessary and sufficient condition for the holomorph of a RIPL to be a Bol loop are deduced. Some algebraic properties and commutative diagrams are established for a RIPL whose holomorph is a GBL.

Bol loops in right alternative ring

Let (G; ) be a Bol loop and let , , be mappings of G into G so that x = x x = x(x x) for all x2 G. We show that the following conditions are equivalent: (a) (xy z)y = x(y(z y )) for all x; y; z2 G, (b) (G; ) is Moufang and x is in the nucleus of (G; ) for all x2 G.

ON THE UNIVERSALITY OF SOME SMARANDACHE LOOPS OF BOL-MOUFANG TYPE

A Smarandache quasigroup(loop) is shown to be universal if all its f, g-principal isotopes are Smarandache f, g-principal isotopes. Also, weak Smarandache loops of Bol-Moufang type such as Smarandache: left(right) Bol, Moufang and extra loops are shown to be universal if all their f, g-principal isotopes are Smarandache f, gprincipal isotopes. Conversely, it is shown that if these weak Smarandache loops of Bol-Moufang type are universal, then some autotopisms are true in the weak Smarandache sub-loops of the weak Smarandache loops of Bol-Moufang type relative to some Smarandache elements. Futhermore, a Smarandache left(right) inverse property loop in which all its f, g-principal isotopes are Smarandache f, g-principal isotopes is shown to be universal if and only if it is a Smarandache left(right) Bol loop in which all its f, g-principal isotopes are Smarandache f, g-principal isotopes. Also, it is established that a Smarandache inverse property loop in which all its f, g-principal isotopes are Smarandache f, g-principal isotopes is universal if and only if it is a Smarandache Moufang loop in which all its f, g-principal isotopes are Smarandache f, g-principal isotopes. Hence, some of the autotopisms earlier mentioned are found to be true in the Smarandache sub-loops of universal Smarandache: left(right) inverse property loops and inverse property loops.

A Representation of a Point Symmetric 2-Structure by a Quasi-Domain

In a symmetric 2-structure \({\Sigma =(P,\mathfrak{G}_1,\mathfrak{G}_2,\mathfrak{K})}\) we fix a chain \({E \in \mathfrak{K}}\) and define on E two binary operations “+” and “·”. Then (E,+) is a K-loop and for \({E^* := E {\setminus}\{o\}}\), (E *,·) is a Bol loop. If \({\Sigma}\) is even point symmetric then (E,+ ,·) is a quasidomain and one has the set \({Aff(E,+,\cdot) := \{a^+\circ b^\bullet | a \in E, b \in E^*\}}\) of affine permutations. From Aff(E, +, ·) one can reproduce via a “chain derivation” the point symmetric 2-structure \({\Sigma}\).

On 2-Engel groups and Bruck loops

Abstract.A classical construction associates a Bruck loop with a Moufang or Bol loop on which the squaring map is a permutation. The idea of that construction is now extended to a

When Is a Bol Loop Moufang?

There are a number of identities which, if satisfied by a Bol loop, imply that the loop is actually Moufang. In this paper we show that in a number of cases, the Moufang identity is also forced not by a single identity, but by giving elements a choice of equations to satisfy.

ON SOME GENERALIZATIONS OF GROUPS WITH TRIALITY

In the present paper we generalize the concept of groups with triality and apply it to the theory of the Moufang, Bol and Bruck loops. Such generalizations allow us to reduce certain problems from the loop theory to problems in the theory of groups.

Inclusion of Loops into Lie Groups: Infinitesimal Characteristics

The infinitesimal characteristics for some class of smooth loops generalizing Bruck loops to be embedded into a Lie group are given. The interest to the problem of inclusion of loops into Lie groups has been shown by various recent publications dedicated to representations of loops as transversals on homogeneous spaces. There is presented an example of a smooth left Bol loop, satisfying the generalized Bruck identity, realized as a section on some reductive homogeneous space.

Incidence properties of cosets in loops

AbstractWe 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.

Special identities for Bol algebras

Bol algebras appear as the tangent algebra of Bol loops. A (left) Bol algebra is a vector space equipped with a binary operation [a,b] and a ternary operation {a,b,c} that satisfy five defining identities. If A is a left or right alternative algebra then Ab is a Bol algebra, where [a,b]≔ab-ba is the commutator and {a,b,c}≔〈b,c,a〉 is the Jordan associator. A special identity is an identity satisfied by Ab for all right alternative algebras A, but not satisfied by the free Bol algebra. We show that there are no special identities of degree ⩽7, but there are special identities of degree 8. We obtain all the special identities of degree 8 in partition six-two.

Algebraic Bol loops

In this paper, we study the category of algebraic Bol loops over an algebraically closed field of definition. On the one hand, we apply techniques from the theory of algebraic groups in order to prove structural theorems for this category. On the other hand, we present some examples showing that these loops lack some nice properties of algebraic groups; for example, we construct local algebraic Bol loops which are not birationally equivalent to global algebaric loops.

Matched Pairs, Permutation Representations, and the Bol Property

The article examines Bol loops that are constructed from matched pairs of groups, including a new simple, non-Moufang Bol loop of order 120. Certain permutation representations of the loops under consideration provide nontrivial examples of doubly stochastic action matrices. A generalization of the matched-pair loop construction yields proper Bol actions of groups.

UNIQUELY 2-DIVISIBLE BOL LOOPS

Although any finite Bol loop of odd prime exponent is solvable, we show there exist such Bol loops with trivial center. We also construct finitely generated, infinite, simple Bruck loops of odd prime exponent for sufficiently large primes. This shows that the Burnside problem for Bruck loops has a negative answer.

On the smallest simple, unipotent Bol loop

Finite simple, unipotent Bol loops have recently been identified and constructed using group theory. However, the purely group-theoretical constructions of the actual loops are indirect, somewhat arbitrary in places, and rely on computer calculations to a certain extent. In the spirit of revisionism, this paper is intended to give a more explicit combinatorial specification of the smallest simple, unipotent Bol loop, making use of concepts from projective geometry and quasigroup theory along with the group-theoretical background. The loop has dual permutation representations on the projective line of order 5, with doubly stochastic action matrices.

A class of finite simple Bol loops of exponent 2

In this paper we give an infinite class of finite simple right Bol loops of exponent 2. The right multiplication group of these loops is an extension of an elementary Abelian $2$-group by $S_5$. The construction uses the description of the structure of such loops given by M. Aschbacher (2005). These results answer some questions of M. Aschbacher.

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.

A class of simple proper Bol loops

The existence of finite simple non-Moufang Bol loops has long been considered to be one of the main open problems in the theory of loops and quasigroups. In this paper, we present a class of simple proper Bol loops. This class contains finite and new infinite simple proper Bol loops.

When is the commutant of a Bol loop a subloop?

A left Bol loop is a loop satisfying x ( y ( x z ) ) = ( x ( y x ) ) z x(y(xz)) = (x(yx))z . 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 2 k 2k , k k 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 3 3 , 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 K K such that K K 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 16 16 with a non-subloop commutant.