Left Bol Research Articles

Algebraic connections between right and middle Bol loops and their cores

To every right or left Bol loop corresponds a middle Bol loop. In this paper, the cores of right Bol loops (RBL) and its corresponding middle Bol loops (MBL) were studied. Their algebraic connections were considered. It was shown that the core of a RBL is elastic and right idempotent. The core of a RBL was found to be alternative (or left idempotent) if and only if its corresponding MBL is right symmetric. If a MBL is right (left) symmetric, then, the core of its corresponding RBL is a medial (semimedial). The core of a middle Bol loop has the left inverse property (automorphic inverse property, right idempotence resp.) if and only if its corresponding RBL has the super anti-automorphic inverse property (automorphic inverse property, exponent 2 resp.). If a RBL is of exponent 2, then, the core of its corresponding MBL is left idempotent. If a RBL is of exponent 2 then: the core of a MBL has the left alternative property (right alternative property) if and only if its corresponding RBL has the cross inverse property (middle symmetry). Some other similar results were derived for RBL of exponent 3.

New algebraic properties of middle Bol loops II

A loop (Q, ·, \, /) is called a middle Bol loop (MBL) if it obeys the identity x(yz\x)=(x/z)(y\x). To every MBL corresponds a right Bol loop (RBL) and a left Bol loop (LBL). In this paper, some new algebraic properties of a middle Bol loop are established in a different style. Some new methods of constructing a MBL by using a non-abelian group, the holomorph of a right Bol loop and a ring are described. Some equivalent necessary and sufficient conditions for a right (left) Bol loop to be a middle Bol loop are established. A RBL (MBL, LBL, MBL) is shown to be a MBL (RBL, MBL, LBL) if and only if it is a Moufang loop.

The first-countability of paratopological left-loops

In this paper, we introduce paratopogical left-loops, as natural generalizations of paratopogical left Bol loops defined by Z. Cai, S. Lin and W. He in [7], and prove that some known results for paratopological groups are still valid for paratopogical left-loops. The main results are: (i) every weakly first-countable paratopological left-loop is first-countable; (ii) a sequential, csf-countable, regular paratopological left-loop is first-countable if and only if it contains no closed copy of Sω.

Differential equations of smooth loops related to some space-time models: Integrability conditions and geometry

The original approach of Lie to the theory of transformation groups acting on smooth manifolds, on the basis of differential equations, being applied to smooth loops, has permitted the development of the infinitesimal theory of smooth loops generalizing the Lie group theory.A loop with the law of associativity verified for its binary operation is a group. It has been shown that the system of differential equations characterizing a smooth loop with the right Bol identity and the integrability conditions lead to the binary-ternary algebra as a proper infinitesimal object, which turns out to be the Bol algebra (i.e. a Lie triple system with an additional bilinear skew-symmetric operation). There exist the analogous considerations for Moufang loops.We will consider the differential equations of smooth loops, generalizing smooth left Bol loops, with the identities that are the characteristic identities for the algebraic description of some relativistic space-time models. Further examinations of the integrability conditions for the differential equations allow us to introduce the proper infinitesimal object for some subclass of loops under consideration.The geometry of corresponding homogeneous spaces can be described in terms of tensors of curvature and torsion.

K-loop Structures Raised by the Direct Limits of Pseudo Unitary $U(p, a_{n})$ and Pseudo Orthogonal $O(p, a_{n})$ Groups

A left Bol loop satisfying the automorphic inverse property is called a K-loop or a gyrocommutative gyrogroup. K-loops have been in the centre of attraction since its first discovery by A.A. Ungar in the context of Einstein's 1905 relativistic theory. In this paper some of the infinite dimensional K-loops are built from the direct limit of finite dimensional group transversals.

Topological gyrogroups: Generalization of topological groups

Left Bol loops with the Aℓ-property or gyrogroups are generalization of groups which do not explicitly have associativity. In this work, we define topological gyrogroups and study some properties of them. In spite of having a weaker algebraic form, topological gyrogroups carry almost the same basic properties owned by topological groups. In particular, we prove that being T0 and T3 are equivalent in topological gyrogroups. Furthermore, a topological gyrogroup is first countable if and only if it is premetrizable. Finally, a direct product of topological gyrogroups is a topological gyrogroup.

A Family of Left Lie Bol Loops

In this paper the left Bol split extension method is used to build left Bol Lie loops from the Lie groups $H$ and $K$ such that $H$ is a Lie subgroup of $Aut(K)$. Furthermore, we investigated some of the properties of those loops constructed in this way. Examples are given for finite and infinite dimensional left Bol Lie loops. Moreover, we showed that the twisted semidirect product of Lie algebras is an Akivis algebra.

On four-dimensional left Bol three-webs with the same core

We found all four-dimensional left Bol three-webs with the same core as that of the unique four-dimensional nonparallelizable group three-web.

On loop extensions satisfying one single identity and cohomology of loops

ABSTRACTIn this paper are defined cohomology-like groups that classify loop extensions satisfying a given identity in three variables for association identities and in two variables for the case of commutativity. It is considered a large amount of identities. These groups generalize those defined in Nishigori [3] and of Kenneth and Leedham-Green [2]. It is computed the number of metacyclic extensions for trivial action of the quotient on the kernel in one particular case for left Bol loops and in general for commutative loops.

On structure equations of a three-web Defined by the core of a given left bol three-web

We give a solution to the problem stated by M. A. Akivis in 1976: Find the structure equations of the 3-web defined by the core of a given Bol 3-web

On the three-web associated to the core of a left Bol three-web

Let B l be a left Bol three-web given on a 2r-dimensional smooth manifold, CB l the left Bol three-web associated to the core of B l, and let CCB l be the left Bol three-web associated to the core of CB l. We prove that the three-webs CB l and CCB l are equivalent.

Six-dimensional left Bol three-webs with common core

We find the conditions under which left Bol three-webs have the same core. These conditions are applied to the finding of six-dimensional left Bol three-webs having the same cores as the webs obtained by the parastrophy transformation from the well known six-dimensional elastic webs E1 and E2.

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.

On isoclinic left Bol three-webs with the trivial core

On isoclinic left Bol three-webs with the trivial core

Left Bol three-webs with the IC-property

In this paper, we describe some properties of the surface V. For a Bol web Bwith the IC-property, we find expansions into a series (up to the third degree) of the equations of the local coordinate loop and the equations of the core. With the use of these expansions, it is proved that if a left Bol web Bhas the IC-property, then each its coordinate loopP , P ∈ V , has zero commutator. We find the equations of the submanifold V in terms of an adapted cobasis and the expansion into a series of the function y = ϕ(x) which determines the surface V in terms of canonical coordinates. We prove (Theorem 2) that the surface V is an isoclinic-geodesic and diagonal surface of the web W in question and that the existence of such surfaces characterizes webs Bwith theIC-property. It turns out that the conditions of existence of the submanifold V are equivalent to the existence of a submanifold on which (and, respectively, on the base of the first foliation of the web) the symmetric connection Γ coincides with the Chern canonical affine connection of B� . The results obtained are illustrated by examples.

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.

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.

A Characterization of Bol left loops

The relationship between the Bol identity, the so-called Left Loop Property LLP and the Left Inverse Property in any left quasigroup is determined. Counterexamples are given whenever two properties are not equivalent. It is shown that a principal isotope of a LLP quasigroup is a left Bol loop. In any LLP left quasigroup the existence of a right identity element is equivalent to the right division.

On the notion of gyrogroup

It is shown that the gyrogroup [1] is just the well known left Bol loop with Bruck identity. This result has been announced in [12]. The other constructions of [1] are also discussed.

Loops as Invariant Sections in Groups, and their Geometry

AbstractWe investigate left conjugacy closed loops which can be given by invariant sections in the group generated by their left translations. These loops are generalizations of the conjugacy closed loops introduced in [13] just as Bol loops generalize Moufang loops. The relations of these loops to common classes of loops are studied. For instance on a connected manifold we construct proper topological left conjugacy closed loops satisfying the left Bol condition but show that any differentiable such loop must be a group. We show that the configurational condition in the 3-net corresponding to an isotopy class of left conjugacy closed loops has the same importance in the geometry of 3-nets as the Reidemeister or the Bol condition.