On infinitary quasigroups.
On left distributive left idempotent groupoids
We study the groupoids satisfying both the left distributivity and the left idempotency laws. We show that they possess a canonical congruence admitting an idempotent groupoid as factor. This congruence gives a construction of left idempotent left distributive groupoids from left distributive idempotent groupoids and right constant groupoids.
On Lie semiheaps and ternary principal bundles
We introduce the notion of a Lie semiheap as a smooth manifold equipped with a para-associative ternary product. For a particular class of Lie semiheaps we establish the existence of left-invariant vector fields. Furthermore, we show how such manifolds are related to Lie groups and establish the analogue of principal bundles in this ternary setting. In particular, we generalise the well-known ‘heapification’ functor to the ambience of Lie groups and principal bundles.
On loops that are abelian groups over the nucleus and Buchsteiner loops
We give sufficient and in some cases necessary conditions for the conjugacy closedness of provided the commutativity of . We show that if for some loop , and are abelian groups, then is a CC loop, consequently has nilpotency class at most three. We give additionally some reasonable conditions which imply the nilpotency of the multiplication group of class at most three. We describe the structure of Buchsteiner loops with abelian inner mapping groups.
On loops whose inner permutations commute
Multiplication groups of (finite) loops with commuting inner permutations are investigated. Special attention is paid to the normal closure of the abelian permutation group.
On Maps x ... xn and the Isotopy-Isomorphy Property of Moufang Loops (Short Communication)
On Maps x-xn and the Isotopy-Isomorphy Property of Moufang Loops
On median point of the system of elements of -structure
On middle universal weak and cross inverse property loops with equal length of inverse cycles.
On Mikheev's construction of enveloping groups
Mikheev, starting from a Moufang loop, constructed a groupoid and reported that this groupoid is in fact a group which, in an appropriate sense, is universal with respect to enveloping the Moufang loop. Later Grishkov and Zavarnitsine gave a complete proof of Mikheev's results. Here we give a direct and self-contained proof that Mikheev's groupoid is a group, in the process extending the result from Moufang loops to Bol loops.
On minimization theorems for polyadic groups H-derived from groups
On Moufang A-loops
In a series of papers from the 1940’s and 1950’s, R.H. Bruck and L.J. Paige developed a provocative line of research detailing the similarities between two important classes of loops: the diassociative A-loops and the Moufang loops ([1]). Though they did not publish any classification theorems, in 1958, Bruck’s colleague, J.M. Osborn, managed to show that diassociative, commutative A-loops are Moufang ([5]). In [2] we relaunched this now over 50 year old program by examining conditions under which...
On multiplication groups of left conjugacy closed loops
A loop is said to be left conjugacy closed (LCC) if the set is closed under conjugation. Let be such a loop, let and be the left and right multiplication groups of , respectively, and let be its inner mapping group. Then there exists a homomorphism determined by , and the orbits of coincide with the cosets of , the associator subloop of . All LCC loops of prime order are abelian groups.
On multiplication groups of relatively free quasigroups isotopic to Abelian groups
If is a quasigroup that is free in the class of all quasigroups which are isotopic to an Abelian group, then its multiplication group is a Frobenius group. Conversely, if is a Frobenius group, a quasigroup, then has to be isotopic to an Abelian group. If is, in addition, finite, then it must be a central quasigroup (a -quasigroup).
On nilpotent n-groups
On nondistributive Steiner quasigroups
A well known result of R. Dedekind states that a lattice is nonmodular if and only if it has a sublattice isomorphic to . Similarly a lattice is nondistributive if and only if it has a sublattice isomorphic to or (see [11]). Recently a few results in this spirit were obtained involving the number of polynomials of an algebra (see e.g. [1], [3], [5], [6]). In this paper we prove that a nondistributive Steiner quasigroup (G,·) has at least 21 essentially ternary polynomials (which improves the...
On Nuclei Of n-Ary Quasigroups
On nuclei of n-ary quasigroups.
On nuclei of regular groupoids with division
On one class of quasigroups