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 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 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
On quadratic loops of Bol-Moufang type
On Quasigroup Varieties Closed Under Isotopy
On quasigroups with the left loop property
Some properties of quasigroups with the left loop property are investigated. In loops we point out that the left loop property is closely related to the left Bol identity and the particular case of homogeneous loops is considered.
On quasiidentities of transitive quasigroups
On quasivarieties of nilpotent Moufang loops. I
In this part the smallest non-abelian quasivarieties for nilpotent Moufang loops are described.
On quasivarieties of nilpotent Moufang loops. II
In this part of the paper we study the quasiidentities of the nilpotent Moufang loops. In particular, we solve the problem of finite basis for quasiidentities in the finitely generated nilpotent Moufang loop.
On Semi-Symmetric Quasigroups.