It is proved that the maximum condition for subloops in a commutative Moufang loop Q is equivalent with the conditions of finite generating of different subloops of the loop Q and different subgroups of the multiplication group of the loop Q. An analogue equivalence is set for the commutative Moufang ZA-loops. Classification: 20N05

The differential equations for a continuous birepresentation of a local analytic Moufang loop are established. The commutation relations for the infinitesimal operators of the birepresentation are found. These commutation relations can be seen as a (minimal) generalization of the Maurer-Cartan equations and do not depend on the particular birepresentation. 2000 MSC: 20N05, 17D10

It has been shown that for distinct odd primes p1, p2, . . . , pn and q, all Moufang loops of order p1p2 · · · pnq are groups if and only if q is not congruent to 1 modulo pi for each i. In this paper, we extend that result to include Moufang loops of order p1p 2 2 · · · pnq. Mathematics Subject Classification: 20N05

Generalized Lie-Cartan theorem for linear birepresentations of an analytic Moufang loop is considered. The commutation relations of the generators of the birepresentation are found. In particular, the Lie algebra of the multiplication group of the birepresentation is explicitly given. 2000 MSC: 20N05, 17D10, 20G05 Dedicated to Maks A. Akivis on the occasion of his 85th birthday and 65 years of ...

In this note we establish connections between the orthogonality of conjugates of a finite or infinite quasigroup and some strictly transitive subsets of the complete multiplication group of this quasigroup. These connections are used for the investigation of orthogonality of distinct pairs of conjugates for quasigroups (loops) from some classes. For finite quasigroups the quasi-identities corre...

In this work, two normal congruences are built on two quasigroups with underlining set Zn relative to the linear-bivariate polynomial P (x, y) = a+ bx+ cy that generates a quasigroup over the ring Zn. Four quasigroups are built using the normal congruences and these are shown to be homomorphic to the quasigroups with underlining set Zn. Some subquasigroups of the quasigroups with underlining se...

Let L be a commutative Moufang loop (CML) with multiplication group M, and let F(L), F(M) be the Frattini subgroup and Frattini subgroup of L and M respectively. It is proved that F(L) = L if and only if F(M) = M and is described the structure of this CLM. Constructively it is defined the notion of normalizer for subloops in CML. Using this it is proved that if F(L) 6= L then L satisfies the no...

Let L1 be a finite simple sloop of cardinality n or the 8-element sloop. In this paper, we construct a subdirectly irreducible (monolithic) sloop L = 2⊗αL1 of cardinality 2n, for each n ≥ 8, with n ≡ 2 or 4 (mod 6), in which each proper homomorphic image is a Boolean sloop. Quackenbush [12] has proved that the variety V (L1) generated by a finite simple planar sloop L1 covers the smallest nontr...

We establish connection between product of two matrices of order k × k over a field and the product of the k-mappings corresponding to the k-operations, defined by these matrices. It is proved that, in contrast to the binary case, for arity k ≥ 3 the components of the k-permutation inverse to a k-permutation, all components of which are polynomial k-quasigroups, are not necessarily k-quasigroup...

The structure of the commutative Moufang loops (CML) with minimum condition for subloops is examined. In particular it is proved that such a CML Q is a finite extension of a direct product of a finite number of the quasicyclic groups, lying in the centre of the CML Q. It is shown that the minimum conditions for subloops and for normal subloops are equivalent in a CML. Moreover, such CML also ch...