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.
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).
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...
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.
In this part the smallest non-abelian quasivarieties for nilpotent Moufang loops are described.
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.