The lattice of quasivarieties of commutative moufang loops

Abstract

It is proved that a lattice of quasivarieties of an arbitrary variety\(\mathfrak{M}\) of commutative Moufang loops either has the power of the continuum or is finite, and that the latter is the case iff\(\mathfrak{M}\) is generated by a finite group. It is also stated that the lattice of all quasivarieties of a least nonassociative variety of commutative Moufang loops contains a quasivariety which is generated by a finite quasigroup and has no covers; hence, it has no independent basis of quasi-identities.