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The lattice of quasivarieties of commutative Moufang loops. (English. Russian original) Zbl 0917.20062
Algebra Logika 37, No. 6, 700-720 (1998); translation in Algebra Logic 37, No. 6, 399-410 (1998).
Previously, the author proved [Algebra Logika 30, No. 6, 726-734 (1991; Zbl 0778.20027)] that a quasivariety generated by a finitely generated commutative Moufang loop $$L$$ has a finite basis of quasi-identities if and only if $$L$$ is a group. In the article under review, it is proven that the lattice of quasivarieties of an arbitrary variety $$\mathfrak M$$ of commutative Moufang loops either has the cardinality of the continuum or is finite, and that the latter holds if and only if $$\mathfrak M$$ is generated by a finite group. Moreover, the author proves that the lattice of all quasivarieties of a minimal nonassociative variety of commutative Moufang loops contains a quasivariety generated by a finite quasigroup and has no covers; hence, it has no independent basis of quasi-identities.

MSC:
 20N05 Loops, quasigroups 08C15 Quasivarieties 08B15 Lattices of varieties 08B05 Equational logic, Mal’tsev conditions
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