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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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