On quasi-identities of finitely generated commutative Moufang loops
Copyright policy )On quasi-identities of finitely generated commutative Moufang loops
\textit{T. Evans} [J. Algebra 31, 508-513 (1974; Zbl 0285.20058)] proved that each finitely generated commutative Moufang loop has a finite basis of identities. This paper now aims at describing the finitely generated commutative Moufang loops with a finite basis of quasi-identities. First, the author adapts the group-theoretic method of \textit{A. Yu. Ol'shanskij} [Sib. Mat. Zh. 15, 1409-1413 (1974; Zbl 0307.20017)] to prove that, if a commutative Moufang loop \(L\) contains an infinite (resp. finite) nonassociative subloop in which all torsion-free subgroups (resp. (3- subgroups) are finitely generated, then the quasivariety generated by \(L\) has no basis of quasi-identities in finitely many variables. The main result of the paper is then: The quasivariety generated by \(L\) with finitely many generators has a finite basis of quasi-identities if and only if \(L\) is a finite group.
finite basis of quasi-identities, Loops, quasigroups, quasivariety, finite group, finite basis of identities, finitely generated commutative Moufang loops, Quasivarieties, Quasivarieties and varieties of groups
finite basis of quasi-identities, Loops, quasigroups, quasivariety, finite group, finite basis of identities, finitely generated commutative Moufang loops, Quasivarieties, Quasivarieties and varieties of groups
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