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The variety of commutative Moufang loops has an independent basis of identities. (English) Zbl 0928.20054
Let \(M\) be the class formed by all commutative Moufang loops (CML), i.e., of all algebras with signature \(\langle\cdot,{}^{-1}\rangle\) in which the following identities are valid: \(xy\cdot zx=x(yz)x\); \(xy=yx\); \(x^{-1}\cdot xy=y\). Extending the result of T. Evans [J. Algebra 31, 508-513 (1974; Zbl 0285.20058)] refering to the nilpotent variety of CML, in this paper the following theorem is proved: Let \(N\) be a variety of CML. Then the following statements are valid: a) any non unity element of the lattice of all subvarieties of \(N\) posseses a covering; b) in \(N\) any subvariety posseses an independent basis of identities; c) in particular, any variety of CML posseses an independent basis of identities.
MSC:
20N05 Loops, quasigroups
08B15 Lattices of varieties
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