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Infinite distributivity in the lattice of varieties of \(\ell\)-groups. (English. Russian original) Zbl 0691.06006

Sib. Math. J. 30, No. 2, 342-345 (1989); translation from Sib. Mat. Zh. 30, No. 2(174), 216-220 (1989).
The author proves that the identity U(\(\bigvee_{i\in I}V_ i)=\bigvee_{i\in I}UV_ i\) does not hold in the lattice of varieties of \(\ell\)-groups for infinite index sets I. This answers in the negative a question posed by A. M. W. Glass, C. W. Holland and S. H. McCleary [Algebra Univers. 10, 1-20 (1980; Zbl 0439.06013)].
Reviewer: B.M.Schein

MSC:

06F15 Ordered groups
08B15 Lattices of varieties

Citations:

Zbl 0439.06013
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References:

[1] A. M. W. Glass, C. W. Holland, and S. H. McCleary, ?The structure of ?-group varieties,? Algebra Univers.,10, No. 1, 1-20 (1980). · Zbl 0439.06013 · doi:10.1007/BF02482885
[2] T. Feil, ?Varieties of representable ?-groups,? in: Ordered Algebraic Structures: Proc. Am. Math. Soc. Meeting, Cincinnati (Ohio), 1982, Lecture Notes Pure Appl. Math., Vol. 99 (1985), pp. 77-87.
[3] V. M. Kopytov and N. Ya. Medvedev, ?Varieties of lattice-ordered groups,? Usp. Mat. Nauk,40, No. 6, 117-128 (1985).
[4] V. M. Kopytov, Lattice-Ordered Groups [in Russian], Nauka, Moscow (1984). · Zbl 0567.06011
[5] M. I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of Group Theory [in Russian], Nauka, Moscow (1982). · Zbl 0508.20001
[6] N. Ya. Medvedev, ?On the lattice of o-approximable ?-varieties,? Czech. Math. J.,34, No. 1, 6-17 (1984). · Zbl 0551.06017
[7] A. I. Mal’tsev, ?On multiplication of classes of algebraic systems,? Sib. Mat. Zh.,8, No. 2, 346-365 (1967).
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