On the lattice of approximable \(\ell\)-varieties. (Russian) Zbl 0551.06017

Medvedev, N. Ya.

Czech. Math. J. 34(109), 6-17 (1984).

An \(\ell\)-variety satisfying \((x\wedge y^{-1} x^{-1}y) \vee e=e\) is called o-approximable. The author shows that the lattice \(L_ o\) of all o-approximable \(\ell\)-varieties has not the covering property (namely, he constructs a variety \(V\in L_ o\) which is not the greatest element of \(L_ o\) and has no cover in \(L_ o)\). Further, he proves that \(L_ o\) is not a Brouwer lattice and finds the base ranks (minimal number of generators of \(\ell\)-groups generating the given variety) for \(\ell\)- varieties given by \((x\wedge y^{-1} x^{-1}y)\vee e=e\) and by \(| x| | y| \wedge | y|^ 2| x|^ 2=| x| | y|\); they are 2 in both cases. Finally, he constructs an \(\ell\)-variety having no independent equational base.

Reviewer: V.Novák

MSC:

06F15	Ordered groups
08B15	Lattices of varieties
06B20	Varieties of lattices

Keywords:

o-approximable \(\ell \)-varieties; covering property; base ranks