Coverings in the lattice of \(\ell\)-varieties (Q791557)

A variety \(V\) of \(\ell\)-groups (\(\ell\)-variety) is said to be 0-approximable if \((x\wedge y^{-1}x^{-1}y)\vee e=e\) holds for every \(x,y\in G\) and \(G\in V\). The class \(L_0\) of all 0-approximable \(\ell\)-varieties is a lattice with respect to the naturally defined operations of sup and inf. Let \(\bar V, V\in L_0\). Then \(\bar V\) is said to cover \(V\) in \(L_0\) if \(\bar V\supseteq V\) and if \(\bar V\supseteq U\supseteq V\) for some \(U\in L_0\) implies \(U=\bar V\) or \(U=V\). In the present paper the existence of \(2^{\omega_0}\) 0-approximable \(\ell\)-varieties is proved which have \(2^{\omega_0}\) different coverings in the lattice \(L_0\).