The lex property of varieties of lattice ordered groups (Q1180702)

Let \(Z\) be the additive group of all integers with the usual linear order and let \(G\) be a lattice-ordered group. The symbol \(G\overleftarrow{\times}Z\) denotes the lexicographic product of \(G\) and \(Z\). For each variety \(V\) of lattice-ordered groups put \(V^ L=\text{Var}(\{G\overleftarrow{\times}Z:\;G\in V\})\). Let us denote by \(A_ 1\) and \(A_ 2\) the class of all \(V\) such that \(V^ L=V\), or \(V^ L\neq V\), respectively. \textit{J. E. Smith} [Trans. Am. Math. Soc. 257, 347- 357 (1980; Zbl 0459.06007)] proved that the representable variety, the normal valued variety and the weakly abelian variety belong to \(A_ 1\). The present author proves that (a) all quasi-representable varieties and all Scrimger varieties belong to \(A_ 1\); (b) the nilpotent Medvedev variety, the solvable Medvedev varieties and the Feil varieties \(U_ r\), \(W_ r\) (\(0<r\leq 1\)) belong to \(A_ 2\). Hence, both \(A_ 1\) and \(A_ 2\) have the cardinality of the continuum.