Whereas pseudovarieties of commutative semigroups are known to be finitely based, the globals of monoidal pseudovarieties of commutative semigroups are shown to be finitely based (or of finite vertex rank) if and only if the index is 0, 1 or $\omega$. Nevertheless, on these pseudovarieties, the operation of taking the global preserves decidability. Furthermore, the gaps between many of these globals are shown to be big in the sense that they contain chains which are order isomorphic to the reals.

AMS 2000 Mathematics subject classification: Primary 20M07; 20M05