  We study the variety ZG of monoids where the elements that belong to a group
are central, i.e., commute with all other elements. We show that ZG is local,
that is, the semidirect product ZG*D of ZG by definite semigroups is equal to
LZG, the variety of semigroups where all local monoids are in ZG. Our main
result is thus: ZG*D = LZG. We prove this result using Straubing's delay
theorem, by considering paths in the category of idempotents. In the process,
we obtain the characterization ZG = MNil \vee Com, and also characterize the ZG
languages, i.e., the languages whose syntactic monoid is in ZG: they are
precisely the languages that are finite unions of disjoint shuffles of
singleton languages and regular commutative languages.
