In an earlier paper, the second author generalized Eilenberg's
variety theory by establishing a basic correspondence between
certain classes of monoid morphisms and families of regular
languages. We extend this theory in several directions. First, we
prove a version of Reiterman's theorem concerning the definition of
varieties by identities, and illustrate this result by describing
the identities associated with languages of the form (a1a2...ak)+, where a1,...,ak are distinct letters. Next, we
generalize the notions of Mal'cev product, positive varieties, and
polynomial closure. Our results not only extend those already known,
but permit a unified approach of different cases that previously
required separate treatment.