A variety of finite monoids is a
class of finite monoids closed under taking submonoids, quotients and finite direct
products. A language L is a subset of a finitely generated free monoid. The variety
theorem of Eilenberg sets up a one to one correspondence between varieties of finite
monoids and classes of languages called, appropriately, varieties of languages. Recent
work in variety theory has been concerned with relating operations on varieties
of languages to operations on the corresponding variety of monoids and
vice versa. For example, passing from a variety V of monoids to the variety
PV generated by the power monoids of members of V corresponds to the
operations of inverse substitution and literal morphism on varieties of languages.
Recall that the power monoid of a monoid M is the power set PM with
the usual multiplication of subsets. In this paper we consider iterating the
operation which assigns PV to V. We show in particular that P3V = P4V for
any variety V and that the exponent 3 is the best possible. In fact if V
contains a non-commutative monoid, then P3V is the variety of all finite
monoids.
The proof of this theorem depends upon a classification of the minimal
noncommutative varieties. A variety is minimal noncommutative if all its
proper subvarieties contain only commutative monoids. We show that such a
variety is either generated by a noncommutative metabelian group or by the
syntactic monoid of one of the languages A∗a, aA∗ or {ab} over the alphabet
A = {a,b}.
