Minimal Noncommutative Varieties and Power Varieties

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 PV = P 4 V for any variety V and that the exponent 3 is the best possible. In fact if V contains a non-commutative monoid, then PV 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).