Vol. 111, No. 1, 1984

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Minimal noncommutative varieties and power varieties

Stuart Wayne Margolis and Jean-Eric Pin

Vol. 111 (1984), No. 1, 125–135

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 Aa, aA or {ab} over the alphabet A = {a,b}.

Mathematical Subject Classification 2000
Primary: 20M07
Secondary: 08C99, 20M35, 68Q70
Received: 24 July 1981
Published: 1 March 1984
Stuart Wayne Margolis
Jean-Eric Pin