Shuffle on positive varieties of languages

We show there is a unique maximal positive variety of languages which does not contain the language (ab)∗. This variety is the unique maximal positive variety satisfying the two following conditions: it is strictly included in the class of rational languages and is closed under the shuffle operation. It is also the largest proper positive variety closed under length preserving morphisms. The ordered monoids of the corresponding variety of ordered monoids are characterized as follows: for every pair (a, b) of mutually inverse elements, and for every element z of the minimal ideal of the submonoid generated by a and b, (abzab)ω ≤ ab. In particular this variety is decidable.