A rigidity theorem for Moor-bialgebras1

We introduce the operad Moor, dual of the operad NAP and the notion of Moor-bialgebras. We warn the reader that the compatibility relation linking the Moor-operation with the Moor-cooperation is not distributive in the sense of Loday. Nevertheless, a rigidity theorem (à la Hopf-Borel) for the category of connected Moorbialgebras is given. We show also that free permutative algebras can be equipped with a Moor-cooperation whose compatibility with the permutative product looks like the infinitesimal relation. Notation: In the sequel K is a characteristic zero field and Σn is the group of permutations over n elements. If A is an operad, then the K-vector space of n-ary operations is denoted as usual by A(n). We adopt Sweedler notation for the binary cooperation ∆ on a K-vector space V and set ∆(x) = x(1) ⊗ x(2).