Free Composition of Paragroups

As a first step we define an extremality condition for paragroups, then we prove that any given extremal paragroup is equivalent to a Popa system constructed with the string algebras associated with the starting paragroup. As a corollary, we get that any extremal paragroup can be thought of as the paragroup coming from a (generally non-hyperfinite) extremal inclusion of II1 factors. Then we define a free composition for paragroups, using as a model the definition by V. Jones and D. Bisch of free composition of subfactors; and we prove that for any two given paragroups P1 and P2 there always exists a third paragroup P which realizes the free composition of P1 and P2 .