Polynomial-time right-ideal morphisms and congruences

We continue with the functional approach to the P-versus-NP problem, begun in [2, 3]. We previously constructed a monoid RM^P that is non-regular iff NP is not P. We now construct homomorphic images of RM^P with interesting properties. In particular, the homomorphic image M^P_poly of RM^P is finitely generated and J^0-simple, and is non-regular iff NP is not P. The group of units of M^P_poly is the famous Richard Thompson group V.