  In this work we present a definition for coherence and compatibility of
multilinear mappings and homogenous polynomial classes. These definitions are
more restricted than the ones proposed before. We began analyzing this new
definition in a technical sense, searching for what it has in common with other
approaches. Then, we moved on to a more practical analysis. Through numerous
examples of different classes of multilinear mappings and homogenous
polynomials we checked the limits on these proposed definitions and propose
there is a vast field in which they apply.
