Matroid isomorphism ($\varphi :E\to E^{\prime }$)

A matroid isomorphism between $(E,\mathcal{I})$ and $(E^{\prime },{\mathcal{I}}^{\prime })$ is a bijection $\varphi$ such that $I\in \mathcal{I}$ if and only if $\varphi (I)\in {\mathcal{I}}^{\prime }$. It relabels the ground set without changing independence. The number of labelled matroids therefore differs from the number of types up to matroid isomorphism.