Lower and upper bounds for the affinity order of transformations of Boolean vector spaces

Let $\Phi_n$ be the set of all transformations of the Boolean vector space $V_n$. Affinity order of a mapping $F\in\Phi_n$ is the least order of the set $V_n$ partition with the property: for every its block there exists an affine mapping $A\colon V_n\to V_n$ being equivalent to $F$ on this block. Affinity order of $\Phi_n$ is the greatest order of $F\in\Phi_n$. Upper and lower bounds for the affinity order of $\Phi_n$ are given in the article. These results can be used for estimating complexity of some techniques in Boolean equations resolving.