Let X be all the objects, and Y all the attributes.
Then using standard notation, the powerset \\(2^X\\) consists of all subsets of \\(X\\), i.e., all sets of objects. And \\(2^Y\\) consists of all subsets of \\(Y\\), all sets of attributes.
Define functions \\(f: 2^X \rightarrow 2^Y\\) and \\(g: 2^Y \rightarrow 2^X\\), called derivation operators, as follows.
Let \\(x \subseteq X\\) be a subset of the objects.
Then define \\(x' = f(x) \subseteq Y\\) as the set of attributes which apply to every object in \\(X\\).
That defines the derivation operator \\(f: 2^X \rightarrow 2^Y\\).