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\$$.