These derivation operators have some nice order-theoretic structure.

First, note the powersets \$$2^X\$$ and \$$2^Y\$$ are of course partially ordered by the inclusion relation.

Next, it is not hard to see that \$$f: 2^X \rightarrow 2^Y\$$ and \$$g: 2^Y \rightarrow 2^X\$$ are _order reversing_ functions.

First consider \$$f: 2^X \rightarrow 2^Y\$$.

What happens if you apply \$$f\$$ to a subset \$$x_1 \subseteq x \subseteq X\$$?

Then, since the \$$x_1\$$ is less than or equal to \$$x\$$, the set of attributes which apply to every object in \$$x_1\$$ must be greater than or equal to the set of attributes which apply to every object in \$$x\$$. We weakened the condition by choosing a subset \$$x_1\$$ of \$$x\$$, which can only increase the set of common attributes.

So \$$x_1 \subseteq x \implies f(x_1) \supseteq f(x)\$$, which says that \$$f\$$ is order-reversing.

Similarly, decreasing a set of attributes \$$y\$$ can only increase the set of objects \$$g(y)\$$ which have all the attributes in \$$y\$$.

So \$$g\$$ is order-reversing too.