I don't understand, are "minimal/minimum/maximal/maxium" elements properties of a partial order or properties of base sets of partial orders? Given any partial order $(X,\leq)$ from what I can gather, the following equivalences seem to hold for any $S\subseteq X$ and any $t\in S$ :
$$t\text{ is a }\leq\text{minimal element of }S\iff t\text{ is a minimal element of }(S,\leq )$$ $$t\text{ is a }\leq\text{minimum element of }S\iff t\text{ is a minimum element of }(S,\leq )$$ $$t\text{ is a }\leq\text{maximal element of }S\iff t\text{ is a maximal element of }(S,\leq )$$ $$t\text{ is a }\leq\text{maximum element of }S\iff t\text{ is a maximum element of }(S,\leq )$$
If this is the case, then why define two different notions of say "maximal elements on subsets" and "maximal elements on orders themselves"? It just seems to complicate the notations even more. Why not just pick one notation and be done with it?