I'm working on a computer program for making decisions that represents the user's preferences as a preorder, as is common in theories of preference. I'd like to formalize the notions of "the best item", or "the n best items", ideally in a way that's consistent with past usage in order theory, but I'm having trouble finding discussion of extrema or extreme elements in preorders.
How I'd define an extremum in a preordered set $S$ is, an element $x ∈ S$ is a maximum of $S$ if $a ≤ x$ for all $a ∈ S$. Notice that this requires $x$ to be comparable to all other elements, although the preorder might not be total, and that $S$ can have more than one maximum (because in a preorder, there can be elements $x$, $y$ with $x ≤ y$ and $y ≤ x$ and yet $x ≠ y$).
I'd define the top-$n$ subset of $S$ to be the set of all elements $x ∈ S$ such that
- $x$ is comparable to all elements of $S$, and
- there are at most $n - 1$ distinct elements $a$ such that $a > x$.
For example, the top-3 subset of the negative integers with their usual order is $\{-3, -2, -1\}$. Notice that the set of all of $S$'s maxima coincides with its top-1 subset. Also, the top-$n$ subset need not have exactly $n$ elements; it can have any cardinality from 0 to $|S|$.
Can you point me to any previously established terms for these notions, or to notions similar to these?