What is the advantage on defining filters on partially ordered sets versus defining them on preordered sets? Almost everywhere where I have seen filters they have been used in the setting of partially ordered sets. Are there any properties of filters defined on partially ordered sets that do not hold if we define them on preordered sets instead?
Here are the definitions that I am referring to:
preorder: relation on a set $X$ that is reflexive and transitive.
partial order: relation on a set $X$ that is reflexive and transitive and antisymmetric (i.e., ($a\leq b$ and $b \leq a$) $\implies a=b$).
filter: a nonempty collection $F$ of elements of $X$ such that:
- $x \in F$ and $y \geq x \implies y \in F$
- $x,y \in F \implies $ there exists $z \in F$ such that $z \leq x$ and $z \leq y.$
Any thoughts will be greatly appreciated!