Maximal and minimal element in preordered set

Question

Generally the notion of maximal and minimal element is defined in a partially ordered set (binary relation is reflexive, antisymmetric and transitive).

A preorder is a binary relation that is reflexive and transitive.

Can we define the notion of maximal and minimal element in a preordered set?

Answer 1

a is a minimal element when for all x <= a, a <= x.

The minimal elements of the preorder
x < y, y < x, x < a, y < a are x and y.

Answer 2

Definition. Let $X$ be a set. Let $\preceq$ be a preorder in $X$. Fix $S\subseteq X$.

A $\preceq$-minimal-element of $S$ is an element $t$ of $S$ such that $(\forall s\in S)[s\preceq t\:\Rightarrow\:s=t]$.

A $\preceq$-maximal-element of $S$ is an element $t$ of $S$ such that $(\forall s\in S)[t\preceq s\:\Rightarrow\:t=s]$.