What are some simple examples of preorders — that is, binary relations that are reflexive and transitive — that are not partial orders (and hence not total orders, either)?
I'm looking for a couple of examples that do not involve graph theory or other less basic ideas in math. And preferably examples simpler than the relation $\preccurlyeq$ on the power set of a given set that declares $A \preccurlyeq B$ iff there exists an injection from $A$ to $B$.
Best Answer
In this answer I do not provide simple examples of preorders, but enable you to find them on base of simple examples of partial orders.
You can just start with a partial order $\langle B,\leq\rangle$ and a surjective function $\nu:A\to B$.
Then $\preceq$ defined by: $$x\preceq y\iff \nu(x)\leq \nu(y)$$ is a preorder on $A$.
Actually every preorder can be described this way. If you start with some preorder $\langle A,\preceq\rangle$ then you can take $B:=A/\sim$ where $\sim$ is the equivalence relation on $A$ characterized by: $$x\sim y\iff x\preceq y\ \wedge y\preceq x$$ For $\nu:A\to B$ you take the natural function prescribed by $a\mapsto[a]$.
This preorder is not a partial order if and only if $\nu$ is not injective.