Do maximal and maxiumum elements coincide for totally preordered sets?

Question

Let $A$ be some finite and countable set, and let $R$ be a total preorder (i.e., a preference relation) on $A$: that is, a binary relation on $A$ satisfying

1. Transitivity: for all $a,b,c\in A$, $aRb$ and $bRc$ imply $aRc$.
2. Reflexivity: for all $a\in A$, $aRa$.
3. Completeness: for all $a,b\in A$, $aRb$ or $bRa$.

Then, $(A,R)$ is a totally preordered set.

Let $P$ be a preorder on $A$: that is, a binary relation on $A$ satisfying

1. Transitivity: for all $a,b,c\in A$, $aPb$ and $bPc$ imply $aPc$.
2. Reflexivity: for all $a\in A$, $aPa$.

Then, $(A,P)$ is a preordered set.

Consider $(A,P)$ and some $B\subseteq A$. Then, $m\in B$ is a maximal element of $B$ with respect to $P$ if: \begin{equation} (\forall b\in B)[(mPb)\Rightarrow(bPm)] \end{equation}

Further, $\tilde{m}\in B$ is a maximum element of $B$ with respect to $P$ if \begin{equation} (\forall b\in B)(bP\tilde{m}) \end{equation}

For partially ordered sets, maximal and maximum elements do not need to coincide. However, do maximal and maxiumum elements coincide for totally preordered sets?