A total preorder is a strongly connected preorder, i.e. a binary relation $\leq$ on $X$ such that:
Reflexivity. For all $x\in X$,
\[x \leq x.\]Transitivity. For all $x$, $y$, $z\in X$,
\[x\leq y \,\land\, y\leq z \enspace\rimp\enspace x\leq z.\]Strong connection. For all $x$, $y\in X$,
\[x\leq y \enspace\lor\enspace y\leq x.\]