Total Preorder

Definition 1. A total preorder is a strongly connected preorder, i.e. a binary relation $\leq$ on a set $X$ such that:

1. (Reflexivity) For all $x \in X$,

   \[x \leq x.\]

2. (Transitivity) For all $x$, $y$, $z \in X$,

   \[x \leq y \,\land\, y \leq z \enspace\rimp\enspace x \leq z.\]

3. (Strong connection) For all $x$, $y \in X$,

   \[x \leq y \enspace\lor\enspace y \leq x.\]