Total Order

Definition 1. A total order is a strongly connected partial order, i.e. a binary relation $\leq$ on $X$ 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. (Antisymmetry) For all $x$, $y \in X$,

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

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

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

Definition 2. A strict total order is a connected strict partial order, i.e. a binary relation $<$ on $X$ such that:

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

   \[x \nless x.\]

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

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

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

   \[x < y \enspace\rimp\enspace y \nless x.\]

4. (Connection) For all $x$, $y \in X$,

   \[x < y \enspace\lor\enspace y < x \enspace\lor\enspace x = y.\]

Proposition 1. $\leq$ is a total order if and only if $<$ is a strict total order.