A total order is a strongly connected partial order, i.e. a binary relation $\leq$ on $X$ such that:
(Reflexivity) For all $x \in X$,
\[x \leq x.\](Transitivity) For all $x$, $y$, $z\in X$,
\[x \leq y \,\land\, y\leq z \enspace\rimp\enspace x \leq z.\](Antisymmetry) For all $x$, $y \in X$,
\[x \leq y \,\land\, y\leq x \enspace\rimp\enspace x = y.\](Strong connection) For all $x$, $y \in X$,
\[x \leq y \enspace\lor\enspace y \leq x.\]
A strict total order is a connected strict partial order, i.e. a binary relation $<$ on $X$ such that:
(Irreflexivity) For all $x \in X$,
\[x \nless x.\](Transitivity) For all $x$, $y$, $z \in X$,
\[x < y \,\land\, y < z \enspace\rimp\enspace x < z.\](Asymmetry) For all $x$, $y \in X$,
\[x < y \enspace\rimp\enspace y \nless x.\](Connection) For all $x$, $y \in X$,
\[x < y \enspace\lor\enspace y < x \enspace\lor\enspace x = y.\]
$\leq$ is a total order if and only if $<$ is a strict total order.