Definition:Total Ordering

Definition

Let $\mathcal R \subseteq S \times S$ be a relation on a set $S$.

$\mathcal R$ is a total ordering on $S$ if and only if:

$(1): \quad \mathcal R$ is an ordering on $S$

$(2): \quad \mathcal R$ is connected

That is, $\mathcal R$ is an ordering with no non-comparable pairs:

$\forall x, y \in S: x \mathop {\mathcal R} y \lor y \mathop {\mathcal R} x$

$\mathcal R$ is a total ordering on $S$ if and only if:

$(1): \quad \mathcal R \circ \mathcal R = \mathcal R$

$(2): \quad \mathcal R \cap \mathcal R^{-1} = \Delta_S$

$(3): \quad \mathcal R \cup \mathcal R^{-1} = S \times S$

Some sources call this a linear ordering, or a simple ordering.

If it is necessary to emphasise that a total ordering $\preceq$ is not strict, then the term weak total ordering may be used.