# Definition:Total Ordering/Definition 1

## 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:

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

## Also see

• Results about total orderings can be found here.