Category:Total Orderings

This category contains results about Total Orderings.
Definitions specific to this category can be found in Definitions/Total Orderings.

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:

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