Category:Definitions/Total Orderings

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This category contains definitions related to Total Orderings.
Related results can be found in Category: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:

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