Definition:Total Ordering/Definition 2

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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:

$(1): \quad \mathcal R \circ \mathcal R = \mathcal R$
$(2): \quad \mathcal R \cap \mathcal R^{-1} = \Delta_S$
$(3): \quad \mathcal R \cup \mathcal R^{-1} = S \times S$

Also see

  • Results about total orderings can be found here.