Definition:Total Ordering/Definition 2

Definition

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

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

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

Also see

• Results about total orderings can be found here.