Definition:Total Ordering
(Redirected from Definition:Linear Ordering)
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Definition
Let $\mathcal R \subseteq S \times S$ be a relation on a set $S$.
Definition 1
$\RR$ is a total ordering on $S$ if and only if:
That is, $\RR$ is an ordering with no non-comparable pairs:
- $\forall x, y \in S: x \mathop \RR y \lor y \mathop \RR x$
Definition 2
$\RR$ 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 known as
Some sources call this a linear ordering, or a simple ordering.
If it is necessary to emphasise that a total ordering $\preceq$ is not strict, then the term weak total ordering may be used.
Also see
- Results about total orderings can be found here.