Category:Axioms/Ordering Axioms
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This category contains axioms related to Ordering Axioms.
Let $S$ be a set.
Definition 1
$\RR$ is an ordering on $S$ if and only if $\RR$ satisfies the ordering axioms:
\((1)\) | $:$ | $\RR$ is reflexive | \(\ds \forall a \in S:\) | \(\ds a \mathrel \RR a \) | |||||
\((2)\) | $:$ | $\RR$ is transitive | \(\ds \forall a, b, c \in S:\) | \(\ds a \mathrel \RR b \land b \mathrel \RR c \implies a \mathrel \RR c \) | |||||
\((3)\) | $:$ | $\RR$ is antisymmetric | \(\ds \forall a, b \in S:\) | \(\ds a \mathrel \RR b \land b \mathrel \RR a \implies a = b \) |
Definition 2
$\RR$ is an ordering on $S$ if and only if $\RR$ satisfies the ordering axioms:
\((1)\) | $:$ | \(\ds \RR \circ \RR \) | |||||||
\((2)\) | $:$ | \(\ds \RR \cap \RR^{-1} = \Delta_S \) |
where:
- $\circ$ denotes relation composition
- $\RR^{-1}$ denotes the inverse of $\RR$
- $\Delta_S$ denotes the diagonal relation on $S$.
Pages in category "Axioms/Ordering Axioms"
The following 4 pages are in this category, out of 4 total.