Axiom:Ordering Axioms/Formulation 1
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Definition
Let $S$ be a set.
Let $\RR \subseteq S \times S$ be a relation on $S$.
$\RR$ is an ordering if and only if $\RR$ satisifes the 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 | \(\ds \forall a, b \in S:\) | \(\ds a \mathrel \RR b \land b \mathrel \RR a \implies a = b \) |
These criteria are called the ordering axioms on $S$.
Also see
- Axiom:Ordering Axioms/Formulation 2 for an alternative formulation of the ordering axioms on a set.
- Axiom:Ordering Axioms/Class Formulation for a formulation of the ordering axioms on a class.
- Definition:Ordering
- Equivalence of Definitions of Ordering
- Results about orderings can be found here.