Axiom:Strict Ordering Axioms
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Definition
Let $\RR$ be a relation on a set $S$.
Formulation 1
$\RR$ satisfies the strict ordering axioms (on $S$) if and only if:
\((1)\) | $:$ | Asymmetry | \(\ds \forall a, b \in S:\) | \(\ds a \mathrel \RR b \) | \(\ds \implies \) | \(\ds \neg \paren {b \mathrel \RR a} \) | |||
\((2)\) | $:$ | Transitivity | \(\ds \forall a, b, c \in S:\) | \(\ds \paren {a \mathrel \RR b} \land \paren {b \mathrel \RR c} \) | \(\ds \implies \) | \(\ds a \mathrel \RR c \) |
Formulation 2
$\RR$ satisfies the strict ordering axioms (on $S$) if and only if:
\((1)\) | $:$ | Antireflexivity | \(\ds \forall a \in S:\) | \(\ds \neg \paren {a \mathrel \RR a} \) | |||||
\((2)\) | $:$ | Transitivity | \(\ds \forall a, b, c \in S:\) | \(\ds \paren {a \mathrel \RR b} \land \paren {b \mathrel \RR c} \implies a \mathrel \RR c \) |
Also see
- Axiom:Strict Ordering Axioms/Formulation 1 for an alternative formulation of the strict ordering axioms
- Definition:Strict Ordering
- Equivalence of Definitions of Strict Ordering