Axiom:Strict Ordering Axioms

Definition

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

$\RR$ 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 $

$\RR$ 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 $

\((1)\)	$:$		Asymmetry	\(\ds \forall a, b \in S:\)		\(\ds a \mathrel \RR b \)	\(\ds \implies \)	\(\ds \neg \paren {b \mathrel \RR a} \)
\((2)\)	$:$		Transitivity	\(\ds \forall a, b, c \in S:\)		\(\ds \paren {a \mathrel \RR b} \land \paren {b \mathrel \RR c} \)	\(\ds \implies \)	\(\ds a \mathrel \RR c \)

\((1)\)	$:$		Antireflexivity	\(\ds \forall a \in S:\)	\(\ds \neg \paren {a \mathrel \RR a} \)
\((2)\)	$:$		Transitivity	\(\ds \forall a, b, c \in S:\)	\(\ds \paren {a \mathrel \RR b} \land \paren {b \mathrel \RR c} \implies a \mathrel \RR c \)