Category:Strict Orderings

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This category contains results about Strict Orderings.
Definitions specific to this category can be found in Definitions/Strict Orderings.


Definition 1

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

Then $\RR$ is a strict ordering (on $S$) if and only if the following two conditions hold:

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


Definition 2

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

Then $\RR$ is a strict ordering (on $S$) if and only if the following two conditions hold:

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