# Category:Axioms/Strict Ordering Axioms

This category contains axioms related to Strict Ordering Axioms.

### Definition 1

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

Then $\RR$ is a strict ordering (on $S$) if and only if $\RR$ satisfies the strict ordering axioms:

 $(1)$ $:$ 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 $\RR$ satisfies the strict ordering axioms:

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

## Pages in category "Axioms/Strict Ordering Axioms"

The following 3 pages are in this category, out of 3 total.