# Category:Strict Orderings

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

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

## Subcategories

This category has the following 2 subcategories, out of 2 total.

## Pages in category "Strict Orderings"

The following 11 pages are in this category, out of 11 total.