# Category:Definitions/Strict Orderings

This category contains definitions related to Strict Orderings.
Related results can be found in Category: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 $\ds \forall a, b, c \in S:$ $\ds \paren {a \mathrel \RR b} \land \paren {b \mathrel \RR c} \implies a \mathrel \RR c$

## Subcategories

This category has only the following subcategory.

## Pages in category "Definitions/Strict Orderings"

The following 14 pages are in this category, out of 14 total.