# Category:Definitions/Orderings

This category contains definitions related to Orderings.
Related results can be found in Category:Orderings.

Let $S$ be a set.

### Definition 1

An ordering on $S$ is a relation $\RR$ on $S$ such that:

 $(1)$ $:$ $\RR$ is reflexive $\displaystyle \forall a \in S:$ $\displaystyle a \mathrel \RR a$ $(2)$ $:$ $\RR$ is transitive $\displaystyle \forall a, b, c \in S:$ $\displaystyle a \mathrel \RR b \land b \mathrel \RR c \implies a \mathrel \RR c$ $(3)$ $:$ $\RR$ is antisymmetric $\displaystyle \forall a \in S:$ $\displaystyle a \mathrel \RR b \land b \mathrel \RR a \implies a = b$

### Definition 2

An ordering on $S$ is a relation $\RR$ on $S$ such that:

$(1): \quad \RR \circ \RR = \RR$
$(2): \quad \RR \cap \RR^{-1} = \Delta_S$

where:

$\circ$ denotes relation composition
$\RR^{-1}$ denotes the inverse of $\RR$
$\Delta_S$ denotes the diagonal relation on $S$.

## Pages in category "Definitions/Orderings"

The following 7 pages are in this category, out of 7 total.