# Category:Orderings

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This category contains results about Orderings.

Definitions specific to this category can be found in Definitions/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 "Orderings"

The following 13 pages are in this category, out of 13 total.