# Definition:Preordering

## Definition

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

### Definition 1

$\RR$ is a **preordering** on $S$ if and only if:

\((1)\) | $:$ | $\RR$ is reflexive | \(\ds \forall a \in S:\) | \(\ds a \mathrel \RR a \) | ||||

\((2)\) | $:$ | $\RR$ is transitive | \(\ds \forall a, b, c \in S:\) | \(\ds a \mathrel \RR b \land b \mathrel \RR c \implies a \mathrel \RR c \) |

### Definition 2

$\mathcal R$ is a **preordering** on $S$ if and only if:

- $(1): \quad \mathcal R \circ \mathcal R = \mathcal R$
- $(2): \quad \Delta_S \subseteq \mathcal R$

where:

- $\circ$ denotes relation composition
- $\Delta_S$ denotes the diagonal relation on $S$.

## Preordered Set

Let $S$ be a set.

Let $\precsim$ be a preordering on $S$.

Then the relational structure $\struct {S, \precsim}$ is called a **preordered set**.

## Notation

Symbols used to denote a general preordering relation are usually variants on $\lesssim$, $\precsim$ or $\precapprox$.

A symbol for a preordering can be reversed, and the sense is likewise inverted:

- $a \precsim b \iff b \succsim a$

The notation $a \sim b$ is defined as:

- $a \sim b$ if and only if $a \precsim b$ and $b \precsim a$

The notation $a \prec b$ is defined as:

- $a \prec b$ if and only if $a \precsim b$ and $a \not \sim b$

## Partial vs. Total Preorderings

Note that this definition of **preordering** does not demand that *every* pair of elements of $S$ is related by $\precsim$.

The way we have defined a **preordering**, they may be, or they may not be, depending on the context.

If it *is* the case that $\precsim$ is a connected relation, that is, that every pair of elements is related by $\precsim$, then $\precsim$ is called a **total preordering**.

If it is specifically *not* the case that $\precsim$ is connected, then $\precsim$ is called a **partial preordering**.

## Also known as

A **preordering** is also known as a **preorder**.

Either name can be seen with a hyphen: **pre-ordering** and **pre-order**.

Some sources use the term **quasiorder** or **quasi-order**.

1964: Steven A. Gaal: *Point Set Topology* uses the term **reflexive partial ordering**, but as this can so easily be confused with the concept of a partial ordering this term is not recommended.

## Examples

### Finite Set Difference on Natural Numbers

Consider the relation $\RR$ on the powerset of the natural numbers:

- $\forall a, b \in \powerset \N: a \mathrel \RR b \iff a \setminus b \text { is finite}$

where $\setminus$ denotes set difference.

Then $\RR$ is a preordering on $\powerset \N$, but not an ordering on $\powerset \N$.

## Also see

- Definition:Preorder Category, interpreting
**preorders**as categories.

- Results about
**preorderings**can be found here.