# Definition:Preordering/Definition 1

## Definition

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

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

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

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

## 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.

## Also see

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

## Sources

- 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 7$: Exercise $7$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: Further exercises: $5$ - 1996: Winfried Just and Martin Weese:
*Discovering Modern Set Theory. I: The Basics*... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations: Definition $6$ - 2010: Steve Awodey:
*Category Theory*(2nd ed.) ... (previous) ... (next): $\S 1.4.7$