# Definition:Preordering/Preordered Set

< Definition:Preordering(Redirected from Definition:Preordered Set)

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## Definition

Let $S$ be a set.

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

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

## Also known as

This is sometimes shortened to **proset**.

Some sources use the term **quasi-ordering** for what we call a **preordering**, and those sources call this a **quasi-ordered set** and may shorten it to **qoset**.

Some sources refer to $\struct {S, \preceq}$ as a **(partial) preorder**, calling $\preceq$ a **(partial) preorder relation**.

## Also see

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

## Sources

- 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 $6$