# Definition:Strictly Ordered Set

## Definition

A **strictly ordered set** is a relational structure $\struct {S, \prec}$ such that the relation $\prec$ is an strict ordering.

## Partial vs. Total Strict Ordering

It is not demanded of a strict ordering $\prec$, defined in its most general form on a set $S$, that *every* pair of elements of $S$ is related by $\prec$.

They may be, or they may not be, depending on the specific nature of both $S$ and $\prec$.

If it *is* the case that $\prec$ is a connected relation, that is, that every pair of distinct elements is related by $\prec$, then $\prec$ is called a strict total ordering.

If it is *not* the case that $\prec$ is connected, then $\prec$ is called a strict partial ordering.

Beware that some sources use the word **partial** for a strict ordering which **may or may not** be connected, while others insist on reserving the word **partial** for one which is specifically **not** connected.

It is wise to be certain of what is meant.

As a consequence, on $\mathsf{Pr} \infty \mathsf{fWiki}$ we resolve any ambiguity by reserving the terms for the objects in question as follows:

**Strict ordering**: a strict ordering whose nature (total or partial) is not specified

**Strict partial ordering**: a strict ordering which is specifically**not**total

**Strict total ordering**: a strict ordering which is specifically**not**partial.

## Also known as

Some sources refer to $\struct {S, \prec}$ as a **strict partial order**, calling $\preceq$ a **strict partial order relation**.

## Also see

- Definition:Ordered Set
- Definition:Partially Ordered Set
- Definition:Totally Ordered Set
- Definition:Well-Ordered Set

- Definition:Strictly Partially Ordered Set
- Definition:Strictly Totally Ordered Set
- Definition:Strictly Well-Ordered Set

- Results about
**strict orderings**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