# Definition:Strictly Totally Ordered Set

## Definition

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

## Also known as

It is common to see the term **linearly ordered set** being used, but some sources use this to refer to a totally ordered set.

As a consequence, on $\mathsf{Pr} \infty \mathsf{fWiki}$ it has been decided that the more cumbersome, but completely descriptive, term **strictly totally ordered set** be used exclusively.

## 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.