# Definition:Strict Total Ordering

## Definition

Let $\struct {S, \prec}$ be a relational structure.

Let $\prec$ be a strict ordering.

Then $\prec$ is a **strict total ordering** on $S$ if and only if $\struct {S, \prec}$ has no non-comparable pairs:

- $\forall x, y \in S: x \ne y \implies x \prec y \lor y \prec x$

That is, if and only if $\prec$ is connected.

## Also known as

Other terms in use are **simple order** and **order relation**.

Some sources, for example 1964: W.E. Deskins: *Abstract Algebra* and 2000: James R. Munkres: *Topology* (2nd ed.), call this a **linear order**.

As this term is also used by other sources to mean **total ordering**, it is preferred that on $\mathsf{Pr} \infty \mathsf{fWiki}$ the terms **partial**, **total** and **well**, and **weak** and **strict**, are the only terms to be used to distinguish between different types of ordering.

## Also see

## 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.

## Sources

- 1964: W.E. Deskins:
*Abstract Algebra*... (previous) ... (next): $\S 1.2$: Definition $1.7 \ \text {(a)}$ - 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets - 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 6.18$ - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations