# Definition:Totally Ordered Class

## Definition

Let $V$ be a basic universe.

Let $\RR \subseteq V \times V$ be a relation.

Let $A$ be a subclass of the field of $\RR$.

Let the restriction of $\RR$ to $A$ be a total ordering on $A$.

Then $A$ is described as being **totally ordered under $\RR$**.

## Also known as

A **totally ordered class** is also called a **simply ordered class** or **linearly ordered class**.

## Partial vs. Total Ordering

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

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

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

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

Beware that some sources use the word **partial** for an 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:

**Partial ordering**: an ordering which is specifically**not**total

**Total ordering**: an ordering which is specifically**not**partial.

## Also see

- Results about
**ordered classes**can be found**here**.

## Sources

- 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering