Definition:Totally Ordered Class

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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:

Ordering: an ordering whose nature (total or partial) is not specified
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.