Definition:Strictly Ordered Set

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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

  • Results about strict orderings can be found here.