Definition:Partially Ordered Set

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Definition

A partially ordered set is a relational structure $\left({S, \preceq}\right)$ such that $\preceq$ is a partial ordering.


The partially ordered set $\left({S, \preceq}\right)$ is said to be partially ordered by $\preceq$.


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 known as

The word poset is frequently to be found in the literature, but this is frequently understood to mean a general ordered set which may be either partial or total.

Some sources use the term partly ordered set.


Also see


Internationalization

Partially ordered set is translated:

In German: teilweise geordnete Menge


Sources