Definition:Partially Ordered Set

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

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

In general, a poset can also be a relational structure $\left({S, \preceq}\right)$ such that $\preceq$ is an ordering which may or may not be partial.

Also known as
Some sources call this an ordered set, and prefer not to use the term partial.

For example, gives:
 * What we have defined as an order on a set many authors call a partial order and what we have called an ordered set is often called a partially ordered set or poset. As there is nothing which can even remotely be called 'partial' in the definition of an order relation, we shall not use this terminology.

However, according to ProofWiki house style, the advantage of being able to specify a difference between the various types of ordering outweighs any possible perceived inaccuracy in terminology.

Thus, on ProofWiki, poset and partially ordered set are the names to be used.

Also see

 * Totally ordered set (toset)
 * Well-ordered set (woset)