The question whether the structure of partially ordered set would cause similar problems is answered by the following result.

A given partially ordered set may have several different completions.

The conditions themselves can be stated in an abstract form, so that they make sense for any partially ordered set.

There are 219 partially ordered sets on four labeled elements.

Every partially ordered set is an upper set of itself.

The partially ordered set is a well partial order.

(This is a an example of how any partially ordered set can be considered as a category).

Consider a partially ordered set (P, ) that is a complete lattice.

Assume that X is also a partially ordered set.

The identity function on any partially ordered set is always an order automorphism.

A closure operator on a partially ordered set is determined by its closed elements.

In mathematics, a filter is a special subset of a partially ordered set.

Order theory is the study of partially ordered sets, both finite and infinite.

A partially ordered set where the meet of any two elements always exists is a meet-semilattice.

It is readily verified that this yields a partially ordered set.

The least upper bound property can be generalized to the setting of partially ordered sets.

B is a partially ordered set and the elements of B are also its bounds.

Partially ordered sets and sets with other relations have applications in several areas.

This definition is extendable to subsets of any partially ordered set.

In order theory, this property can be generalized to a notion of completeness for any partially ordered set.

In mathematics, join and meet are dual binary operations on the elements of a partially ordered set.

Mathematically, in its most general form, a hierarchy is a partially ordered set or poset.

Such partially ordered sets are therefore completely described by their Hasse diagrams.

This partially ordered set is always a distributive lattice.

Let S be a partially ordered set.