Can a partially ordered set be complete?

Question

I learned the definition a set being complete as below.

An ordered set (X, =<) is said to be complete if for every non-empty subset of X which is bounded above (or below), there exists a supremum (or infimum).

But I'm not sure if an "ordered set" means only a totally ordered set or both a totally ordered set a partially ordered set. If it means the latter, I don't understand how a partially ordered set can be complete. Because I understood a partially ordered set can have elements which are not comparable and so there may not exist a supremum or infimum.

Does an "ordered set" mean only a totally ordered set or both a totally ordered set and a partially ordered set? And could you give me an example if it is the latter?

Answer 1

A typical example of a partially ordered, non-linearly ordered set is $\mathcal{P}(X)$, all subsets of $X$ (a set with at least $2$ elements) ordered by inclusion.

If $A$ and $B$ are non-comparable sets, they still have a lower bound $\emptyset$ and upper bound $X$. Consider what $\sup \{A,B\}$ is, we cannot have maximum as the sets are incomparable, but $A \cup B$ is a common upper bound and some thought reveals that if $C$ is an upper bound, so $A \subseteq C$ and $B \subseteq C$, then $A \cup B \subseteq C$ as well. So $A \cup B$ is the minimal element among all upper bounds of $\{A,B\}$ and is thus its supremum by definition.

In fact all subsets $\mathcal{A} \subseteq \mathcal{P}(X)$ have $\sup \mathcal{A} = \bigcup \mathcal{A}$. Likewise with infima and intersections.

Answer 2

No, a (partial) order S cannot always be complete.
The rationals for example.
It can however be embedded into a complete order
of the lower sets of S with the subset order.
A lower set is a subset A with
for all x in S, a in A, (x <= a implies x in A).
The construction is simular to Dedekind's completion of the rationals except instead of using pairs of subsets, only the lower subset is used.

Orders were originally linear. Then partial orders were considered.
Thus one needed to explicate partial. In time the streamlined term ordered became used for partial order (includes linear orders) whereupon one has to explicate linear or total order. As often, terms vary from writer to writer and order is one of those context sensitive terms.