Completeness (order theory)

Relationships between completeness properties

It was already observed that binary meets/joins yield all non-empty finite meets/joins. Likewise, many other (combinations) of the above conditions are equivalent.

• The best-known example is the existence of all suprema, which is in fact equivalent to the existence of all infima. Indeed, for any subset X of a poset, one can consider its set of lower bounds B. The supremum of B is then equal to the infimum of X: since each element of X is an upper bound of B, sup B is smaller than all elements of X, i.e. sup B is in B. It is the greatest element of B and hence the infimum of X. In a dual way, the existence of all infima implies the existence of all suprema.
• Bounded completeness can also be characterized differently. By an argument similar to the above, one finds that the supremum of a set with upper bounds is the infimum of the set of upper bounds. Consequently, bounded completeness is equivalent to the existence of all non-empty infima.
• A poset is a complete lattice if and only if it is a cpo and a join-semilattice. Indeed, for any subset X, the set of all finite suprema (joins) of X is directed and the supremum of this set (which exists by directed completeness) is equal to the supremum of X. Thus every set has a supremum and by the above observation we have a complete lattice. The other direction of the proof is trivial.
• Assuming the axiom of choice, a poset is chain complete if and only if it is a dcpo.

Completeness in terms of universal algebra

As explained above, the presence of certain completeness conditions allows to regard the formation of certain suprema and infima as total operations of a partially ordered set. It turns out that in many cases it is possible to characterize completeness solely by considering appropriate algebraic structures in the sense of universal algebra, which are equipped with operations like ${\displaystyle \vee }$ or ${\displaystyle \wedge }$. By imposing additional conditions (in form of suitable identities) on these operations, one can then indeed derive the underlying partial order exclusively from such algebraic structures. Details on this characterization can be found in the articles on the "lattice-like" structures for which this is typically considered: see semilattice, lattice, Heyting algebra, and Boolean algebra. Note that the latter two structures extend the application of these principles beyond mere completeness requirements by introducing an additional operation of negation.

Completeness in terms of adjunctions

Another interesting way to characterize completeness properties is provided through the concept of (monotone) Galois connections, i.e. adjunctions between partial orders. In fact this approach offers additional insights both into the nature of many completeness properties and into the importance of Galois connections for order theory. The general observation on which this reformulation of completeness is based is that the construction of certain suprema or infima provides left or right adjoint parts of suitable Galois connections.

Consider a partially ordered set (X, ≤). As a first simple example, let 1 = {*} be a specified one-element set with the only possible partial ordering. There is an obvious mapping j: X → 1 with j(x) = * for all x in X. X has a least element if and only if the function j has a lower adjoint j^*: 1 → X. Indeed the definition for Galois connections yields that in this case j^*(*) ≤ x if and only if * ≤ j(x), where the right hand side obviously holds for any x. Dually, the existence of an upper adjoint for j is equivalent to X having a greatest element.

Another simple mapping is the function q: X → X × X given by q(x) = (x, x). Naturally, the intended ordering relation for X × X is just the usual product order. q has a lower adjoint q^* if and only if all binary joins in X exist. Conversely, the join operation ${\displaystyle \vee }$: X × X → X can always provide the (necessarily unique) lower adjoint for q. Dually, q allows for an upper adjoint if and only if X has all binary meets. Thus the meet operation ${\displaystyle \wedge }$, if it exists, always is an upper adjoint. If both ${\displaystyle \vee }$ and ${\displaystyle \wedge }$ exist and, in addition, ${\displaystyle \wedge }$ is also a lower adjoint, then the poset X is a Heyting algebra—another important special class of partial orders.

Further completeness statements can be obtained by exploiting suitable completion procedures. For example, it is well known that the collection of all lower sets of a poset X, ordered by subset inclusion, yields a complete lattice D(X) (the downset-lattice). Furthermore, there is an obvious embedding e: X → D(X) that maps each element x of X to its principal ideal {y in X | y ≤ x}. A little reflection now shows that e has a lower adjoint if and only if X is a complete lattice. In fact, this lower adjoint will map any lower set of X to its supremum in X. Composing this lower adjoint with the function that maps any subset of X to its lower closure (again an adjunction for the inclusion of lower sets in the powerset), one obtains the usual supremum map from the powerset 2^X to X. As before, another important situation occurs whenever this supremum map is also an upper adjoint: in this case the complete lattice X is constructively completely distributive. See also the articles on complete distributivity and distributivity (order theory).

The considerations in this section suggest a reformulation of (parts of) order theory in terms of category theory, where properties are usually expressed by referring to the relationships (morphisms, more specifically: adjunctions) between objects, instead of considering their internal structure. For more detailed considerations of this relationship see the article on the categorical formulation of order theory.