The motivation for considering completeness properties derives from the great importance of suprema (least upper bounds, joins, "
Finding a supremum means to single out one distinguished least element from the set of upper bounds.
On the other hand, the knowledge that certain types of subsets are guaranteed to have suprema or infima enables us to consider the evaluation of these elements as total operations on a partially ordered set.
For this reason, posets with certain completeness properties can often be described as algebraic structures of a certain kind.
In addition, studying the properties of the newly obtained operations yields further interesting subjects.
Hence every completeness property has its dual, obtained by inverting the order-dependent definitions in the given statement.
The dual notion, the empty lower bound, is the greatest element, top, or unit (1).
Further simple completeness conditions arise from the consideration of all non-empty finite sets.
An order in which all non-empty finite sets have both a supremum and an infimum is called a lattice.
However, using the given order, one can restrict to further classes of (possibly infinite) subsets, that do not yield this strong completeness at once.
The term is used widely with this definition that focuses on suprema and there is no common name for the dual property.
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.
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.
Note that the latter two structures extend the application of these principles beyond mere completeness requirements by introducing an additional operation of negation.
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.
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.
is also a lower adjoint, then the poset X is a Heyting algebra—another important special class of partial orders.
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).
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.