Editing Completeness (order theory) (section)

==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 structure]]s in the sense of [[universal algebra]], which are equipped with operations like <math>\vee</math> or <math>\wedge</math>. By imposing additional conditions (in form of suitable [[Identity (mathematics)|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 (order)|lattice]], [[Heyting algebra]], and [[Boolean algebra (structure)|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''.