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 an 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 or . 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.
Read more about this topic: Completeness (order Theory)
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