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)
Famous quotes containing the words completeness, terms, universal and/or algebra:
“Poetry presents indivisible wholes of human consciousness, modified and ordered by the stringent requirements of form. Prose, aiming at a definite and concrete goal, generally suppresses everything inessential to its purpose; poetry, existing only to exhibit itself as an aesthetic object, aims only at completeness and perfection of form.”
—Richard Harter Fogle, U.S. critic, educator. The Imagery of Keats and Shelley, ch. 1, University of North Carolina Press (1949)
“One of your biggest jobs as a parent of multiples is no bigger than simply talking to your children individually and requiring that they respond to you individually as well. The benefits of this kind of communication can be enormous, in terms of the relationship you develop with each child, in terms of their language development, and eventually in terms of their sense of individuality, too.”
—Pamela Patrick Novotny (20th century)
“We are now going through a period of demolition. In morals, in social life, in politics, in medicine, and in religion there is a universal upturning of foundations. But the day of reconstruction seems to be looming, and now the grand question is: Are there any sure and universal principles that will evolve a harmonious system in which we shall all agree?”
—Catherine E. Beecher (1800–1878)
“Poetry has become the higher algebra of metaphors.”
—José Ortega Y Gasset (1883–1955)