in the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set (poset). the most familiar example is the completeness of the real numbers. a special use of the term refers to complete partial orders or complete lattices. however, many other interesting notions of completeness exist. the motivation for considering completeness properties derives from the great importance of suprema (least upper bounds, joins, "\vee") and infima (greatest lower bounds, meets, "\wedge") to the theory of partial orders. finding a supremum means to single out one distinguished least element from the set of upper bounds. on the one hand, these special elements often embody certain concrete properties that are interesting for the given application (such as being the least common multiple of a set of numbers or the union of a collection of sets). on the other hand, the knowledge that certain types of subsets are guaranteed to have suprema or infima enables us to consider the computation 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. ==types of completeness properties== all completeness properties are described along a similar scheme: one describes a certain class of subsets of a partially ordered set that are required to have a supremum or infimum. hence every completeness property has its dual, obtained by inverting the order-dependent definitions in the given statement. some of the notions are usually not dualized while others may be self-dual (i.e. equivalent to their dual statements). ===least and greatest elements=== the easiest example of a supremum is the empty one, i.e. the supremum of the empty set. by definition, this is the least element among all elements that are greater than each member of the empty set. but this is just the least element of the whole poset, if it has one, since the empty subset of a poset p is conventionally considered to be both bounded from above and from below, with every element of p being both an upper and lower bound of the empty subset. other common names for the least element are bottom and zero (0). the dual notion, the empty lower bound, is the greatest element, top, or unit (1). posets that have a bottom are sometimes called pointed, while posets with a top are called unital or topped. an order that has both a least and a greatest element is bounded. however, this should not be confused with the notion of bounded completeness given below. ===finite completeness=== further simpl
Order_theory