Complete lattice in the context of Infimum

In mathematics, specifically order theory, the join of a subset ${\displaystyle S}$ of a partially ordered set ${\displaystyle P}$ is the supremum (least upper bound) of ${\displaystyle S,}$ denoted ${\textstyle \bigvee S,}$ and similarly, the meet of ${\displaystyle S}$ is the infimum (greatest lower bound), denoted ${\textstyle \bigwedge S.}$ In general, the join and meet of a subset of a partially ordered set need not exist. Join and meet are dual to one another with respect to order inversion.

A partially ordered set in which all pairs have a join is a join-semilattice. Dually, a partially ordered set in which all pairs have a meet is a meet-semilattice. A partially ordered set that is both a join-semilattice and a meet-semilattice is a lattice. A lattice in which every subset, not just every pair, possesses a meet and a join is a complete lattice. It is also possible to define a partial lattice, in which not all pairs have a meet or join but the operations (when defined) satisfy certain axioms.

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, "${\displaystyle \vee }$") and infima (greatest lower bounds, meets, "${\displaystyle \wedge }$") 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 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.