Editing Infimum and supremum (section)

== Existence and uniqueness ==

Infima and suprema do not necessarily exist. Existence of an infimum of a subset <math>S</math> of <math>P</math> can fail if <math>S</math> has no lower bound at all, or if the set of lower bounds does not contain a greatest element. (An example of this is the subset <math>\{ x \in \mathbb{Q} : x^2 < 2 \}</math> of <math>\mathbb{Q}</math>. It has upper bounds, such as 1.5, but no supremum in <math>\mathbb{Q}</math>.)

Consequently, partially ordered sets for which certain infima are known to exist become especially interesting. For instance, a [[Lattice (order)|lattice]] is a partially ordered set in which all {{em|nonempty finite}} subsets have both a supremum and an infimum, and a [[complete lattice]] is a partially ordered set in which {{em|all}} subsets have both a supremum and an infimum. More information on the various classes of partially ordered sets that arise from such considerations are found in the article on [[Completeness (order theory)|completeness properties]].

If the supremum of a subset <math>S</math> exists, it is unique.  If <math>S</math> contains a greatest element, then that element is the supremum; otherwise, the supremum does not belong to <math>S</math> (or does not exist).  Likewise, if the infimum exists, it is unique.  If <math>S</math> contains a least element, then that element is the infimum; otherwise, the infimum does not belong to <math>S</math> (or does not exist).