structural:summarization, linguistic:translation
Existence and uniqueness

Infima and suprema do not necessarily exist. Existence of an infimum of a subset S {\displaystyle S} of P {\displaystyle P} can fail if S {\displaystyle S} 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 { x ∈ Q: x 2 < 2 } {\displaystyle \{x\in \mathbb {Q}:x^{2}<2\}} of Q {\displaystyle \mathbb {Q} }. It has upper bounds, such as 1.5, but no supremum in Q {\displaystyle \mathbb {Q} }.)

Consequently, partially ordered sets for which certain infima are known to exist become especially interesting. For instance, a lattice is a partially ordered set in which all nonempty finite subsets have both a supremum and an infimum, and a complete lattice is a partially ordered set in which 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 properties.

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