linguistic:voice_change, structural:expansion, information:filtering
Relation to maximum and minimum elements

The infimum of a subset S {\displaystyle S} of a partially ordered set P, {\displaystyle P,} assuming it exists, does not necessarily belong to S. {\displaystyle S.} If it does, it is a minimum or least element of S. {\displaystyle S.} Similarly, if the supremum of S {\displaystyle S} belongs to S, {\displaystyle S,} it is a maximum or greatest element of S. {\displaystyle S.}

For example, consider the set of negative real numbers (excluding zero). This set has no greatest element, since for every element of the set, there is another, larger, element. For instance, for any negative real number x, {\displaystyle x,} there is another negative real number x 2, {\displaystyle {\tfrac {x}{2}},} which is greater. On the other hand, every real number greater than or equal to zero is certainly an upper bound on this set. Hence, 0 {\displaystyle 0} is the least upper bound of the negative reals, so the supremum is 0. This set has a supremum but no greatest element.

However, the definition of maximal and minimal elements is more general. In particular, a set can have many maximal and minimal elements, whereas infima and suprema are unique.

Whereas maxima and minima must be members of the subset that is under consideration, the infimum and supremum of a subset need not be members of that subset themselves.