than or equal to any other upper bound). Existence and uniqueness Infima and suprema do not necessarily exist. Existence of an infimum of a subset of can fail if has no lower bound at all, or if the set of lower bounds does not contain a greatest element. However, if an infimum or supremum does exist, it is unique. 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 subsets have both a supremum and an infimum, and a complete lattice is a partially ordered set in which 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 exists, it is unique. If contains a greatest element, then that element is the supremum; otherwise, the supremum does not belong to (or does not exist). Likewise, if the infimum exists, it is unique. If contains a least element, then that element is the infimum; otherwise, the infimum does not belong to (or does not exist). Relation to maximum and minimum elements The infimum of a subset of a partially ordered set assuming it exists, does not necessarily belong to If it does, it is a minimum or least element of Similarly, if the supremum of belongs to it is a maximum or greatest element of 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 there is another negative real number 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, 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. Minimal upper bounds Finally, a