Infima in Partially Ordered Sets
The definition of infima easily generalizes to subsets of arbitrary partially ordered sets and as such plays a vital role in order theory. In this context, especially in lattice theory, greatest lower bounds are also called meets.
Formally, the infimum of a subset S of a partially ordered set (P, ≤) is an element a of P such that
- a ≤ x for all x in S, (a is a lower bound) and
- for all y in P, if for all x in S, y ≤ x, then y ≤ a (a larger than any other lower bound).
Any element with these properties is necessarily unique, but in general no such element needs to exist. Consequently, orders for which certain infima are known to exist become especially interesting. More information on the various classes of partially ordered sets that arise from such considerations are found in the article on completeness properties.
The dual concept of infimum is given by the notion of a supremum or least upper bound. By the duality principle of order theory, every statement about suprema is thus readily transformed into a statement about infima. For this reason, all further results, details, and examples can be taken from the article on suprema.
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