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An infimum of a set is always and only defined relative to a superset of the set in question. For example, there is no infimum of the positive real numbers inside the positive real numbers (as their own superset), nor any infimum of the positive real numbers inside the complex numbers with positive real part.
For example, in lattice theory, one is interested in orders where all finite non-empty sets have both a least upper bound and a greatest lower bound. In domain theory, on the other hand, one focuses on partially ordered sets in which every directed subset has a supremum. Complete lattices and orders with a least element (the "empty supremum ...
The supremum of B is then equal to the infimum of X: since each element of X is an upper bound of B, sup B is smaller than all elements of X, i.e. sup B is in B. It is the greatest element of B and hence the infimum of X. In a dual way, the existence of all infima implies the existence of all suprema.
The infimum of two elements may be written as inf{x,y} or x ∧ y. If the set X is finite, one speaks of a finite infimum. The dual notion is called supremum. Interval. For two elements a, b of a partially ordered set P, the interval [a,b] is the subset {x in P | a ≤ x ≤ b} of P. If a ≤ b does not hold the interval will be empty. Interval ...
This concept is also called supremum or join, and for a set S one writes sup(S) or for its least upper bound. Conversely, the greatest lower bound is known as infimum or meet and denoted inf(S) or . These concepts play an important role in many applications of order theory.
In mathematics, specifically in order theory and functional analysis, a subset of an ordered vector space is said to be order complete in if for every non-empty subset of that is order bounded in (meaning contained in an interval, which is a set of the form [,]:= {:}, for some ,), the supremum ' and the infimum both exist and are elements of .
In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers.Since the supremum and infimum of an unbounded set of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the ...
An example is given by the power set of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection. Another example is given by the natural numbers, partially ordered by divisibility, for which the supremum is the least common multiple and the infimum is the greatest common divisor.