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If the supremum of exists, it is unique, and if b is an upper bound of , then the supremum of is less than or equal to b. Consequently, the supremum is also referred to as the least upper bound (or LUB). [1] The infimum is, in a precise sense, dual to the concept of a
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.
In mathematics, the least-upper-bound property (sometimes called completeness, supremum property or l.u.b. property) [1] is a fundamental property of the real numbers. More generally, a partially ordered set X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound (supremum) in X .
An element of is called the meet (or greatest lower bound or infimum) of and is denoted by , if the following two conditions are satisfied: m ≤ x and m ≤ y {\displaystyle m\leq x{\text{ and }}m\leq y} (that is, m {\displaystyle m} is a lower bound of x and y {\displaystyle x{\text{ and }}y} ).
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 .
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . A conditionally complete lattice satisfies at least one of these properties for bounded subsets. For comparison, in a general lattice, only pairs of elements need to have a supremum and an infimum. Every non-empty finite ...