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  2. Infimum and supremum - Wikipedia

    en.wikipedia.org/wiki/Infimum_and_supremum

    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

  3. Completeness (order theory) - Wikipedia

    en.wikipedia.org/wiki/Completeness_(order_theory)

    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.

  4. Order theory - Wikipedia

    en.wikipedia.org/wiki/Order_theory

    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.

  5. Join and meet - Wikipedia

    en.wikipedia.org/wiki/Join_and_meet

    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} ).

  6. Limit inferior and limit superior - Wikipedia

    en.wikipedia.org/wiki/Limit_inferior_and_limit...

    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 ...

  7. Complete lattice - Wikipedia

    en.wikipedia.org/wiki/Complete_lattice

    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 ...

  8. Order complete - Wikipedia

    en.wikipedia.org/wiki/Order_complete

    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 .

  9. Upper and lower bounds - Wikipedia

    en.wikipedia.org/wiki/Upper_and_lower_bounds

    A set with upper bounds and its least upper bound. In mathematics, particularly in order theory, an upper bound or majorant [1] of a subset S of some preordered set (K, ≤) is an element of K that is greater than or equal to every element of S.