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There is a corresponding greatest-lower-bound property; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least ...
Thus, the infimum or meet of a collection of subsets is the greatest lower bound while the supremum or join is the least upper bound. In this context, the inner limit, lim inf X n, is the largest meeting of tails of the sequence, and the outer limit, lim sup X n, is the smallest joining of tails of the sequence. The following makes this precise.
Likewise, a greatest element of a partially ordered set (poset) is an upper bound of the set which is contained within the set, whereas a maximal element m of a poset A is an element of A such that if m ≤ b (for any b in A), then m = b. Any least element or greatest element of a poset is unique, but a poset can have several minimal or maximal ...
If (,) is a partially ordered set, such that each pair of elements in has a meet, then indeed = if and only if , since in the latter case indeed is a lower bound of , and since is the greatest lower bound if and only if it is a lower bound. Thus, the partial order defined by the meet in the universal algebra approach coincides with the original ...
The example function (,) = (+) illustrates that the equality does not hold for every function. A theorem giving conditions on f, W, and Z which guarantee the saddle point property is called a minimax theorem.
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).
For example, 5 is a lower bound for the set S = {5, 8, 42, 34, 13934} (as a subset of the integers or of the real numbers, etc.), and so is 4. On the other hand, 6 is not a lower bound for S since it is not smaller than every element in S. 13934 and other numbers x such that x ≥ 13934 would be an upper bound for S.
In calculus and mathematical analysis the limits of integration (or bounds of integration) of the integral () of a Riemann integrable function f {\displaystyle f} defined on a closed and bounded interval are the real numbers a {\displaystyle a} and b {\displaystyle b} , in which a {\displaystyle a} is called the lower limit and b {\displaystyle ...