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A real number x is the least upper bound (or supremum) for S if x is an upper bound for S and x ≤ y for every upper bound y of S. The least-upper-bound property states that any non-empty set of real numbers that has an upper bound must have a least upper bound in real numbers.
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 ...
An upper bound is said to be a tight upper bound, a least upper bound, or a supremum, if no smaller value is an upper bound. Similarly, a lower bound is said to be a tight lower bound, a greatest lower bound, or an infimum, if no greater value is a lower bound.
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 seldom-considered dual notion to a dcpo is the filtered-complete poset. Dcpos with a least element ("pointed dcpos") are one of the possible meanings of the phrase complete partial order (cpo). If every subset that has some upper bound has also a least upper bound, then the respective poset is called bounded complete. The term is used ...
That is, the least upper bound (sup or supremum) of the empty set is negative infinity, while the greatest lower bound (inf or infimum) is positive infinity. By analogy with the above, in the domain of the extended reals, negative infinity is the identity element for the maximum and supremum operators, while positive infinity is the identity ...