For the case of a join-semilattice note that any set can be turned into a directed set with the same supremum by closing under finite ( non-empty ) suprema.
32.
Then just argue that the least upper bound of the resulting directed set is a least upper bound of the original set .-- talk ) 00 : 24, 30 May 2011 ( UTC)
33.
Naturally, one has a special interest in those domains of computations in which all consistent specifications " converge ", i . e . in orders in which all directed sets have a least upper bound.
34.
From the article on Power domains : " P " [ D ] is the collection of downward-closed subsets of domain D that are also closed under existing least upper bounds of directed sets in D.
35.
One says it is a "'directed family of seminorms "'if the family is a directed set with addition as the join, in other words if for every and, there is a such that.
36.
A topology is order consistent with respect to a certain order d " if it induces d " as its specialization order and it has the above property of inaccessibility with respect to ( existing ) suprema of directed sets in d ".
37.
This generalization exhibits the Riemann Stieltjes integral as the Moore Smith limit on the directed set of partitions of [ " a ", " b " ] . calls it the "'Pollard Moore Stieltjes integral " '.
38.
In general, however, it has an approximate identity which is a net ( or generalized sequence ) \ { e _ i \ } _ { i \ in I } indexed on a directed set I such that f * e _ i \ to f.
39.
In the mathematical area of order theory, the "'compact "'or "'finite elements "'of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not already contain members above the compact element.
40.
According to what was said above, this would be an element that is the most general piece of information that extends the information of all elements of the directed set, i . e . the unique element that contains " exactly " the information that was present in the directed set-and nothing more.
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