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.
42.
"By a directed set in a poset ( P, < = ), we mean a nonempty subset D such that any pair { x, y } of elements of D has an upper bound in D . We say ( P, < = ) is directed-complete if each directed subset D ?" P has a least upper bound in P.
43.
We define a directed set in a poset P by a subset D such that every pair of elements of D has an upper bound in D . We say P is directed complete if each directed subset of P has a least upper bound in P . I wish to show that a poset is complete iff it is directed complete and has supremums for all its finite subsets . ( Sup = least upper bound)
44.
Directed sets are a more general concept than ( join ) semilattices : every ordered bitwise ( e . g . 1000 d " 1011 holds, but 0001 d " 1000 does not, since in the last bit 1 > 0 ), where { 1000, 0001 } has three upper bounds but no " least " upper bound, cf . picture . ( Also note that without 1111, the set is not directed .)
45.
Since it is certainly the case that there can be non directed-complete subsets, I tried to say'for each pair of elements there is a least upper bound'since it is a finite subset, then add these least upper bounds to get a directed set : but then you may be lacking upper bounds between the elements you have just added, and then you need to keep on adding more and more elements, so I don't think that line of reasoning is going anywhere.
46.
""'Dead End " "'is a 1969 Hong Kong action drama film directed by Chang Cheh and starring Ti Lung, Lee Ching, David Chiang, Chan Hung-lit and Angela Yu Chien . " Dead End " marks the first film that Chang directed set in modern day, Ti's first leading role and the first collaboration of director Chang with stars Ti and Chiang, dubbed " The Iron Triangle, " because of their successful line of films together which would last several years.
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