this entry is about the notion of colimits in posets. For the concepts of join of topological spaces, join of simplicial sets, join of categories and join of quasi-categories see there.
If and are elements of a poset , then their join (or supremum, abbreviated sup, or least upper bound, abbreviated lub), is an element of the poset such that:
(These may be combined as: for all , iff and .) Such a join may not exist; if it does, then it is unique.
If is a proset, then join may be defined similarly, but it need not be unique. (However, it is still unique up to the natural equivalence in .)
The above definition is for the join of two elements of , but it can easily be generalised to any number of elements. It may be more common to use ‘join’ for a join of finitely many elements and ‘supremum’ for a join of (possibly) infinitely many elements, but they are the same concept. The join may also be called the maximum if it equals one of the original elements.
A poset that has all finite joins is a join-semilattice. A poset that has all suprema is a suplattice.
A join of subsets or subobjects is called a union.
A join of zero elements is a bottom element. Any element is a join of that one element.
As a poset is a special kind of category, so a join is simply a coproduct in that category.
In constructive analysis, we sometimes want a stronger notion of supremum. (Dual remarks apply to infima.)
Let be a set of real numbers, and let be a real number. We say (as above) that is a least upper bound (lub) of if for each real number , iff for each member of , . But we say that is a supremum of if for each real number , iff for some member of , . In constructive mathematics, we can prove that lubs and suprema are both unique when they exist and that every supremum is an lub, but we cannot prove that every lub is a supremum. (We can prove that, if is an lub of and , then there is not not some member of such that , but not that there is such a member . For a specific weak counterexample, let be any truth value, and let be the subsingleton . Then is a supremum of iff is true, while is an lub of iff is not not true.)
This generalizes to any set equipped with a relation (better written in the general case) that is an irreflexive connected comparison (properties dual to the properties that define a partial order) if is defined as the negation of (which forces to be a partial order). It's not even necessary for to be a comparison, as long as its negation is a partial order (which still forces to be irreflexive and connected).
Still more generally, let be a set equipped with the antithesis interpretation of a partial order. This consists of two binary relations and such that is a partial order, is irreflexive, and and are compatible:
Then we have two versions of a join :
Then neither of these implies the other, and we probably really want to demand both at once. The extended MacNeille real numbers provide a good example here.
join
Last revised on October 16, 2024 at 01:18:12. See the history of this page for a list of all contributions to it.