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 x≤ax \leq a and y≤ay \leq a, then x∨y≤ax \vee y \leq a. Such a join may not exist; if it does, then it is unique.
The above definition is for the join of two elements of a poset, 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 aa is a join of that one element.
As a poset is a special kind of category, a join is simply a coproduct in that category.
Last revised on November 20, 2015 at 12:11:33. See the history of this page for a list of all contributions to it.