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 $x$ and $y$ are elements of a poset $P$, then their join (or supremum, abbreviate sup, or least upper bound, abbreviated lub), is an element $x \vee y$ of the poset such that:
(These may be combined as: for all $a$, $x \vee y \leq a$ iff $x \leq a$ and $y \leq a$.) Such a join may not exist; if it does, then it is unique.
If $P$ 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 $P$.)
The above definition is for the join of two elements of $P$, 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 $a$ 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 $S$ be a set of real numbers, and let $M$ be a real number. We say (as above) that $M$ is a least upper bound (lub) of $S$ if for each real number $a$, $M \leq a$ iff for each member $x$ of $S$, $x \leq a$. But we say that $M$ is a supremum of $S$ if for each real number $a$, $M \gt a$ iff for some member $x$ of $S$, $x \gt a$. 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 $M$ is an lub of $S$ and $M \gt a$, then there is not not some member $x$ of $S$ such that $x \gt a$, but not that there is such an member $x$. For a specific weak counterexample, let $p$ be any truth value, and let $S$ be the subsingleton $\{0 \;|\; p\}$. Then $0$ is a supremum of $S$ iff $p$ is true, while $0$ is an lub of $S$ iff $p$ is not not true.)
This generalizes to any set $P$ equipped with a relation $\gt$ (better written $\nleq$ in the general case) that is an irreflexive connected comparison (properties dual to the properties that define a partial order) if $\leq$ is defined as the negation of $\nleq$ (which forces $\leq$ to be a partial order). It's not even necessary for $\nleq$ to be a comparison, as long as its negation is a partial order (which still forces $\nleq$ to be irreflexive and connected).
Still more generally, let $P$ be a set equipped with the antithesis interpretation of a partial order. This consists of two binary relations $\leq$ and $\nleq$ such that $\leq$ is a partial order, $\nleq$ is irreflexive, and $\leq$ and $\nleq$ are compatible:
Then we have two versions of a join $M$:
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 February 15, 2020 at 19:35:23. See the history of this page for a list of all contributions to it.