nLab join

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.

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Definition

If $x$ and $y$ are elements of a poset $P$, then their join (or supremum, abbreviated sup, or least upper bound, abbreviated lub), is an element $x \vee y$ of the poset such that:

• $x \leq x \vee y$ and $y \leq x \vee y$;
• if $x \leq a$ and $y \leq a$, then $x \vee y \leq a$.

(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.

Special cases

A join of zero elements is a bottom element. Any element $a$ is a join of that one element.

Properties

As a poset is a special kind of category, so a join is simply a coproduct in that category.

In constructive mathematics

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 a 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:

• $y \geq x \nleq z$ implies $y \nleq z$,
• $x \nleq z \geq y$ implies $x \nleq y$.

Then we have two versions of a join $M$:

• for each element $a$ of $P$, $M \leq a$ iff for each member $x$ of $S$, $x \leq a$;
• for each element $a$ of $P$, $M \nleq a$ iff for some member $x$ of $S$, $x \nleq a$.

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.

Related concepts

• join

• meet