nLab
join

Contents

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.


Context

Limits and colimits

(0,1)(0,1)-Category theory

Contents

Definition

If xx and yy are elements of a poset PP, then their join (or supremum, abbreviate sup, or least upper bound, abbreviated lub), is an element xyx \vee y of the poset such that:

  • xxyx \leq x \vee y and yxyy \leq x \vee y;
  • if xax \leq a and yay \leq a, then xyax \vee y \leq a.

(These may be combined as: for all aa, xyax \vee y \leq a iff xax \leq a and yay \leq a.) Such a join may not exist; if it does, then it is unique.

If PP 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 PP.)

The above definition is for the join of two elements of PP, 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 aa 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 SS be a set of real numbers, and let MM be a real number. We say (as above) that MM is a least upper bound (lub) of SS if for each real number aa, MaM \leq a iff for each member xx of SS, xax \leq a. But we say that MM is a supremum of SS if for each real number aa, M>aM \gt a iff for some member xx of SS, x>ax \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 MM is an lub of SS and M>aM \gt a, then there is not not some member xx of SS such that x>ax \gt a, but not that there is such an member xx. For a specific weak counterexample, let pp be any truth value, and let SS be the subsingleton {0|p}\{0 \;|\; p\}. Then 00 is a supremum of SS iff pp is true, while 00 is an lub of SS iff pp is not not true.)

This generalizes to any set PP 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 PP 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:

  • yxzy \geq x \nleq z implies yzy \nleq z,
  • xzyx \nleq z \geq y implies xyx \nleq y.

Then we have two versions of a join MM:

  • for each element aa of PP, MaM \leq a iff for each member xx of SS, xax \leq a;
  • for each element aa of PP, MaM \nleq a iff for some member xx of SS, xax \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.

Last revised on February 15, 2020 at 19:35:23. See the history of this page for a list of all contributions to it.