arbitrary join

Let P be a poset and AP. The join of A is the supremumMathworldPlanetmathPlanetmath of A, if it exists. It is denoted by

A   or   iIai   or   {aiiI},

if the elements of A are indexed by a set I:


In other words, A=supA, where the equality is directed in the sense that one side is defined iff the other side is, and when this is the case, both sides are equal. Dually, one defines the meet of A to be the infimumMathworldPlanetmath of A, if it exists. The meet of A is denoted by A.

Remark. The concepts of and sup of an ordered set are identical. Besides being notationally distinct, is often used in order theory, while sup is more prevalent in analysisMathworldPlanetmath. Moreover, is generally being viewed as a (partial) function on the powerset 2P of the poset P, while sup is frequently seen as an operationMathworldPlanetmath on sequences (or more generally nets) of elements of P.

  1. 1.

    As was remarked, given a poset P, let us view :2PP as a partial functionMathworldPlanetmath. Then is defined for all singletons. In fact {a}=a for all aP. Dually, {a}=a.

  2. 2.

    If is defined for all doubletons, then it is defined for all finite subsets of P. In this case, P is called a join-semilattice. Dually, P is a meet-semilattice is is defined for all doubletons.

  3. 3.

    If AB, then AB, provided that both joins exist. We also have a dual statement: AB implies that BA, provided that both meets exist.

  4. 4.

    exists iff P has a bottom 0, and when this is the case, =0. This is essentially the result of the previous bulleted statement. Dually, P has a top 1 iff exists, and when this is the case =1.

  5. 5.

    Simiarly P=1 and P=0, where the equality is directed on both sides.

  6. 6.

    It can be shown that if is a total function from 2P to P, then P is a complete latticeMathworldPlanetmath. (see proof here (

  7. 7.

    Let P be a poset such that is defined for all subsets (of P) of cardinality 𝔪, is it true that is defined for all subsets of cardinality 𝔪? The answer is no, even when 𝔪 is finite. A counterexampleMathworldPlanetmath can be constructed as follows.


    Let C be an infinite chain with a top element 1 (this can be found by taking the set of natural numbers and dualize the usual order). Adjoin three elements a,b,c to C so that a and b are below all elements of C, and c is covered by 1, and no two of a,b,c are comparablePlanetmathPlanetmath. This new poset P has the property that any three distinct elements have a join. For example, {a,b,c}=1. However, {a,b} does not exist.

Title arbitrary join
Canonical name ArbitraryJoin
Date of creation 2013-03-22 17:27:53
Last modified on 2013-03-22 17:27:53
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Definition
Classification msc 06A06
Related topic CompleteLattice
Related topic CompleteSemilattice
Defines arbitrary meet
Defines infinite join
Defines infinite meet