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Join (mathematics)

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Join (mathematics)


In a partially ordered set P, the join and meet of a subset S are respectively the supremum (least upper bound) of S, denoted ⋁S, and infimum (greatest lower bound) of S, denoted ⋀S. If they exist, the join and meet are elements of P. Join and meet can also be defined as a commutative, associative and idempotent partial binary operation on pairs of elements from P. If a and b are elements from P, the join is denoted as ab and the meet is denoted ab.

Join and meet are symmetric duals with respect to order inversion. The join/meet of a subset of a totally ordered set is simply its maximal/minimal element.

It is not assumed that all pairs of elements have either a meet or a join. A partially ordered set in which all pairs have a join is a join-semilattice: dually, a partially ordered set in which all pairs have a meet is meet-semilattice. A partially ordered set that is both a join-semilattice and a meet-semilattice is a lattice. A lattice in which every subset, not just every pair, possesses a meet and a join is a complete lattice.

In the following we dispense discussing joins, because they become meet when considering the reverse partial order, thanks to duality.

Partial order approach

Let A be a set with a partial order ≤, and let x and y be two elements in A. An element z of A is the meet (or greatest lower bound or infimum) of x and y, if the following two conditions are satisfied:

  1. zx and zy (i.e., z is a lower bound of x and y).
  2. For any w in A, such that wx and wy, we have wz (i.e., z is greater than or equal to any other lower bound of x and y).

If there is a meet of x and y, then indeed it is unique, since if both z and z′ are greatest lower bounds of x and y, then zz and z′ ≤ z, whence indeed z = z. If the meet does exist, it is denoted xy. Some pairs of elements in A may lack a meet, either since they have no lower bound at all, or since none of their lower bounds is greater than all the others. If all pairs of elements have meets, then indeed the meet is a binary operation on A, and it is easy to see that this operation fulfils the following three conditions: For any elements x, y, and z in A,

a. xy = yx (commutativity),
b. x ∧ (yz) = (xy) ∧ z (associativity), and
c. xx = x (idempotency).

Universal algebra approach

By definition, a binary operation ∧ on a set A is a meet, if it satisfies the three conditions a, b, and c. The pair (A,∧) then is a meet-semilattice. Moreover, we then may define a binary relation ≤ on A, by stating that xy if and only if xy = x. In fact, this relation is a partial order on A. Indeed, for any elements x, y, and z in A,

  • xx, since xx = x by c;
  • if xy and yx, then x = xy = yx = y by a; and
  • if xy and yz, then xz, since then xz = (xy) ∧ z = x ∧ (yz) = xy = x by b.

Note that both meets and joins equally satisfy this definition: a couple of associated meet and join operations yield partial orders which are the reverse of each other. When choosing one of these orders as the main ones, one also fixes which operation is considered a meet (the one giving the same order) and which is considered a join (the other one).

Equivalence of approaches

If (A,≤) is a partially ordered set, such that each pair of elements in A has a meet, then indeed xy = x if and only if xy, since in the latter case indeed x is a lower bound of x and y, and since clearly x is the greatest lower bound if and only if it is a lower bound. Thus, the partial order defined by the meet in the universal algebra approach coincides with the original partial order.

Conversely, if (A,∧) is a meet-semilattice, and the partial order ≤ is defined as in the universal algebra approach, and z = xy for some elements x and y in A, then z is the greatest lower bound of x and y with respect to ≤, since

zx = xz = x ∧ (xy) = (xx) ∧ y = xy = z

and therefore zx. Similarly, zy, and if w is another lower bound of x and y, then wx = wy = w, whence

wz = w ∧ (xy) = (wx) ∧ y = wy = w.

Thus, there is a meet defined by the partial order defined by the original meet, and the two meets coincide.

In other words, the two approaches yield essentially equivalent concepts, a set equipped with both a binary relation and a binary operation, such that each one of these structures determines the other, and fulfil the conditions for partial orders or meets, respectively.

Meets of general subsets

If (A,∧) is a meet-semilattice, then the meet may be extended to a well-defined meet of any non-empty finite set, by the technique described in iterated binary operations. Alternatively, if the meet defines or is defined by a partial order, some subsets of A indeed have infima with respect to this, and it is reasonable to consider such an infimum as the meet of the subset. For non-empty finite subsets, the two approaches yield the same result, whence either may be taken as a definition of meet. In the case where each subset of A has a meet, in fact (A,≤) is a complete lattice; for details, see completeness (order theory).

References

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