Join and meet

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This Hasse diagram depicts a partially ordered set with four elements: a, b, the maximal element a b equal to the join of a and b, and the minimal element a b equal to the meet of a and b. The join/meet of a maximal/minimal element and another element is the maximal/minimal element and conversely the meet/join of a maximal/minimal element with another element is the other element. Thus every pair in this poset has both a meet and a join and the poset can be classified as a lattice.

In mathematics, specifically order theory, the join of a subset of a partially ordered set is the supremum (least upper bound) of denoted and similarly, the meet of is the infimum (greatest lower bound), denoted In general, the join and meet of a subset of a partially ordered set need not exist. Join and meet are dual to one another with respect to order inversion.

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 a 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. It is also possible to define a partial lattice, in which not all pairs have a meet or join but the operations (when defined) satisfy certain axioms.[1]

The join/meet of a subset of a totally ordered set is simply its maximal/minimal element, if such an element exists.

If a subset of a partially ordered set is also an (upward) directed set, then its join (if it exists) is called a directed join or directed supremum. Dually, if is a downward directed set, then its meet (if it exists) is a directed meet or directed infimum.

Definitions[edit]

Partial order approach[edit]

Let be a set with a partial order and let An element of is called the meet (or greatest lower bound or infimum) of and is denoted by if the following two conditions are satisfied:

  1. (that is, is a lower bound of ).
  2. For any if then (that is, is greater than or equal to any other lower bound of ).

If there is a meet of then it is unique, since if both are greatest lower bounds of then and thus If the meet does exist then it is denoted Some pairs of elements in 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 from have a meet, then the meet is a binary operation on and it is easy to see that this operation fulfills the following three conditions: For any elements

  1. (commutativity),
  2. (associativity), and
  3. (idempotency).

Joins are defined dually with the join of if it exists, denoted by An element of is the join (or least upper bound or supremum) of in if the following two conditions are satisfied:

  1. (that is, is a upper bound of ).
  2. For any if then (that is, is less than or equal to any other upper bound of ).

If not all pairs of elements from have a meet (respectively, join), then the meet (respectively, join) can still be seen as a partial binary operation on

Universal algebra approach[edit]

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

  • since by c;
  • if then by a; and
  • if then since then by b.

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[edit]

If is a partially ordered set, such that each pair of elements in has a meet, then indeed if and only if since in the latter case indeed is a lower bound of and since 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 is a meet-semilattice, and the partial order is defined as in the universal algebra approach, and for some elements then is the greatest lower bound of with respect to since

and therefore Similarly, and if is another lower bound of then whence
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 fulfill the conditions for partial orders or meets, respectively.

Meets of general subsets[edit]

If 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 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, and so either may be taken as a definition of meet. In the case where each subset of has a meet, in fact is a complete lattice; for details, see completeness (order theory).

Examples[edit]

If some power set is partially ordered in the usual way (by ) then joins are unions and meets are intersections; in symbols, (where the similarity of these symbols may be used as a mnemonic for remembering that denotes the join/supremum and denotes the meet/infimum[note 1]).

More generally, suppose that is a family of subsets of some set that is partially ordered by If is closed under arbitrary unions and arbitrary intersections and if belong to then

But if is not closed under unions then exists in if and only if there exists a unique -smallest such that For example, if then whereas if then does not exist because the sets are the only upper bounds of in that could possibly be the least upper bound but and If then does not exist because there is no upper bound of in

See also[edit]

Notes[edit]

  1. ^ Grätzer 1996, p. 52.
  1. ^ It can be immediately determined that supremums and infimums in this canonical, simple example are respectively. The similarity of the symbol to and of to may thus be used as a mnemonic for remembering that in the most general setting, denotes the supremum (because a supremum is a bound from above, just like is "above" and ) while denotes the infimum (because an infimum is a bound from below, just like is "below" and ). This can also be used to remember whether meets/joins are denoted by or by Intuition suggests that "join"ing two sets together should produce their union which looks similar to so "join" must be denoted by Similarly, two sets should "meet" at their intersection which looks similar to so "meet" must be denoted by

References[edit]

  • Davey, B.A.; Priestley, H.A. (2002). Introduction to Lattices and Order (2nd ed.). Cambridge: Cambridge University Press. ISBN 0-521-78451-4. Zbl 1002.06001.
  • Vickers, Steven (1989). Topology via Logic. Cambridge Tracts in Theoretic Computer Science. 5. ISBN 0-521-36062-5. Zbl 0668.54001.