Join and meet

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.

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.

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.

DefinitionsEdit

Partial order approachEdit

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  ).

The meet need not exist, either since the pair has no lower bound at all, or since none of the lower bounds is greater than all the others. However, if there is a meet of   then it is unique, since if both   are greatest lower bounds of   then   and thus  [2] If not all pairs of elements from   have a meet, then the meet can still be seen as a partial binary operation on  [1]

If the meet does exist then it is denoted   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  ).

Universal algebra approachEdit

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 approachesEdit

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 subsetsEdit

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).

ExamplesEdit

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 alsoEdit

NotesEdit

  1. ^ a b Grätzer 1996, p. 52.
  2. ^ Hachtel, Gary D.; Somenzi, Fabio (1996). Logic synthesis and verification algorithms. Kluwer Academic Publishers. p. 88. ISBN 0792397460.
  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  

ReferencesEdit