Join and meet

In mathematics, specifically order theory, the join of a subset of a partially ordered set is the supremum of denoted and similarly, the meet of is the infimum, 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 satisfy certain axioms.
The join/meet of a subset of a totally ordered set is simply the maximal/minimal element of that subset, if such an element exists.
If a subset of a partially ordered set is also an directed set, then its join is called a directed join or directed supremum. Dually, if is a downward directed set, then its meet is a directed meet or directed infimum.

Definitions

Partial order approach

Let be a set with a partial order and let An element of is called the ' of and is denoted by if the following two conditions are satisfied:
.
For any if then .
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 If not all pairs of elements from have a meet, then the meet can still be seen as a partial binary operation on
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

,
, and
.
Joins are defined dually with the join of if it exists, denoted by
An element of is the ' of in if the following two conditions are satisfied:

.
For any if then .
Universal algebra approach

By definition, a binary operation on a set is a 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 and which is considered a join.

Equivalence of approaches

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 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

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 subset of has a meet, in fact is a complete lattice; for details, see completeness.

Examples

If some power set is partially ordered in the usual way then joins are unions and meets are intersections; in symbols, .
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 upper bound but and
If then does not exist because there is no upper bound of in