In mathematics, specifically order theory, the join of a subsetS of a partially ordered setP is the supremum of S, denoted ⋁S, and similarly, the meet of S is the infimum, denoted ⋀S. 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 its maximal/minimal element, if such an element exists. If a subset S of a partially ordered set P is also an directed set, then its join is called a directed join or directed supremum. Dually, if S is a downward directed set, then its meet is a directed meet or directed infimum.
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 of x and y, if the following two conditions are satisfied:
z ≤ x and z ≤ y.
For any w in A, such that and, we have .
If there is a meet of x and y, then it is unique, since if both z and z′ are greatest lower bounds of x and y, then and, and thus z = z′. If the meet does exist, it is denoted. 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 from A have a meet, then 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, Joins are defined dually, and the join of x and y in A is denoted by x ∨ y. If not all pairs of elements from A have a meet, then the meet can still be seen as a partial binary operation on A.
By definition, a binary operation ∧ on a set A 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 ifx ∧ y = x. In fact, this relation is a partial order on A. Indeed, for any elements x, y, and z in A,
x ≤ x, since x ∧ x = x by c;
if x ≤ y and y ≤ x, then x = x ∧ y = y ∧ x = y by a; and
if x ≤ y and y ≤ z, then x ≤ z, since then x ∧ z = ∧ z = x ∧ = x ∧ y = 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 and which is considered a join.
Equivalence of approaches
If is a partially ordered set, such that each pair of elements in A has a meet, then indeed x ∧ y = x if and only if, 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 is a meet-semilattice, and the partial order ≤ is defined as in the universal algebra approach, and z = x ∧ y for some elements x and y in A, then z is the greatest lower bound of x and y with respect to ≤, since and therefore. Similarly,, and if w is another lower bound of x and y, then w ∧ x = w ∧ y = w, 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 fulfil 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-emptyfinite 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, and so either may be taken as a definition of meet. In the case where each subset of A has a meet, in fact is a complete lattice; for details, see completeness.
