Join and meet
All definitions tacitly require the homogeneous relation
A term's definition may require additional properties that are not listed in this table.
In mathematics, specifically order theory, the join of a subset
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 the maximal/minimal element of that subset, if such an element exists.
is called the meet (or greatest lower bound or infimum) of
if the following two conditions are satisfied: 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.
and it is easy to see that this operation fulfills the following three conditions: For any elements
is the join (or least upper bound or supremum) of
if the following two conditions are satisfied: By definition, a binary operation
is a meet if it satisfies the three conditions a, b, and c. The pair
In fact, this relation is a partial order on
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).
is a partially ordered set, such that each pair of elements in
Thus, the partial order defined by the meet in the universal algebra approach coincides with the original partial order.
is defined as in the universal algebra approach, and
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.
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.
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.
) 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
