# Join and meet

In mathematics, specifically order theory, the join of a subset $S$ of a partially ordered set $P$ is the supremum (least upper bound) of $S,$ denoted ${\textstyle \bigvee S,}$ and similarly, the meet of $S$ is the infimum (greatest lower bound), denoted ${\textstyle \bigwedge S.}$ 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. This Hasse diagram depicts a partially ordered set with four elements: a, b, the maximal element a ∨ {\displaystyle \vee } b equal to the join of a and b, and the minimal element a ∧ {\displaystyle \wedge } 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.

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 (upward) directed set, then its join (if it exists) is called a directed join or directed supremum. Dually, if $S$ is a downward directed set, then its meet (if it exists) is a directed meet or directed infimum.