Join and meet

In a partially ordered set P, the join and meet of a subset S are respectively the supremum (least upper bound) of S, denoted ⋁S, and infimum (greatest lower bound) of S, denoted ⋀S. In general, the join and meet of a subset of a partially ordered set need not exist; when they do exist, they are elements of P.

Join and meet can also be defined as a commutative, associative and idempotent partial binary operation on pairs of elements from P. If a and b are elements from P, the join is denoted as a ∨ b and the meet is denoted a ∧ b.

Join and meet are symmetric duals with respect to order inversion. The join/meet of a subset of a totally ordered set is simply its maximal/minimal element.

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.

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 (or greatest lower bound or infimum) of x and y, if the following two conditions are satisfied:

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