Join and meet
In mathematics, specifically order theory, the join of a subset S of a partially ordered set P 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 the maximal/minimal element of that subset, if such an element exists. If a subset S of a partially ordered set P is also an (upward) 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.
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