Join and meet

In mathematics, specifically order theory, the join of a subset {\displaystyle S}S of a partially ordered set {\displaystyle P}P is the supremum (least upper bound) of {\displaystyle S,}S, denoted {\textstyle \bigvee S,}{\textstyle \bigvee S,} and similarly, the meet of {\displaystyle S}S is the infimum (greatest lower bound), denoted {\textstyle \bigwedge S.}{\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.

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.

If a subset {\displaystyle S}S of a partially ordered set {\displaystyle P}P is also an (upward) directed set, then its join (if it exists) is called a directed join or directed supremum. Dually, if {\displaystyle S}S is a downward directed set, then its meet (if it exists) is a directed meet or directed infimum.

This Hasse diagram depicts a partially ordered set with four elements: a, b, the maximal element a {\displaystyle \vee }\vee b equal to the join of a and b, and the minimal element a {\displaystyle \wedge }\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.

1 Definitions

1.1 Partial order approach

1.2 Universal algebra approach

1.3 Equivalence of approaches

2 Meets of general subsets

3 Examples

4 See also