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Join and meet
Concept in order theory / From Wikipedia, the free encyclopedia
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In mathematics, specifically order theory, the join of a subset of a partially ordered set
is the supremum (least upper bound) of
denoted
and similarly, the meet of
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.
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![]() ![]() All definitions tacitly require the homogeneous relation |
![](http://upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Join_and_meet.svg/220px-Join_and_meet.svg.png)
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 of a partially ordered set
is also an (upward) directed set, then its join (if it exists) is called a directed join or directed supremum. Dually, if
is a downward directed set, then its meet (if it exists) is a directed meet or directed infimum.