Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.
Concept
Join and meet
Summary
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.
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 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.
Let be a set with a partial order and let An element of is called the (or or ) of and is denoted by if the following two conditions are satisfied:
(that is, is a lower bound of ).
For any if then (that is, is greater than or equal to any other lower bound of ).
The meet need not exist, either since the pair has no lower bound at all, or since none of the lower bounds is greater than all the others. However, if there is a meet of then it is unique, since if both are greatest lower bounds of then and thus If not all pairs of elements from have a meet, then the meet can still be seen as a partial binary operation on
If the meet does exist then it is denoted If all pairs of elements from have a meet, then the meet is a binary operation on and it is easy to see that this operation fulfills the following three conditions: For any elements
(commutativity),
(associativity), and
(idempotency).
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.