In mathematics, specifically order theory, the join of a subset of a partially ordered set is the supremum and similarly, the meet is the infimum 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.

is also an (upward) directed set, then its join (if it exists) is called a directed join or directed supremum is a downward directed set, then its meet (if it exists) is a directed meet or directed infimum.

Definitions

Partial order approach

be a set with a partial order is called the (or or if the following two conditions are satisfied:

1. is a lower bound).

2. ).

have a meet, then the meet can still be seen as a partial

have a meet, then the meet is a binary operation (commutativity), (associativity), and (idempotency).

Joins are defined dually is the (or or if the following two conditions are satisfied:

1. is an upper bound).

2. ).

Universal algebra approach

By definition, a binary operation is a if it satisfies the three conditions a, b, and c is then a meet-semilattice. Moreover, we then may define a binary relation on A In fact, this relation is a partial order

• by c;

• by a; and

• by b.

Both meets and joins equally satisfy this definition: a couple of associated meet and join operations yield partial orders which are the reverse of each other. When choosing one of these orders as the main ones, one also fixes which operation is considered a meet (the one giving the same order) and which is considered a join (the other one).

Equivalence of approaches

is a partially ordered set is the lower bound if and only if it is a lower bound. Thus, the partial order defined by the meet in the universal algebra approach coincides with the original partial order.

is a meet-semilattice Thus, there is a meet defined by the partial order defined by the original meet, and the two meets coincide.

In other words, the two approaches yield essentially equivalent concepts, a set equipped with both a binary relation and a binary operation, such that each one of these structures determines the other, and fulfill the conditions for partial orders or meets, respectively.

Meets of general subsets

is a meet-semilattice, then the meet may be extended to a well-defined meet of any non-empty finite set, by the technique described in iterated binary operations indeed have infima with respect to this, and it is reasonable to consider such an infimum as the meet of the subset. For non-empty finite subsets, the two approaches yield the same result, and so either may be taken as a definition of meet. In the case where is a complete lattice; for details, see completeness (order theory).

Examples

If some power set denotes the meet/infimum).

is a family of subsets that is partially ordered that could possibly be the

See also

Notes

References