In mathematical order theory, join is a binary operation on a partially ordered set that gives the supremum (least upper bound) of its arguments, provided the least upper bound exists. The join of elements and is denoted . A partially ordered set in which each pair of elements has a join is called a join-semilattice.
Join operations can be abstractly described as commutative and associative binary operations satisfying an idempotency law. In the study of complete lattices, the join operation is extended to return the least upper bound of an arbitrary set of elements.
Let A be a set with a partial order , and let and be two elements in A. An element of A is the join (or least upper bound or supremum) of and , if the following two conditions are satisfied:
By definition, a binary operation on a set A is a join, if it satisfies the three conditions a, b, and c supra. The pair (A,) then is a join-semilattice. Moreover, we then may define a binary relation on A, by stating that if and only if . In fact, this relation is a partial order on A. Indeed, for any elements , , and in A,
Conversely, if (A,) is a join-semilattice, and the partial order is defined as in the universal algebra approach, and for some elements and in A, then is the least upper bound of and with respect to , since , similarly , and if is another upper bound of and , then , whence . Thus, there is a join defined by the partial order defined by the original join, and the two joins coincide.
In other words, the two approaches yield essentially equivalent concepts, a set equipped with both a binary order relation and a binary join operation, such that each one of these structures determines the other, and fulfil the conditions for partial orders or joins, respectively.
If (A,) is a join-semilattice, then the join may be extended to a well-defined join of any non-empty finite set, by the technique described in iterated binary operations. Alternatively, the join defines or is defined by a partial order, some subsets of A indeed have suprema with respect to this, and it is reasonable to consider such a supremum as the join of the subset. For non-empty finite subsets, the two approaches yield the same result, whence either may be taken as a definition of join. In the case where each subset of A has a join, in fact (A,) is a complete lattice; for details, see completeness (order theory).