Definitions

# Join (mathematics)

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 $x$ and $y$ is denoted $x lor y$. A partially ordered set in which each pair of elements has a join is called a join-semilattice.

There is a dual operation, called meet, that returns the greatest lower bound of its arguments. Each pair of elements in a lattice has both a join (least upper bound) and meet (greatest lower bound).

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.

## The partial order approach

Let A be a set with a partial order $leq$, and let $x,!$ and $y,!$ be two elements in A. An element $z,!$ of A is the join (or least upper bound or supremum) of $x,!$ and $y,!$, if the following two conditions are satisfied:

1. $x leq z$ and $y leq z$ (i.e., $z,!$ is an upper bound of $x,!$ and $y,!$); and
2. for any $w,!$ in A, such that $x leq w$ and $y leq w$, we have $z leq w$ (i.e., $z,!$ is less than any other upper bound of $x,!$ and $y,!$).
If there is a join of $x,!$ and $y,!$, then indeed it is unique, since if both $z,!$ and $z\text{'},!$ are least upper bounds of $x,!$ and $y,!$, then $z leq z\text{'} leq z$, whence indeed $z = z\text{'},!$. If the join does exist, it is denoted $x lor y$. Some pairs of elements in A may lack a join, either because they have no upper bound at all, or because none of their upper bounds is less than or equal to all the others. If all pairs of elements have joins, then indeed the join is a binary operation on A, and it is easy to see that this operation fulfils the following three conditions: For any elements $x,!$, $y,!$, and $z,!$ in A,
a. $x lor y = y lor x$ (commutativity),
b. $x lor \left(y lor z\right) =\left(x lor y\right) lor z$ (associativity), and
c. $x lor x = x$ (idempotency).

## The universal algebra approach

By definition, a binary operation $lor$ on a set A is a join, if it satisfies the three conditions a, b, and c supra. The pair (A,$lor$) then is a join-semilattice. Moreover, we then may define a binary relation $leq$ on A, by stating that $x leq y$ if and only if $x lor y = y$. In fact, this relation is a partial order on A. Indeed, for any elements $x,!$, $y,!$, and $z,!$ in A,

$x leq x$, since $x lor x = x$ by c;
if $x leq y$ and $y leq x$, then $y = x lor y = y lor x = x$ by a; and
if $x leq y$ and $y leq z$, then $x leq z$, since then $x lor z = x lor \left(y lor z\right) = \left(x lor y\right) lor z = y lor z = z$ by b.

## Equivalence of approaches

If (A,$leq$) is a partially ordered set, such that each pair of elements in A has a join, then indeed $x lor y = y$ if and only if $x leq y$, since in the latter case indeed $y$ is an upper bound of $x$ and $y$, and since clearly $y$ is the least upper bound if and only if it is an upper bound. Thus, the partial order defined by the join in the universal algebra approach coincides with the original partial order.

Conversely, if (A,$lor$) is a join-semilattice, and the partial order $leq$ is defined as in the universal algebra approach, and $z = x lor y$ for some elements $x$ and $y$ in A, then $z$ is the least upper bound of $x$ and $y$ with respect to $leq$, since $x lor z = \left(x lor x\right) lor y = z ;Rightarrow; x leq z$, similarly $y leq z$, and if $w$ is another upper bound of $x$ and $y$, then $x lor w = y lor w = w$, whence $z lor w = \left(x lor y\right) lor w = x lor \left(y lor w\right) = x lor w = w$. 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.

## Joins of general subsets

If (A,$lor$) 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,$leq$) is a complete lattice; for details, see completeness (order theory).