Join (mathematics)

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In mathematics, a join on a set is defined either as unique suprema (least upper bounds) with respect to a partial order on the set, provided such suprema exist, or (abstractly) as a commutative and associative binary operation satisfying an idempotency law. In either case, the set together with the join is a join-semilattice. The two definitions yield equivalent results, except that in the partial order approach it may be possible directly to define joins of more general sets of elements. The most common context in which to find a join is as one of the operations in a lattice.

Usually, the join of $x$ and $y$ is denoted $x \lor y$ .

1 The partial order approach
2 The universal algebra approach
3 Equivalence of approaches
4 Joins of general subsets
5 See also

[edit] 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 $z'$ are least upper bounds of $x$ and $y$ , then $z \leq z' \leq z$ , whence indeed $z = z'\,$ . If the join does exist, it is denoted $x \lor y$ . Some pairs of elements in A may lack a join, either since they have no upper bound at all, or since none of their upper bounds is less than 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. $x \lor (y \lor z) =(x \lor y) \lor z$ (associativity), and

c. $x \lor x = x$ (idempotency).

[edit] 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 $x \lor z = x \lor (y \lor z) = (x \lor y) \lor z = y \lor z = z$ by b.

[edit] 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 a lower 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 $x \lor z = (x \lor x) \lor y = z \;\Rightarrow\; x \leq z$ , similarly $y \leq z$ , and if $w$ is another upper bound of $x$ and $y$ , then $x \lor w = y \lor w = w$ , whence $z \lor w = (x \lor y) \lor w = x \lor (y \lor w) = x \lor w = w$ . 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 relation and a binary operation, such that each one of these structures determines the other, and fulfil the conditions for partial orders or joins, respectively.

[edit] 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).