# Join and meet

Let \(\langle P, \leq \rangle\) be a partially ordered set, and let \(S \subseteq P\). The **join** of \(S\) in \(P\), denoted by \(\bigvee_P S\), is an element \(p \in P\) satisfying the following two properties:

p is an

*upper bound*of \(S\); that is, for all \(s \in S\), \(s \leq p\).For all upper bounds \(q\) of \(S\) in \(P\), \(p \leq q\).

\(\bigvee_P S\) does not necessarily exist, but if it does then it is unique. The notation \(\bigvee S\) is typically used instead of \(\bigvee_P S\) when \(P\) is clear from context. Joins are often called *least upper bounds* or *supremums*. For \(a, b\) in \(P\), the join of \(\{a,b\}\) in \(P\) is denoted by \(a \vee_P b\), or \(a \vee b\) when \(P\) is clear from context.

The dual concept of the join is that of the meet. The **meet** of \(S\) in \(P\), denoted by \(\bigwedge_P S\), is defined an element \(p \in P\) satisfying.

p is a

*lower bound*of \(S\); that is, for all \(s\) in \(S\), \(p \leq s\).For all lower bounds \(q\) of \(S\) in \(P\), \(q \leq p\).

Meets are also called *infimums*, or *greatest lower bounds*. The notations \(\bigwedge S\), \(p \wedge_P q\), and \(p \wedge q\) are all have meanings that are completely analogous to the aforementioned notations for joins.

## Basic example

The above Hasse diagram represents a poset with elements \(a\), \(b\), \(c\), and \(d\). \(\bigvee \{a,b\}\) does not exist because the set \(\{a,b\}\) has no upper bounds.\(\bigvee \{c,d\}\) does not exist for a different reason: although \(\{c, d\}\) has upper bounds \(a\) and \(b\), these upper bounds are incomparable, and so \(\{c, d\}\) has no *least* upper bound. There do exist subsets of this poset which possess joins; for example, \(a \vee c = a\), \(\bigvee \{b,c,d\} = b\), and \(\bigvee \{c\} = c\).

Now for some examples of meets.\(\bigwedge \{a, b, c, d\}\) does not exist because \(c\) and \(d\) have no common lower bounds. However, \(\bigwedge \{a,b,d\} = d\) and \(a \wedge c = c\).

## Additional Material

## Further reading

Children:

Parents:

- Order theory
The study of binary relations that are reflexive, transitive, and antisymmetic.

Examples?