Let $\langle P, \leq \rangle$ be a poset, 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 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 $P$ , $p \leq q$ .

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

Basic example

Joins Failing to exist in a finite lattice

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

Now that you know what the join and meet operators are, you may wish to head over to examples or exercises. A lattice is a poset in which every pair of elements has both a join and a meet.