Join and meet

[summary: 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. Meets are greatest lower bounds, and are related to joins by duality. ]

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

• Examples
• Exercises

Further reading

• Lattices