# Definition:Join Semilattice/Definition 1

## Definition

Let $\struct {S, \preceq}$ be an ordered set.

Suppose that for all $a, b \in S$:

$a \vee b \in S$

where $a \vee b$ is the join of $a$ and $b$ with respect to $\preceq$.

Then the ordered structure $\struct {S, \vee, \preceq}$ is called a join semilattice.