Definition:Join Semilattice

Definition 1
Let $\left({S, \preceq}\right)$ 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 $\left({S, \vee, \preceq}\right)$ is called a join semilattice.

That it is indeed an ordered structure is proved on Join Semilattice is Ordered Structure.

That a join semilattice is a semilattice is proved on Join Semilattice is Semilattice.

Definition 2
Let $(S, \vee)$ be a semilattice.

Let $\preceq$ be the ordering on $S$ defined by:
 * $a \preceq b \iff (a \vee b) = b$.

Then the ordered structure $(S, \vee, \preceq)$ is called a join semilattice.

Also see

 * Definition:Meet Semilattice
 * Definition:Semilattice (Abstract Algebra)