# Definition:Join Semilattice

## Definition

### 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 $\left({S, \vee}\right)$ be a semilattice.

Let $\preceq$ be the ordering on $S$ defined by:

$a \preceq b \iff \left({a \vee b}\right) = b$

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