# Definition:Join Semilattice

 It has been suggested that this page or section be merged into Definition:Upper Semilattice on Classical Set. (Discuss)

## Definition

### Definition 1

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.

### Definition 2

Let $\struct {S, \vee}$ be a semilattice.

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

$a \preceq b \iff \paren {a \vee b} = b$

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

## Also known as

A join semilattice is also known as an upper semilattice.