# Definition:Join Semilattice

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## 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** or a **$\vee$-semilattice**.

Some sources hyphenate: **join semi-lattice**.

## Also see

- Results about
**join semilattices**can be found**here**.

## Sources

- Semi-lattice.
*Encyclopedia of Mathematics*. URL: https://www.encyclopediaofmath.org/index.php?title=Semi-lattice&oldid=39737