### Table of Contents

## Ordered semilattices

Abbreviation: **OSlat**

### Definition

An \emph{ordered semilattice} is a ordered semigroup $\mathbf{A}=\langle A,\cdot,\le\rangle$ that is

\emph{commutative}: $x\cdot y = y\cdot x$ and

\emph{idempotent}: $x\cdot x = x$

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be ordered semigroups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a orderpreserving homomorphism: $h(x \cdot y)=h(x) \cdot h(y)$, $x\le y\Longrightarrow h(x)\le h(y)$.

### Examples

Example 1:

### Basic results

### Properties

### Finite members

$\begin{array}{rr}
f(1)=&1

f(2)=&2

f(3)=&5

f(4)=&14

f(5)=&42

f(6)=&132

f(7)=&

f(8)=&

\end{array}$

This sequence is the Catalan numbers http://oeis.org/A000108