## Algebraic semilattices

Abbreviation: **ASlat**

### Definition

An ** algebraic semilattice** is a complete semilattice $\mathbf{P}=\langle P,\leq \rangle $
such that

the set of compact elements below any element is directed and

every element is the join of all compact elements below it.

An element $c\in P$ is ** compact** if for every subset $S\subseteq P$ such that $c\le\bigvee S$, there exists
a finite subset $S_0$ of $S$ such that $c\le\bigvee S_0$.

The set of compact elements of $P$ is denoted by $K(P)$.

##### Morphisms

Let $\mathbf{P}$ and $\mathbf{Q}$ be algebraic semilattices. A morphism from $\mathbf{P}$ to
$\mathbf{Q}$ is a function $f:P\rightarrow Q$ that is ** Scott-continuous**, which means that $f$ preserves all directed joins:

$z=\bigvee D\Longrightarrow f(z)= \bigvee f[D]$

### Examples

Example 1:

### Basic results

### Properties

Classtype | second-order |
---|---|

Amalgamation property | |

Strong amalgamation property | |

Epimorphisms are surjective |

### Finite members

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

### Subclasses

### Superclasses

### References

Trace: » algebraic_semilattices