=====Algebraic Lattices=====

Abbreviation: **ALat**

====Definition====
An \emph{algebraic lattice} is a [[complete lattice]] $\mathbf{A}=\langle A,\bigvee,\bigwedge\rangle$ such that
every element is a join of compact elements.

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

==Morphisms==
Let $\mathbf{A}$ and $\mathbf{B}$ be algebraic lattices. 
A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a complete homomorphism: 

$h(\bigvee S)=\bigvee h[S] \mbox{ and } h(\bigwedge S)=\bigwedge h[S]$

====Examples====
Example 1: 

====Basic results====


====Properties====
^[[Classtype]]  |second-order |
^[[Amalgamation property]]  |yes |
^[[Strong amalgamation property]]  |yes |
^[[Epimorphisms are surjective]]  |yes |


====Subclasses====
[[Algebraic distributive lattices]] 


====Superclasses====
[[Complete lattices]] 

[[Algebraic semilattices]] 


====References====

[(Ln19xx>
)]