Abbreviation: ALat

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 $c\le\bigvee S_0$.

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:

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

Example 1:

Algebraic distributive lattices

Complete lattices

Algebraic semilattices