## Complete lattices

Abbreviation: CLat

### Definition

A complete lattice is a structure $\mathbf{L}=\langle L,\bigvee,\bigwedge\rangle$ such that $\bigvee,\bigwedge$ map subsets of $L$ to elements of $L$ and

$\langle L,\vee,\wedge\rangle$ is a lattice where $x\vee y=\bigvee\{x,y\}$, $x\wedge y=\bigwedge\{x,y\}$ and

$\bigvee S$ is the least upper bound of $S$,

$\bigwedge S$ is the greatest lower bound of $S$.

##### Morphisms

Let $\mathbf{L}$ and $\mathbf{M}$ be complete lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a complete homomorphism:

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

### Examples

Example 1: $\langle \mathcal{P}(X),\bigcup,\bigcap\rangle$, the set of all subsets of a set $X$, with union and intersection of families of sets.

### Properties

Classtype Second-order Yes Yes Yes