## Directed complete partial orders

Abbreviation: **DCPO**

### Definition

A ** directed complete partial order** is a poset $\mathbf{P}=\langle P,\leq \rangle $
such that every directed subset of $P$ has a least upper bound:
$\forall D\subseteq P\ (D\ne\emptyset\mbox{and}\forall x,y\in D\ \exists z\in D
(x,y\le z)\Longrightarrow \exists z\in P(z=\bigvee D))$.

##### Morphisms

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

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

### Examples

Example 1: $\langle \mathbb{R},\leq \rangle $, the real numbers with the standard order. Example 1: $\langle P(S),\subseteq \rangle $, the collection of subsets of a sets $S$, ordered by inclusion.

### Basic results

### Properties

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

Amalgamation property | |

Strong amalgamation property | |

Epimorphisms are surjective |

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ \end{array}$

### Subclasses

### Superclasses

### References

Trace: » directed_complete_partial_orders