Table of Contents
Directed partial orders
Abbreviation: DPO
Definition
A directed partial order is a poset $\mathbf{P}=\langle P,\leq \rangle $ that is directed, i.e. every finite subset of $P$ has an upper bound in $P$, or equivalently, $P\ne\emptyset$, $\forall xy\exists z (x\le z$ and $y\le z)$.
Morphisms
Let $\mathbf{P}$ and $\mathbf{Q}$ be directed partial orders. A morphism from $\mathbf{P}$ to $\mathbf{Q}$ is a function $f:Parrow Q$ that is order preserving:
$x\le y\Longrightarrow f(x)\le f(y)$
Examples
Example 1:
Basic results
Properties
Classtype | first-order |
---|---|
Amalgamation property | |
Strong amalgamation property | |
Epimorphisms are surjective |
Finite members
$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &2\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ \end{array}$
Subclasses
Superclasses
References
Trace: » directed_partial_orders