Algebraic posets
Abbreviation: APos
Definition
An algebraic poset is a directed complete partial orders $\mathbf{P}=\langle P,\leq \rangle $ such that
the set of compact elements below any element is directed and
every element is the join of all compact elements below it.
An element $c\in P$ is compact if for every subset $S\subseteq P$ such that $c\le\bigvee S$, there exists a finite subset $S_0$ of $S$ such that $c\le\bigvee S_0$.
The set of compact elements of $P$ is denoted by $K(P)$.
Morphisms
Let $\mathbf{P}$ and $\mathbf{Q}$ be algebraic posets. A morphism from $\mathbf{P}$ to $\mathbf{Q}$ is a function $f:P\rightarrow Q$ that is Scott-continuous, which means that $f$ preserves all directed joins:
$z=\bigvee D\Longrightarrow f(z)= \bigvee f[D]$
Examples
Example 1:
Basic results
Properties
Classtype | second-order |
---|---|
Amalgamation property | |
Strong amalgamation property | |
Epimorphisms are surjective |
Finite members
$\begin{array}{lr} f(1)= &1\\ \end{array}$
Subclasses
Superclasses
References
Trace: » algebraic_posets