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algebraic_posets [2010/07/29 15:11]
jipsen created
algebraic_posets [2010/09/04 16:55] (current)
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-f+=====Algebraic posets===== 
 + 
 +Abbreviation: **APos** 
 + 
 +====Definition==== 
 +An \emph{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 \emph{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 \emph{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==== 
 +[[Algebraic semilattices]]  
 + 
 + 
 +====Superclasses==== 
 +[[Directed complete partial orders]]  
 + 
 + 
 +====References==== 
 + 
 +[(Ln19xx> 
 +)]