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algebraic_semilattices [2010/07/29 15:12]
jipsen created
algebraic_semilattices [2010/09/04 16:55] (current)
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-f+=====Algebraic semilattices=====
+
+Abbreviation: **ASlat**
+
+====Definition====
+An \emph{algebraic semilattice} is a [[complete semilattice]] $\mathbf{P}=\langle P,\leq \rangle$
+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 semilattices. 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 lattices]]
+
+
+====Superclasses====
+[[Algebraic posets]]
+
+
+====References====
+
+[(Ln19xx>
+)]