# Differences

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complete_semilattices [2010/07/29 15:46] (current)
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+=====Complete semilattices=====
+Abbreviation: **CSlat**
+====Definition====
+A \emph{complete semilattice} is a [[directed complete partial orders]] $\mathbf{P}=\langle P,\leq \rangle$
+such that every nonempty subset of $P$ has a greatest lower bound:
+$\forall S\subseteq P\ (S\ne\emptyset\Longrightarrow \exists z\in P(z=\bigwedge S))$.
+==Morphisms==
+Let $\mathbf{P}$ and $\mathbf{Q}$ be complete semilattices. A morphism from $\mathbf{P}$ to
+$\mathbf{Q}$ is a function $f:P\rightarrow Q$ that preserves all nonempty meets and all directed joins:
+
+$z=\bigwedge S\Longrightarrow f(z)=\bigwedge f[S]$ for all nonempty $S\subseteq P$ and
+$z=\bigvee D\Longrightarrow f(z)= \bigvee f[D]$
+
+====Examples====
+Example 1:
+
+====Basic results====
+
+====Properties====
+^[[Classtype]]  |second-order |
+^[[Amalgamation property]]  | |
+^[[Strong amalgamation property]]  | |
+^[[Epimorphisms are surjective]]  | |
+====Finite members====
+
+$\begin{array}{lr} +f(1)= &1\\ +f(2)= &\\ +f(3)= &\\ +f(4)= &\\ +f(5)= &\\ +f(6)= &\\ +\end{array}$
+
+====Subclasses====
+[[Complete lattices]]
+
+====Superclasses====
+[[Directed complete partial orders]]
+
+
+====References====
+
+[(Ln19xx>
+)]

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