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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>
 +)]