# Differences

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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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