# Differences

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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> | ||

+ | )] |

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