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+ | =====Chains===== | ||
+ | |||
+ | ====Definition==== | ||
+ | A \emph{chain} is a [[partially ordered set]] $\mathbf{C}=\langle C,\le\rangle$ such that | ||
+ | |||
+ | |||
+ | $\le$ is a total order: $x\le y \mbox{ or } y\le x$ | ||
+ | |||
+ | |||
+ | Remark: | ||
+ | |||
+ | ==Morphisms== | ||
+ | Let $\mathbf{C}$ and $\mathbf{D}$ be chains. A morphism from $\mathbf{C}$ to $\mathbf{D}$ is a function $h:C\rightarrow D$ that is a orderpreserving: | ||
+ | |||
+ | $x\le y\Longrightarrow h(x)\le h(y)$ | ||
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+ | ====Examples==== | ||
+ | Example 1: | ||
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+ | ====Basic results==== | ||
+ | |||
+ | |||
+ | ====Properties==== | ||
+ | ^[[Classtype]] |Universal | | ||
+ | ^[[Quasiequational theory]] | | | ||
+ | ^[[First-order theory]] | | | ||
+ | ^[[Amalgamation property]] | | | ||
+ | ^[[Strong amalgamation property]] | | | ||
+ | ^[[Epimorphisms are surjective]] | | | ||
+ | ====Finite members==== | ||
+ | |||
+ | $\begin{array}{lr} | ||
+ | f(1)= &1\\ | ||
+ | f(2)= &1\\ | ||
+ | f(3)= &1\\ | ||
+ | f(4)= &1\\ | ||
+ | f(5)= &1\\ | ||
+ | f(6)= &1\\ | ||
+ | \end{array}$ | ||
+ | |||
+ | ====Subclasses==== | ||
+ | [[Well-ordered chains]] | ||
+ | |||
+ | [[Dense linear orders]] | ||
+ | |||
+ | ====Superclasses==== | ||
+ | [[Trees]] | ||
+ | |||
+ | |||
+ | ====References==== | ||
+ | |||
+ | [(Ln19xx> | ||
+ | )] | ||
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