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chains [2010/07/29 15:46] (current)
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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:
 +
 +$x\le y\Longrightarrow h(x)\le h(y)$
 +
 +====Examples====
 +Example 1:
 +
 +====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>
 +)]
 +
 +
 +
 +
 +