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preordered_sets [2010/07/29 15:46] (current)
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 +=====Preordered sets=====
 +Abbreviation: **Qoset**
 +
 +====Definition====
 +A \emph{preordered set} (also called a \emph{quasi-ordered set} or \emph{qoset} for short) is a structure $\mathbf{P}=\langle P,\preceq\rangle$
 +such that $P$ is a set and $\preceq $ is a binary relation on $P$ that is
 +
 +reflexive:  $x\preceq x$ and
 +
 +transitive:  $x\preceq y \text{ and } y\preceq z\Longrightarrow x\preceq z$
 +
 +Remark:
 +
 +==Morphisms==
 +Let $\mathbf{P}$ and $\mathbf{Q}$ be qosets. A morphism from $\mathbf{P}$ to
 +$\mathbf{Q}$ is a function $f:P\rightarrow Q$ that is preorder-preserving:
 +
 +$x\preceq y\Longrightarrow f(x)\preceq f(y)$
 +
 +====Examples====
 +Example 1:
 +
 +====Basic results====
 +
 +
 +====Properties====
 +^[[Classtype]]  |Universal Horn class |
 +^[[Universal theory]]  |Decidable |
 +^[[First-order theory]]  |Undecidable |
 +^[[Amalgamation property]]  | |
 +^[[Strong amalgamation property]]  | |
 +^[[Epimorphisms are surjective]]  | |
 +====Finite members====
 +
 +$\begin{array}{lr}
 +f(1)= &1\\
 +f(2)= &2\\
 +f(3)= &\\
 +f(4)= &\\
 +f(5)= &\\
 +f(6)= &\\
 +f(7)= &\\
 +\end{array}$
 +
 +====Subclasses====
 +[[Posets]]
 +
 +[[Connected qosets]]
 +
 +====Superclasses====
 +[[Binary relational structures]]
 +
 +
 +====References====
 +
 +[(Ln19xx>
 +)]