# Differences

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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>
+)]