Abbreviation: Qoset
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:
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:
$x\preceq y\Longrightarrow f(x)\preceq f(y)$
Example 1:
| Classtype | Universal Horn class |
|---|---|
| Universal theory | Decidable |
| First-order theory | Undecidable |
| Amalgamation property | |
| Strong amalgamation property | |
| Epimorphisms are surjective |
$\begin{array}{lr}
f(1)= &1
f(2)= &2
f(3)= &
f(4)= &
f(5)= &
f(6)= &
f(7)= &
\end{array}$