=====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>
)]