### Table of Contents

## 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}$