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:

$\begin{array}{lr} f(1)= &1 f(2)= &2 f(3)= & f(4)= & f(5)= & f(6)= & f(7)= & \end{array}$

Posets

Connected qosets

Binary relational structures