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

Finite members

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

Subclasses

Superclasses

References


QR Code
QR Code preordered_sets (generated for current page)