Ordered monoids with zero

Abbreviation: OMonZ

Definition

An \emph{ordered monoid with zero} is of the form $\mathbf{A}=\langle A,\cdot,1,0,\le\rangle$ such that $\mathbf{A}=\langle A,\cdot,1,\le\rangle$ is an ordered monoid and

$0$ is a \emph{zero}: $x\cdot 0 = 0$ and $0\cdot x = 0$

Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be ordered monoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a orderpreserving homomorphism: $h(x \cdot y)=h(x) \cdot h(y)$, $h(1)=1$, $h(0)=0$, $x\le y\Longrightarrow h(x)\le h(y)$.

Example 1:

Properties

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

Classtype universal

Finite members

$f(n)=$ number of members of size $n$.

$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= &3 f(4)= &15 f(5)= &84 f(6)= &575 f(7)= &4687 f(8)= &45223 f(9)= & \end{array}$

Superclasses

Ordered monoids reduced type

Ordered semigroups with zero reduced type