This is an old revision of the document!

Ordered monoids with zero

Abbreviation: OMonZ


An 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 zero: $x\cdot 0 = 0$ and $0\cdot x = 0$

Remark: This is a template. If you know something about this class, click on the ``Edit text of this page'' link at the bottom and fill out this page.

It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.


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: $h(x \cdot y)=h(x) \cdot h(y)$, $h(1)=1$, $h(0)=0$, $x\le y\Longrightarrow h(x)\le h(y)$.


A is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that

$...$ is …: $axiom$

$...$ is …: $axiom$


Example 1:

Basic results


Finite members

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

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


[[Commutative ordered monoids]]


[[Ordered monoids]] reduced type
[[Ordered semigroups with zero]] reduced type
[[Representable residuated lattices]]