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

====Examples====
Example 1: 

====Basic results====


====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  |
^[[Equational theory]]                | |
^[[Quasiequational theory]]           | |
^[[First-order theory]]               | |
^[[Locally finite]]                   | |
^[[Residual size]]                    | |
^[[Congruence distributive]]          | |
^[[Congruence modular]]               | |
^[[Congruence $n$-permutable]]        | |
^[[Congruence regular]]               | |
^[[Congruence uniform]]               | |
^[[Congruence extension property]]    | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]      | |
^[[Amalgamation property]]            | |
^[[Strong amalgamation property]]     | |
^[[Epimorphisms are surjective]]      | |

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

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

====Subclasses====
[[Commutative ordered monoids]]


====Superclasses====
[[Ordered monoids]] reduced type

[[Ordered semigroups with zero]] reduced type

[[Representable residuated lattices]]


====References====

[(Lastname19xx>
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]] 
)]