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## Partially ordered monoids

Abbreviation: PoMon

### Definition

A partially ordered monoid is a structure $\mathbf{A}=\langle A,\cdot,1,\le\rangle$ such that

$\langle A,\cdot,1\rangle$ is a monoid

$\langle G,\le\rangle$ is a partially ordered set

$\cdot$ is orderpreserving: $x\le y\Longrightarrow wxz\le wyz$

##### Morphisms

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

Example 1:

### Basic results

Every monoid with the discrete partial order is a po-monoid.

### Properties

Classtype quasivariety

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &4\\ f(3)= &37\\ f(4)= &549\\ f(5)= &\\ \end{array}$

### Subclasses

[[Commutative partially ordered monoids]]
[[Lattice-ordered monoids]] expanded type

### Superclasses

[[Partially ordered semigroups]] reduced type