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Partially ordered monoids

Abbreviation: PoMon


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: $x\le y\Longrightarrow wxz\le wyz$

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


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

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$ $\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$


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


[[Partially ordered semigroups]] reduced type