=====Residuated partially ordered monoids=====

Abbreviation: **RpoMon**

====Definition====
A \emph{residuated partially ordered monoid} (or \emph{rpo-monoid}) is a structure $\mathbf{A}=\langle A,\le,\cdot,1,\backslash,/\rangle$ such that

$\langle A,\le\rangle$ is a [[partially ordered set]],

$\langle A,\cdot,1\rangle$ is a [[monoid]] and

$\backslash$ is the left residual of $\cdot$:  $x\cdot y\le z\iff y\le x\backslash z$

$/$ is the right residual of $\cdot$:  $x\cdot y\le z\iff x\le z/y$.

==Morphisms==
Let $\mathbf{A}$ and $\mathbf{B}$ be residuated po-monoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is an order-preserving homomorphism: 
$x\le y\implies h(x)\le h(y)$, $h(x \cdot y)=h(x) \cdot h(y)$, $h(x \backslash y)=h(x) \backslash h(y)$, $h(x / y)=h(x) / h(y)$.

====Examples====

====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]]                        |order variety [(Ln19xx)]  |
^[[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====

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


====Subclasses====

[[Commutative residuated partially ordered monoids]]

[[Involutive residuated partially ordered monoids]]

[[Residuated lattices]]


====Superclasses====

[[Partially ordered monoids]]

[[Residuated partially ordered semigroups]]


====References====

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