commutative_residuated_partially_ordered_semigroups

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=====Commutative residuated partially ordered semigroups=====

Abbreviation: **CRPoSgrp**

====Definition====
A \emph{commutative residuated partially ordered semigroup} is a [[residuated partially ordered semigroup]] $\mathbf{A}=\langle A, \cdot, \to, \le\rangle$ such that

$\cdot$ is \emph{commutative}:  $xy=yx$

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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.

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

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

$...$ is ...:  $axiom$
  
$...$ is ...:  $axiom$

====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]]                        |quasivariety  |
^[[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 lattice-ordered semigroups]] expanded type


====Superclasses====
  [[Residuated partially ordered semigroups]] same type

  [[Commutative partially ordered semigroups]] reduced type


====References====

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

Trace: • trees • banach_spaces • vector_spaces_over_a_field • finite_monoids • boolean_lattices • ordered_semilattices • boolean_algebras_with_operators • congruence_e-regular • compact_topological_spaces • chains