partially_ordered_monoids - MathStructures

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=====Partially ordered monoids=====

Abbreviation: **PoMon**

====Definition====
A \emph{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 \emph{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: 
$h(x \cdot y)=h(x) \cdot h(y)$, $h(1)=1$,
$x\le y\Longrightarrow h(x)\le h(y)$

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

====Basic results====

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

====Properties====


^[[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====

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


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

[[Lattice-ordered monoids]] expanded type


====Superclasses====
[[Partially ordered semigroups]] reduced type


====References====

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

Trace: • directed_partial_orders • properties • sequential_algebras • commutative_rings • semifields • quantales • abelian_partially_ordered_groups • abelian_ordered_groups • commutative_regular_rings • completely_regular_hausdorff_spaces