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Cancellative commutative monoids
Abbreviation: CanCMon
Definition
A cancellative commutative monoid is a cancellative monoid $\mathbf{M}=\langle M,\cdot ,e\rangle $ such that
$\cdot $ is commutative: $x\cdot y=y\cdot x$
Morphisms
Let $\mathbf{M}$ and $\mathbf{N}$ be cancellative commutative monoids. A morphism from $\mathbf{M}$ to $\mathbf{N}$ is a function $h:Marrow N$ that is a homomorphism:
$h(x\cdot y)=h(x)\cdot h(y)$, $h(e)=e$
Examples
Example 1: $\langle\mathbb{N},+,0\rangle$, the natural numbers, with addition and zero.
Basic results
All commutative free monoids are cancellative.
All finite commutative (left or right) cancellative monoids are reducts of abelian groups.
Properties
Finite members
$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &2\\ f(5)= &1\\ f(6)= &1\\ f(7)= &1\\ \end{array}$
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Subclasses
Superclasses
References
Trace: » cancellative_commutative_monoids