Cancellative monoids
Abbreviation: CanMon
Definition
A cancellative monoid is a monoid $\mathbf{M}=\langle M, \cdot, e\rangle$ such that
$\cdot $ is left cancellative: $z\cdot x=z\cdot y\Longrightarrow x=y$
$\cdot $ is right cancellative: $x\cdot z=y\cdot z\Longrightarrow x=y$
Morphisms
Let $\mathbf{M}$ and $\mathbf{N}$ be cancellative monoids. A morphism from $\mathbf{M}$ to $\mathbf{N}$ is a function $h:M\rightarrow N$ 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 free monoids are cancellative.
All finite (left or right) cancellative monoids are reducts of groups.
Properties
Finite members
$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &2\\ f(5)= &1\\ f(6)= &2\\ f(7)= &1\\ \end{array}$
Subclasses
Superclasses
References
Trace: » cancellative_monoids