Abbreviation: CanMon

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: $x\cdot z=y\cdot z\Longrightarrow x=y$

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:

$h(x\cdot y)=h(x)\cdot h(y)$, $h(e)=e$

Example 1: $\langle\mathbb{N},+,0\rangle$, the natural numbers, with addition and zero.

All free monoids are cancellative.

All finite (left or right) cancellative monoids are reducts of groups.

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

Groups

Cancellative residuated lattices

Cancellative semigroups

Monoids