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

Classtype quasivariety undecidable no unbounded no

Finite members

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