nLab GCD monoid

Definition

A monoid (or semigroup) $M$ is a left GCD monoid if for all elements $m,n \in M$, there is a unique minimal left ideals in $M$ generated by $m$ and $n$. Similarily, right GCD monoid if for all elements $m,n \in M$, there is a unique minimal right ideals in $M$ generated by $m$ and $n$. Finally, $M$ is a two-sided GCD monoid if it is both a left GCD monoid and a right GCD monoid, and $M$ is a GCD monoid if $M$ is a two-sided GCD monoid and for all elements $m,n \in M$ the minimal left and right ideals generated by $m$ and $n$ coincide.

Related concepts

• ideal of a monoid

• principal ideal of a monoid

• principal ideal monoid