Normal submonoid

Definition
A submonoid $$H$$ of a monoid $$M$$ is said to be normal if $$Hx = xH \forall x \in M$$.

Analogy
The notion of normal submonoid is the generalization to monoids of the notion of normality for subgroups. Recall that a subgroup $$H$$ of a group $$M$$ is said to be normal if $$xH = Hx \forall x \in M$$.