Normal submonoid

ANALOGY: This is an analogue in monoid of a property encountered in group. Specifically, it is a submonoid property analogous to the subgroup property: normal subgroup
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Definition

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

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 $xH=Hx\mathrm{\forall }x\in M$.