Definition:Normal Subgroup of Monoid
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Definition
Let $\struct {S, \circ}$ be a monoid whose identity element is $e$.
Let $\struct {H, \circ}$ be a subgroup of $\struct {S, \circ}$.
$H$ is a normal subgroup of $S$ if and only if:
- $e$ is the identity element of $H$
- $\forall s \in S: s \circ H = H \circ s$
where $s \circ H$ denotes the subset product of $s$ with $H$.
Also defined as
It will be noted that this is the same definition as a normal subgroup of a group, which is the usual context in which to find this definition.
Also see
- Results about normal subgroups can be found here.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Exercise $11.16$