Definition:Normal Subgroup of Monoid

It has been suggested that this page or section be merged into Definition:Normal Subgroup. In particular: It is strange that these pages do not at least link to each other Actually they do -- see the "also defined as" section. Deliberately wanted to keep this separate from the definition for a normal subgroup of a group, as I have run into considerable controversy as to whether or not it should be "allowed" to make this definition. Some sources suggest that it is invalid to define a normal subgroup of any structure but a group. But Warner does it, as part of his approach to make everything as general as can be, with a view to establishing the limits of what you can do I suppose. Understood and agreed. I would just like to see a reference to this page from Definition:Normal Subgroup, I guess. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Mergeto}} from the code.

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 $11.16$