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You say the category is not the same as the category of abelian groups, which is true enough, but isn’t it equivalent to the category of pointed abelian groups, equivalently the undercategory or co-slice $\mathbb{Z} \downarrow Ab$? “Pointed commutative invertible semigroup” just sounds so long-winded, if “pointed abelian group” would do. Do other people use this phrase?
There is (or maybe was) a trend of anonymous guests not reacting to substantial remarks on their edits (similarly here). I think we should wait maybe a couple more days but then go ahead and re-edit.
I renamed this page to pointed abelian group, defining the object as an abelian group with an additional element, and I moved the definition as a “pointed commutative invertible semigroup” into a remark which states that the neutral element is not needed in the definition of pointed abelian group.
Joachim Joszef
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