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There is a statement at semigroup which looks wrong: that the functor $(-)^+: Semigrp \to Mon$ that adjoins an identity element to form a monoid is fully faithful. It’s faithful of course, but the monoid morphism $S^+ \to T^+$ that factors through the inclusion $\{e\} \hookrightarrow T^+$ is not of the form $f^+$ for any semigroup morphism $f: S \to T$.
The exercise mentioned immediately after at semigroup might need to be re-examined, but it reminds me vaguely of a remark out of Categories, Allegories: the category of rings without unit is equivalent to the category of rings over $\mathbb{Z}$ (consider augmentation ideals), an example “where adding in more structure results in less structure”. But I don’t see how to adapt this example to make it fit for semirings without unit, much less semigroups without unit.
Yes, you’re right. I see you changed the exercise to “describe this structure”. I think one solution would be to say that a semigroup is a monoid equipped with a set of “nonidentity elements” which contains exactly the elements that are not the identity. That might not be describable in such a clever way as in the ring case, though.
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