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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 9th 2013
    • (edited Oct 9th 2013)

    There is a statement at semigroup which looks wrong: that the functor () +:SemigrpMon(-)^+: 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 +T +S^+ \to T^+ that factors through the inclusion {e}T +\{e\} \hookrightarrow T^+ is not of the form f +f^+ for any semigroup morphism f:STf: 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.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeOct 9th 2013

    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.