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There are at least two standard ways of unitizing a (small) semigroup $\mathbb A$:

(i) We add an identity regardless that $\mathbb A$ is already unital.

(ii) We add an identity only if none is already available.

In the former case, the unitization process is functorial, as it amounts to the existence of a left adjoint to the canonical forgetful functor from the category of small categories to the category of small semicategories (in the sense of B. Mitchell).

Question. Is there any standard terminology to differentiate (i) from (ii)? I would be content with something like "(i) is occasionally called the unitization à la X" or "(ii) is referred to by some authors as Y's unitization".

Thanks in advance.

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  • $\begingroup$ I have seen (ii) called the canonical unitization. $\endgroup$
    – Aaron Meyerowitz
    Jan 16, 2013 at 16:11
  • $\begingroup$ Actually the place I saw it was a well written paper whose author you may know: arxiv.org/abs/1208.3233 ! $\endgroup$
    – Aaron Meyerowitz
    Jan 16, 2013 at 16:18
  • $\begingroup$ In the context of Banach algebras, (i) has sometimes been called the forced unitization and (ii) has been called the conditonal unitization -- I think I have seen this terminology in the pink book of Helemskii, for instance, but my memory is rusty. $\endgroup$
    – Yemon Choi
    Jan 16, 2013 at 19:49
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    $\begingroup$ I believe that there is no commonly used noun for these things. I would call the process in (i) adjoining an identity and call the result A with an adjoined identity. Personally I would not give (ii) a name. I know of several results that are incorrect because people used (ii) in adapting monoid constructions to semigroups when they should have used (i). My book with Rhodes has a diatribe on this. $\endgroup$
    – Benjamin Steinberg
    Jan 16, 2013 at 22:31
  • $\begingroup$ In the context of rings, at least one former MO user (and former vocal MSE user) has referred to (i) as the Dorroh extension mathoverflow.net/questions/31358/… $\endgroup$
    – Yemon Choi
    Jan 18, 2013 at 3:24

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