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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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I have seen (ii) called the canonical unitization. –  Aaron Meyerowitz Jan 16 '13 at 16:11
    
Actually the place I saw it was a well written paper whose author you may know: arxiv.org/abs/1208.3233 ! –  Aaron Meyerowitz Jan 16 '13 at 16:18
    
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. –  Yemon Choi Jan 16 '13 at 19:49
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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. –  Benjamin Steinberg Jan 16 '13 at 22:31
    
@Aaron. Thanks, but that paper has some naivety and a lot of typos (hopefully they will be fixed 'shortly'). And yes, the author refers to (ii) as the canonical unitization of a sgrp, but I beg to disagree with him, for reasons related to the categorial construction that I mentioned in reference to Mitchell's semicats. On another hand, I like Yemon's suggestion, but it seems that it is not very popular (out of the context of Banach algebras). @Benjamin. I'm citing that excerpt from your book at some point. –  Salvo Tringali Jan 17 '13 at 21:28

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