The following is motivated by the fact that I'd like to have a way, much better if canonical, to isometrically embed a normed group into a normed divisible group. But semigroups are a much more natural setting for posing the question.

Q1. What is known about the embeddability of a (possibly non-commutative) semigroup into a divisible semigroup? Q2. What are some relevant references for the question?

EDIT (05/04/2013). As pointed out by Benjamin Steinberg below, that every semigroup can be embedded into a divisible semigroup was proved by B.H. Neumann; see [1, Theorem 6.2] and [2, Sect. 3]. However, I'm not really happy with Neumann's construction, and this is why I'd like to add the following:

Q3. Is there a functorial way to embed an arbitrary semigroup into a divisible semigroup?

On a related note:

Q4. What is known about the existence of adjoints to the inclusion of $\bf DivSgrp$ into $\bf Sgrp$?

Here, $\bf Sgrp$ is the usual cat of small semigroups (say, with respect to a fixed universe $\mathcal U$, in TG), and $\bf DivSgrp$ the full subcat of $\bf Sgrp$ of divisible semigroups. Feel free to switch questions from semigroups to groups if answers are known for the latter (since then it is likely that they can be adapted to the former).

Thanks in advance for any hint.


[1] B.H. Neumann, Adjunction of elements to groups, JLMS, 18 (1943), 4-11.

[2] B.H. Neumann, Some remarks on semigroup presentations, Canad. J. Math., 19 (1967), 1018–1026.


I believe from what I saw in a survey article that Šutov, È. G. Embeddings of semigroups into simple and complete semigroups Mat. Sb. (N.S.) 62 (104) 1963 496–511 proves every semigroup embeds in a semigroup which is both congruence-free and divisible. I don't have access to the journal and don't know if an English translation exists.

Neumann, B. H. Some remarks on semigroup presentations. Canad. J. Math. 19 1967 1018–1026. Shows how to embed in a divisible semigroup by adjoining roots of elements.

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    $\begingroup$ The Russian original of Shutov's paper is freely available: mathnet.ru/php/… . $\endgroup$ Apr 4 '13 at 14:41
  • $\begingroup$ Thanks, Benjamin, I've just retrieved Neumann's paper. For the record, the manuscript comes with a corrigendum (cms.math.ca/cjm/a145888), for "an error in the first proof, p. 1020, of Theorem 3.1", and an addendum addressing the reader to Šutov's work for an alternative proof. $\endgroup$ Apr 4 '13 at 17:18
  • $\begingroup$ Good to know.... $\endgroup$ Apr 4 '13 at 23:03
  • $\begingroup$ Well, I don't know about Šutov's paper (my Russian doesn't go beyond the alphabet), but I read Neumann's, and strictly speaking Neumann doesn't prove, not in the paper referred to in the above answer, that any given sgrp can be embedded into a divisible sgrp. However, he mentions that this can be done (p. 1021), and addresses the reader to [1, Theorem 6.2], where an analogous result is established for groups. But his construction doesn't look very canonical, so let me edit the OP and add another question. References: [1] B.H. Neumann, Adjunction of elements to groups, JLMS, 18 (1943), 4-11. $\endgroup$ Apr 5 '13 at 12:32
  • $\begingroup$ I believe what Neumann does is show how to adjoin roots of an element and then he refers to the paper doing it for groups because the same iteration method will work. $\endgroup$ Apr 5 '13 at 17:34

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