# Embedding a semigroup into a divisible semigroup

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.

References.

[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.

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