I asked this on MSE, but I was told to ask it here because it is a difficult question. Consider the additive magma of the real numbers, $(\mathbb{R};+)$. Does there exist a subset $S$ of the reals which additively generates the reals, but such that no proper subset of $S$ additively generates the reals?

(Alex Kruckman showed that there is not a minimal subset generating the group of reals under addition, but that answers a weaker question; separately, Keith Kearnes pointed out some obstacles to the question as phrased.)

  • $\begingroup$ I've added a link and a bit of context. $\endgroup$ Sep 14, 2022 at 16:58
  • $\begingroup$ I think second paragraph is that he showed there is no minimal subset... $\endgroup$ Sep 14, 2022 at 17:08
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    $\begingroup$ @bof: The answer is negative for (β„š,+). If π‘†βŠ†β„š existed, each element of 𝑆 could be isolated from the others by a maximal subsemigroup (βˆ€π‘ βˆˆπ‘† βˆƒ maximal π‘€β‰€β„š such that π‘†βˆ’π‘€={𝑠}). But the only maximal subsemigroups of (β„š,+) are [0,∞) and (βˆ’βˆž,0]. Altogether this means that (β„š,+) is not finitely generated, so any irredundant generating subset would have to be infinite, but there aren't enough maximal subsemigroups to isolate each element of an infinite subset from the others in that set. $\endgroup$ Sep 15, 2022 at 9:51
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    $\begingroup$ It's a question about semigroups, nothing serious to do with general magmas. $\endgroup$
    – YCor
    Sep 15, 2022 at 9:57
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    $\begingroup$ The argument for nonzero divisible abelian groups appears at the foot of page 462 of 'Abelian groups possessing irreducible systems of generators' by Alexander Soifer, Siberian Mathematical Journal volume 12, pages 461-468 (1971). $\endgroup$ Sep 15, 2022 at 15:51


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