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.)

nominimal subset... $\endgroup$7more comments