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Let $\mathfrak B$ be a Hamel basis for $\mathbb R$ over $\mathbb Q$. Then the set $\mathfrak{M} = \left\{ e^b | b \in \mathfrak{B} \right\}$ has the property that any $r\in\mathbb R^+$ can be written uniquely as a product of the form $$\prod_{v \in \mathfrak M} v^{c_v} $$ where each $c_v \in \mathbb Q$ and only finitely many values of $c_v$ are different from $0$. Conversely, any set with this property can be obtained by exponentiating a Hamel basis -- so in a sense the two ideas are really the same thing.

Is there a name for this "multiplicative analogue" of a Hamel basis?

mweiss
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    I doubt it has a special name. It's just a Hamel basis for the vector space $\mathbb{R}^+$ over $\mathbb{Q}$ (with multiplication and exponentiation as the operations). – Eric Wofsey Aug 30 '19 at 20:16
  • @EricWofsey that's a very good point. – mweiss Aug 30 '19 at 20:21

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