Term for "field-like" algebraic object with infinitely-many "scaled" multiplication" operations parameterized by its elements?

Mathematics Asked on January 1, 2022

The motivation is an object which generalizes the notion of percentages.

Consider the the set $mathbb{R}$ along with the usual binary addition operation $+$ and infinitely-many binary multiplication operations $boldsymbol{cdot}_alpha$ where $alpha in mathbb{R}$ and $aboldsymbol{cdot}_alpha b = (alpha a) cdot b$.

For instance, $50boldsymbol{cdot}_{0.01} 6 = (0.01cdot50) cdot b$

You can easily prove that the set $mathbb{R}$ with $+$ and any fixed $boldsymbol{cdot}_alpha$ is a field. The $alpha = 1$ case corresponds to the usual definition of field $mathbb{R}$ and $alpha=0.01$ corresponds to a field with a "percentage of" as it’s product operation. If, for some twisted reason, one wanted to work with "perpentages" one would consider the $alpha = 0.2$ case.

This structure represents arbitrarily scaled multiplication and satisfies the field axioms for any fixed scalar. Working with this structure was motivated by the following question: "How can I most effectively procrastinate studying for the GRE and punish myself for making stupid math mistakes on easy percentage problems?"

Are there references to comparable objects anywhere/what would you call this type of thing?

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