# 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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