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This is likely the same as a distributive law of monads... https://ncatlab.org/nlab/show/distributive+law on good days i might understand this stuff..

Another less high-brow connection: associativity of matrix multiplication relies on the usual distributivity law..

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Yes: the requirement of commutativity in the "addition" for distributivity to work with a noncommutative "multiplication" is exactly why my current model for code* is based on an (anathema to practicing programmers) unordered choice.

* and also data; with suitable choices of definitions there are very few differences between the left and right hand arguments to application.

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