Can, in a very general sense, a sum be viewed as combination without interaction, and a product as a combination with interaction/interdependence? In all "kinds" of addition, the result is based on some "independent" combination of the inputs, while a product involves almost a "fusion"/"dependent" combination of the inputs.
Sum Examples: arithmetic addition (figuratively: just arrange line segments one after another), set union (just take all elements from both sets), logical OR (if any input is true, result is true).
Product Examples: arithmetic multiplication (assuming integral interpretation, for each unit of the first place units of a quantity equal to the second), set intersection (elements must be in BOTH), logical each (EVERY input must be true).
I just realized that the product must always be computed "globally" with knowledge of all the inputs, while the sum can be computed by "locally" by considering only one input at a time; i.e., the general algorithm for summation would process each input alone without any memory of any of the other inputs, while the general algorithm for multiplication would have to process the inputs as a group - each part of one input would be processed in relation to each part of the other input.
So is this general interpretation of independent vs. dependent combination corresponding to sum, product respectively correct?
I only have a minor in math FYI.