Also if we want to get all math abusey (using graduate level concepts to do basic work): how about something like "moments are a co-product"? The emphasized p

Also if we want to get all math abusey (using graduate level concepts to do basic work): how about something like "moments are a co-product"? The emphasized property in the article is calculations on the raw data can be done on the moments. So all the calculations of interest map through the moment summaries (hence you can think of them as a co-product). Surely that is more exotic than just saying "monoid." And monoid is kind of used up by the crowd that says "free monoid" when all they mean is "set of all strings."