When thinking about monoids, the word sum and product aren't distinguishable unless you specific the precise operation, as neither one is a canonical analogy to

When thinking about monoids, the word sum and product aren't distinguishable unless you specific the precise operation, as neither one is a canonical analogy to integer sum/product. Pointwise combination is an unattractive definition, as it ignores a lot of interesting structure in a connected space like real numbers. Indeed, convolution, the common way of combining functions, is defined as a sum of products at each point.