Technically yeah but there is a natural isomorphism between struct { int A; float B; } and struct { float B; int A; } so if you're willing to do everything up

Technically yeah but there is a natural isomorphism between struct { int A; float B; } and struct { float B; int A; } so if you're willing to do everything up to isomorphism (which is usually completely fine and standard) then you get a commutative semiring. You have to work up to isomorphism anyway even to get a noncomutative semiring because technically the multiplication isn't exactly associative only associative up to an isomorphism.