Comment by JadeNB

As Sniffnoy (https://news.ycombinator.com/item?id=9753381) mentions, you are talking about formal power series; but, although certain power series (like `x`) admit closures analogous to those above, not all (like `1`) do, and so I don't think that the formal-power-series ring (not just semiring!) is closed in any useful version of this sense.