[mcguire]

To the first question, a Semiring has two operations, + and × conventionally (along with ...). That seems to be more than length needs. A Monoid is just an operation and its identity, which should be exactly what you need for length.

For the second question, traditional (+,×,...) is a Semiring for integers; so is (&,|,...) (bitwise boolean operations). How can you define both Semiring implementations without getting them confused?

There is another reply to my comment that I haven't digested yet. Maybe it answers the second.

[mcguire]

"A Monoid is just an operation and its identity, which should be exactly what you need for length."

And I'm wrong about that. You need a zero and a one, right. What would that structure be?