meta numbers

Let's define the Meta type. In grammar terms, Meta -> [ Meta* ].

Meta is a number type. Every meta number is an mset of meta numbers.

The sum of a meta number is the number constructed by taking each number in each number in the original number.

The product of a meta number is the number constructed by taking each sum of numbers taken one from each number in the original number.

Suppose for example that a b c d e f are meta numbers. Then:
sum [] = []
sum [a] = a
sum [a []] = a
sum [[a b] [c d e] [f]] = [a b c d e f]
prod [] = [[]]
prod [a] = a
prod [a [[]]] = a
prod [[a b] [c d e] [f]] = [(sum [a c f]) (sum [a d f]) (sum [a e f]) (sum [b c f]) (sum [b d f]) (sum [b e f])]

Here's how we connect to existing notation:
0 = []
1 = [[]]
2 = [[][]]
3 = [[][][]]

Please convince yourself by using the definitions above that:
prod [0 a] = sum [] = 0
prod [1 a] = sum [a] = a
prod [2 a] = sum [a a]
prod [3 a] = sum [a a a].

This is exciting!