Ivory: Adjoins And Quotients

We’ve seen how sums and products determine the structure of Ivory expressions. Now we’ll define the elements of these sums and products.

Adjoining

Each level of the Ivory tower includes the levels above. In algebraic terms, each is an extension of the level above, with extra values adjoined. The result of this “adjoining” is a level containing:

All of the values of the level above.
The value we adjoined.
All sums and products of those values, AKA the closure of addition and multiplication.

The power of this approach comes from taking the closure. For example, we can’t turn natural into integer by appending each negative number separately. Yet if we allow arbitrary products, we only need to append (- 1) directly: every other negative integer will arise as the product of (- 1) with an existing natural.

The following levels are extensions in this way:

zero adjoins the value 0 to void.
natural adjoins the value 1 to zero.
integer adjoins the value (- 1) to natural.
dyadic adjoins the value (^ 2 (- 1)) to integer.
sexagesimal adjoins the values (^ 3 (- 1)) and (^ 5 (- 1)) to dyadic.
geometric adjoins values of the form (i n), (h n) and (d n) to scalar.
`

On its own, these

New element symbols are adjoined which introduces new symbols; and quotients.