Canonical forms

Using the following lemma, it is an easy exercise to see that 0 and 1 have only one signed-digit binary representation, that each binary rational 49#49 has countably infinite representations, and that every other number has uncountably many representations. Let 50#50 be the equivalence relation on numerals induced by the denotation function:

51#51 iff 52#52.

Lemma 8.1   The following identities hold for all 53#53 and 54#54:

55#55,          56#56.

But each number has two canonically associated representations. Recall that

57#57 holds in the lexicographical order iff there is an integer k such that 58#58 and 15#15 for each i < k.

Proposition 8.2   Every number is denoted by a smallest and by a largest signed-digit numeral in the lexicographical order.

59#59
However, there are no effectively determinable canonical forms:

Proposition 8.3   There is no continuous, denotation-preserving idempotent map 60#60 such that 61#61 is a singleton for each 62#62.

63#63