[FOM] 605: Integer and Real Functions

[W.Taylor at math.canterbury.ac.nz]

Quoting Mitchell Spector <spector at alum.mit.edu>:

> W.Taylor at math.canterbury.ac.nz wrote:
>> None of any of this addresses the fact that ordinal ADDITION is also
>> non-commutative - and I know of no other examples of "addition"
>> where this is so.   Are there any?
>>
>> -- Bill Taylor
>
> How about string concatenation?  Even though often notated via
> juxtaposition like multiplication, it's quintessentially additive in
> nature, and it's not commutative.

Well, I'm not sure what is especially additive about it,
OR multiplicative, in itself, either.  I guess the thing is
that many single 2-input operations could be regarded as either
one; BUT that when there are two such operations, and exactly
one distributes over the other, then it's clear which is
"really" the additive and which the multiplicative one.
This is clearly the case with ordinal numbers, for example.

I think it is also the case with strings.  Looking at regular
languages, a very natural context with its connection to finite
automata, and extensions to other automata, we see that there
is a very natural way to define "add" and "multiply".  To add
is merely to union two (sets of) words, and to multiply is merely
to concatenate two (sets of) words (in all possible combinations).
One easily identifies a word with the singleton set of words
that contains only itself.

Even better, one standardly distinguishes the empty set of words,
denoted by 0, and the singleton set of solely the empty word,
denoted by 1.  Then one gets a whole semi-ring with 0 and 1
and all the usual properties, including exactly one pair of
distributive laws, but not the other.

So I would suggest this all adds up to a convincing demonstration
that concatenation of strings is "really" a multiplicative thing.

- Bill Taylor


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