[FOM] 605: Integer and Real Functions

[Mitchell Spector]

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.


This is closely connected to other examples.

(1) Concatenation of finite strings on an alphabet of size 1 is isomorphic to addition of natural 
numbers, which is commutative.

But if you increase either the length of the strings or the size of the alphabet, it becomes 
non-commutative:

2) Concatenation of strings of well-ordered (not necessarily finite) length, on an alphabet of size 
1, is isomorphic to addition of ordinals, which is not commutative.

3) Concatenation of finite strings on an alphabet of size 2 or larger is not commutative.


Mitchell Spector