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In semi-response to Organic's post I thought I'd half take up one of his challenges:
Let S = NxT be the product of the natural numbers, N, with the set of all rooted finite trees (or directed graphs satisfying the obvious conditions), embedded in the plane, with the natural ordering on the branches/leaves.
Let C be the subset of all possible { (n,t) | n in N, t a tree with exactly n edges}
We define an operation + on elements of c:
(n,t)+(m,r) = (n+m,s) where s is the tree obtained by gluing the tree r onto the end of the first leaf.
The subset (n,t) with t the trivial tree with 1 leaf and n edges, forms a copy of N under addition.
We define * to be (n,t)*(m,s) by
(((...(((n,t)+(n,t))+(n,t))+...)+(n,t))
where there are m-1 addition signs.
again (n,t) with t the tree with n edges and 1 leaf, is a subset that forms a copy of N under multiplication.
neither + nor * are in general commutative, and I doubt they are associative either, but I can't be bothered to check, they are both well defined binary operations from CxC to C.
Now shall I claim that C is the new non-commutative natural numbers or not?
Let S = NxT be the product of the natural numbers, N, with the set of all rooted finite trees (or directed graphs satisfying the obvious conditions), embedded in the plane, with the natural ordering on the branches/leaves.
Let C be the subset of all possible { (n,t) | n in N, t a tree with exactly n edges}
We define an operation + on elements of c:
(n,t)+(m,r) = (n+m,s) where s is the tree obtained by gluing the tree r onto the end of the first leaf.
The subset (n,t) with t the trivial tree with 1 leaf and n edges, forms a copy of N under addition.
We define * to be (n,t)*(m,s) by
(((...(((n,t)+(n,t))+(n,t))+...)+(n,t))
where there are m-1 addition signs.
again (n,t) with t the tree with n edges and 1 leaf, is a subset that forms a copy of N under multiplication.
neither + nor * are in general commutative, and I doubt they are associative either, but I can't be bothered to check, they are both well defined binary operations from CxC to C.
Now shall I claim that C is the new non-commutative natural numbers or not?