On 11/16/2006 01:20 AM, Gabriel Dos Reis wrote:
Ralf Hemmecke <[EMAIL PROTECTED]> writes:
| > The definition you gave is it: least fixed point of
| > X |-> 1 + T × X × X
|
| Hmmm, good question. In Aldor-combinat (AC) we deal with combinatorial
| species. They encode actual structures. The corresponding generating
| series G(x) for binary trees given by your X above has to fulfil the
| equation
|
| G(x) = 1 + x * G(x) * G(x) (+)
|
| As a quadratic formula it has at most 2 solutions. And only one of
| those solution is a power series with only non-negative
| coefficients. Since I don't know what it should mean to say "there are
| -5 trees with 3 nodes", it is clear which solution I choose for the
| generating series.
|
| Assuming that I understand a bit of the theory of species, then there
| is only *one* solution to
|
| X = 1 + T * X * X.
|
| We are not yet dealing with "virtual species" which would allow
| negative coefficients in the generating series.
I realize my sentence could be ambiguous: I meant "least fixed point
in the category of continuous partial orders (CPO)."
Well, it seems you have to be even more precise. I still cannot
understand that. You mean X, 1, T are continuous partial orders?
And what does then + and × stand for?
Aha! Good point. Now, I just need you to point me to a good
introductory point on species.
The short one you find at
http://en.wikipedia.org/wiki/Combinatorial_species
and the book I currently like best is given as the second reference on
the above page:
François Bergeron, Gilbert Labelle, Pierre Leroux, Théorie des espèces
et combinatoire des structures arborescentes, LaCIM, Montréal (1994).
English version: Combinatorial Species and Tree-like Structures,
Cambridge University Press (1998).
Ralf
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