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[Axiom-developer] Re: FW: data structure vs. mathematical structure

From: Ralf Hemmecke
Subject: [Axiom-developer] Re: FW: data structure vs. mathematical structure
Date: Thu, 16 Nov 2006 01:39:08 +0100
User-agent: Thunderbird (X11/20061025)

On 11/16/2006 01:20 AM, Gabriel Dos Reis wrote:
Ralf Hemmecke <address@hidden> 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

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).


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