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% $Id: FreeOperad.tex,v 1.1.1.1 2004/10/04 17:25:13 nthiery Exp $
% File:		lib/EXPERIMENTAL/DOC/FreeOperad.tex
% Authors:	Nicolas M. Thiéry <nthiery@users.sourceforge.net>
% Purpose:	Documentation for the domain experimental::FreeOperad
% Created:	Tue Dec  9 21:59:27 2003
% License:	LGPL
% Project:	MuPAD-Combinat, http://mupad-combinat.sf.net
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{Mhelppage}{FreeOperad}{Nicolas M. Thiéry}
  [ImplState=beta,TestState=beta,DocState=beta]
%\Mproofread{user}{date}{comment}

\begin{MInfo}
  \eng{domain for the free operad}
  \de{}
\end{MInfo}

\begin{MIntro}
  \eng{\Mthis\Mex{(R)} creates a domain for the free operad on one
    binary generator over the ground ring $R$.}
  
  \eng{\Mthis\Mex{(R, [f,g,\Mseq])}: similar-wise with several binary
    generators. }

  
  \eng{\Mthis\Mex{(R, [f=[Arity=3, Commutative],g=[Arity=1],\Mseq])}:
    similar-wise, but specifying further properties of the
    generators (to be implemented).}
\end{MIntro}

\begin{MExample}{ex-basic}
  \eng{We create the free operad over two binary operators \Mex{f} and
    \Mex{g}:}
  \begin{Mexin}
A := experimental::FreeOperad(Dom::Rational, [f, g]):
  \end{Mexin}
  \begin{Mexout}
  \end{Mexout}
  \eng{The basis of this operad consists of all the possible
    compositions of \Mex{f} and \Mex{g}; here are those compositions
    which are ternary:}
  \begin{Mexin}
A::basis(3)
  \end{Mexin}
  \begin{Mexout}
--  f  ,  f  ,  f  ,  f  ,  f  ,  f  ,  f  ,  f  ,  f  ,  f  ,
|  / \   / \   / \   / \   / \   / \   / \   / \   / \   / \
|  1 f   1 f   1 g   1 g   2 f   2 f   2 g   2 g   3 f   3 f
|   / \   / \   / \   / \   / \   / \   / \   / \   / \   / \
--  2 3   3 2   2 3   3 2   1 3   3 1   1 3   3 1   1 2   2 1

    f  ,  f  ,   f ,   f ,   f ,   f ,   f ,   f ,   f ,   f ,
   / \   / \    / \   / \   / \   / \   / \   / \   / \   / \
   3 g   3 g    f 3   f 3   g 3   g 3   f 2   f 2   g 2   g 2
    / \   / \  / \   / \   / \   / \   / \   / \   / \   / \
    1 2   2 1  1 2   2 1   1 2   2 1   1 3   3 1   1 3   3 1

     f ,   f ,   f ,   f ,  g  ,  g  ,  g  ,  g  ,  g  ,  g  ,
    / \   / \   / \   / \  / \   / \   / \   / \   / \   / \
    f 1   f 1   g 1   g 1  1 f   1 f   1 g   1 g   2 f   2 f
   / \   / \   / \   / \    / \   / \   / \   / \   / \   / \
   2 3   3 2   2 3   3 2    2 3   3 2   2 3   3 2   1 3   3 1

    g  ,  g  ,  g  ,  g  ,  g  ,  g  ,   g ,   g ,   g ,   g ,
   / \   / \   / \   / \   / \   / \    / \   / \   / \   / \
   2 g   2 g   3 f   3 f   3 g   3 g    f 3   f 3   g 3   g 3
    / \   / \   / \   / \   / \   / \  / \   / \   / \   / \
    1 3   3 1   1 2   2 1   1 2   2 1  1 2   2 1   1 2   2 1

     g ,   g ,   g ,   g ,   g ,   g ,   g ,   g  --
    / \   / \   / \   / \   / \   / \   / \   / \  |
    f 2   f 2   g 2   g 2   f 1   f 1   g 1   g 1  |
   / \   / \   / \   / \   / \   / \   / \   / \   |
   1 3   3 1   1 3   3 1   2 3   3 2   2 3   3 2  --
  \end{Mexout}
\end{MExample}

\begin{MChanges}
  \version{3.1}
  \eng[New domain]{\Mthis{} is a new domain} %
\end{MChanges}

\end{Mhelppage}
