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% $Id: OperadWithBasis.tex,v 1.1.1.1 2004/10/04 17:25:13 nthiery Exp $
% File:		lib/DOMAINS/CATEGORY/DOC/OperadWithBasis.tex
% Authors:	Nicolas M. Thiéry <nthiery@users.sourceforge.net>
% Purpose:	Documentation for the category Cat::OperadWithBasis
% Created:	Tue Dec  9 20:54:18 2003
% License:	LGPL
% Project:	MuPAD-Combinat, http://mupad-combinat.sf.net
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\begin{Mhelppage}{OperadWithBasis}{Nicolas M. Thiéry}
  [ImplState=beta,TestState=beta,DocState=beta]
\Mproofread{nthiery}{28/09/2004}{reasonably complete, but not proofread}

\begin{MInfo}
  \eng{the category of operads over a distinguished basis}
  \de{}
\end{MInfo}

\begin{MIntro}
  \eng{\Mthis{} represents the category of operads over a distinguished basis.}
  \de{}
\end{MIntro}

\begin{MCatConst}
  \MCall{}
\end{MCatConst}

\begin{MCategories}
  \Mup{}
  \Mupc{}{}
\end{MCategories}

\begin{MDetails}
  \eng{A \Mthis{} is an operad (\Mup{Cat::Operad}) with a
    distinguished basis. The multiplication is typically implemented
    by extending by linearity its definition on the basis elements.}

  \de{}
\end{MDetails}

\begin{MDomBasEntries}
  \MDomEntry{oneBasis}
  \eng{If the unit of the algebra is a basis element, the index of
    this basis element can be provided here, instead of defining
    \Mslot{one}.}
\end{MDomBasEntries}

\begin{MMethodDetails}{compositionBasis}
  \infoEng{composition of basis elements (optional)}
  \infoDe{}
  \call{\Mtype{dom::basisIndices} x, \Mtype{dom::basisIndices} y,
    \Mtype{positive integer} i}
  \eng{Returns the composition of the basis elements indexed by \Mpar{x}
    and \Mpar{y} at position \Mpar{i}. }
\end{MMethodDetails}

\begin{MMethodDetails}{ground}
  \infoEng{ground coefficient }
  \infoDe{}
  \call{\Mopt{\Mtype{dom} x}}
  \eng{Returns the ground coefficient of x, that is the coefficient of
    \Mpar{x} on the unit of the algebra.}
  \eng{This only makes sense if the unit of the algebra is a basis
    element, so this method is optional. A default implementation is
    provided when \Mslot{oneBasis} is defined.}
  \de{}
\end{MMethodDetails}

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

\end{Mhelppage}
