Quelle smallantimagmas.tex Sprache: Latech

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\begin{document}

\logpage{[ 0, 0, 0 ]}
\begin{titlepage}
\mbox{}\vfill

\begin{center}{\maintitlesize \textbf{ smallantimagmas \mbox{}}}\\
\vfill

\hypersetup{pdftitle= smallantimagmas }
\markright{\scriptsize \mbox{}\hfill  smallantimagmas  \hfill\mbox{}}
{\Huge \textbf{ A library of antiassociative magmas of small order \mbox{}}}\\
\vfill

{\Huge  0.5.1 \mbox{}}\\[1cm]
{ 26 September 2025 \mbox{}}\\[1cm]
\mbox{}\\[2cm]
{\Large \textbf{ Kamil Zabielski\\
  \mbox{}}}\\
{\Large \textbf{ Ryszard Mazurek\\
  \mbox{}}}\\
\hypersetup{pdfauthor= Kamil Zabielski\\
  ;  Ryszard Mazurek\\
  }
\end{center}\vfill

\mbox{}\\
{\mbox{}\\
\small \noindent \textbf{ Kamil Zabielski\\
  }\\
  Address: \begin{minipage}[t]{8cm}\noindent
Department of Theoretical Computer Science\\
Bia{\l}ystok University of Technology\\
Wiejska 45A\\
15\texttt{\symbol{45}}325 Bia{\l}ystok\\
Poland\\
\end{minipage}
}\\
{\mbox{}\\
\small \noindent \textbf{ Ryszard Mazurek\\
  }\\
  Address: \begin{minipage}[t]{8cm}\noindent
Department of Theoretical Computer Science\\
Bia{\l}ystok University of Technology\\
Wiejska 45A\\
15\texttt{\symbol{45}}325 Bia{\l}ystok\\
Poland\\
\end{minipage}
}\\
\end{titlepage}

\newpage\setcounter{page}{2}
{\small
\section*{Copyright}
\logpage{[ 0, 0, 1 ]}
\index{License} {\copyright} 2024 by Kamil Zabielski

\textsf{smallantimagmas} package is free software; you can redistribute it and/or modify it under the
terms of the \href{http://www.fsf.org/licenses/gpl.html} {GNU General Public License} as published by the Free Software Foundation; either version 2 of the License,
or (at your option) any later version. \mbox{}}\\[1cm]
{\small
\section*{Acknowledgements}
\logpage{[ 0, 0, 2 ]}
We appreciate very much all past and future comments, suggestions and
contributions to this package and its documentation provided by \textsf{GAP} users and developers. \mbox{}}\\[1cm]
\newpage

\def\contentsname{Contents\logpage{[ 0, 0, 3 ]}}

\tableofcontents
\newpage


\chapter{\textcolor{Chapter }{smallantimagmas automatic generated documentation}}\label{Chapter_smallantimagmas_automatic_generated_documentation}
\logpage{[ 1, 0, 0 ]}
\hyperdef{L}{X7EB7BB817CC15891}{}
{

\section{\textcolor{Chapter }{smallantimagmas automatic generated documentation of properties}}\label{Chapter_smallantimagmas_automatic_generated_documentation_Section_smallantimagmas_automatic_generated_documentation_of_properties}
\logpage{[ 1, 1, 0 ]}
\hyperdef{L}{X78F902077A6714C4}{}
{


\subsection{\textcolor{Chapter }{IsAntiassociative (for IsMagma)}}
\logpage{[ 1, 1, 1 ]}\nobreak
\hyperdef{L}{X7B8537C57970AE37}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsAntiassociative({\mdseries\slshape M})\index{IsAntiassociative@\texttt{IsAntiassociative}!for IsMagma}
\label{IsAntiassociative:for IsMagma}
}\hfill{\scriptsize (property)}}\\
\textbf{\indent Returns:\ }
\texttt{true} or \texttt{false}

identifies whether magma \mbox{\texttt{\mdseries\slshape M}} is antiassociative.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@IsAntiassociative(OneSmallGroup(16));|
  false
  !gapprompt@gap>| !gapinput@IsAntiassociative(OneSmallAntimagma(2));|
  true
  !gapprompt@gap>| !gapinput@IsAntiassociative(OneSmallAntimagma(3));|
  true
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{IsLeftCyclic (for IsMagma)}}
\logpage{[ 1, 1, 2 ]}\nobreak
\hyperdef{L}{X8392BFEE78284DE1}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsLeftCyclic({\mdseries\slshape M})\index{IsLeftCyclic@\texttt{IsLeftCyclic}!for IsMagma}
\label{IsLeftCyclic:for IsMagma}
}\hfill{\scriptsize (property)}}\\
\textbf{\indent Returns:\ }
\texttt{true} or \texttt{false}

if magma is left cyclic \mbox{\texttt{\mdseries\slshape m}}. }

\subsection{\textcolor{Chapter }{IsRightCyclic (for IsMagma)}}
\logpage{[ 1, 1, 3 ]}\nobreak
\hyperdef{L}{X7BE3D1C0801AE7EA}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsRightCyclic({\mdseries\slshape M})\index{IsRightCyclic@\texttt{IsRightCyclic}!for IsMagma}
\label{IsRightCyclic:for IsMagma}
}\hfill{\scriptsize (property)}}\\
\textbf{\indent Returns:\ }
\texttt{true} or \texttt{false}

if magma is left cyclic \mbox{\texttt{\mdseries\slshape m}}. }

\subsection{\textcolor{Chapter }{IsLeftDistributive (for IsMagma)}}
\logpage{[ 1, 1, 4 ]}\nobreak
\hyperdef{L}{X7D377A068350EE46}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsLeftDistributive({\mdseries\slshape M})\index{IsLeftDistributive@\texttt{IsLeftDistributive}!for IsMagma}
\label{IsLeftDistributive:for IsMagma}
}\hfill{\scriptsize (property)}}\\
\textbf{\indent Returns:\ }
\texttt{true} or \texttt{false}

if magma is left distributive \mbox{\texttt{\mdseries\slshape m}}.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@List(AllSmallAntimagmas(3), M -> IsLeftDistributive(M) );|
  [ true, false, false, false, false, false, false, false, false, true ]
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{IsRightDistributive (for IsMagma)}}
\logpage{[ 1, 1, 5 ]}\nobreak
\hyperdef{L}{X856E2BBA7B87F209}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsRightDistributive({\mdseries\slshape M})\index{IsRightDistributive@\texttt{IsRightDistributive}!for IsMagma}
\label{IsRightDistributive:for IsMagma}
}\hfill{\scriptsize (property)}}\\
\textbf{\indent Returns:\ }
\texttt{true} or \texttt{false}

if magma is right distributive \mbox{\texttt{\mdseries\slshape m}}.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@List(AllSmallAntimagmas(3), M -> IsRightDistributive(M) );|
  [ false, false, false, false, true, false, false, false, true, false ]
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{IsLeftCancellative (for IsMagma)}}
\logpage{[ 1, 1, 6 ]}\nobreak
\hyperdef{L}{X7D994F7382127E77}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsLeftCancellative({\mdseries\slshape M})\index{IsLeftCancellative@\texttt{IsLeftCancellative}!for IsMagma}
\label{IsLeftCancellative:for IsMagma}
}\hfill{\scriptsize (property)}}\\
\textbf{\indent Returns:\ }
\texttt{true} or \texttt{false}

if magma is left cancellative \mbox{\texttt{\mdseries\slshape m}}.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@M := SmallAntimagma(2, 1);|
  <magma with 2 generators>
  !gapprompt@gap>| !gapinput@Display( MultiplicationTable(M) );|
  [ [  2,  1 ],
    [  2,  1 ] ]
  !gapprompt@gap>| !gapinput@IsRightCancellative(M);|
  false
  !gapprompt@gap>| !gapinput@IsLeftCancellative(M);|
  true
  !gapprompt@gap>| !gapinput@List(AllSmallAntimagmas(2), M -> IsLeftCancellative(M));|
  [ true, false ]
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{IsRightCancellative (for IsMagma)}}
\logpage{[ 1, 1, 7 ]}\nobreak
\hyperdef{L}{X85C01ECF83CD5484}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsRightCancellative({\mdseries\slshape M})\index{IsRightCancellative@\texttt{IsRightCancellative}!for IsMagma}
\label{IsRightCancellative:for IsMagma}
}\hfill{\scriptsize (property)}}\\
\textbf{\indent Returns:\ }
\texttt{true} or \texttt{false}

if magma is right cancellative \mbox{\texttt{\mdseries\slshape m}}.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@List(AllSmallAntimagmas(2), M -> IsRightCancellative(M));|
  [ false, true ]
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{IsCancellative (for IsMagma)}}
\logpage{[ 1, 1, 8 ]}\nobreak
\hyperdef{L}{X7E58570787350F43}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsCancellative({\mdseries\slshape M})\index{IsCancellative@\texttt{IsCancellative}!for IsMagma}
\label{IsCancellative:for IsMagma}
}\hfill{\scriptsize (property)}}\\
\textbf{\indent Returns:\ }
\texttt{true} or \texttt{false}

if magma is cancellative \mbox{\texttt{\mdseries\slshape m}}.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@List(AllSmallAntimagmas(2), M -> IsCancellative(M));|
  [ false, false ]
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{IsLeftFPFInducted (for IsMagma)}}
\logpage{[ 1, 1, 9 ]}\nobreak
\hyperdef{L}{X7965D94579F05D6F}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsLeftFPFInducted({\mdseries\slshape M})\index{IsLeftFPFInducted@\texttt{IsLeftFPFInducted}!for IsMagma}
\label{IsLeftFPFInducted:for IsMagma}
}\hfill{\scriptsize (property)}}\\
\textbf{\indent Returns:\ }
\texttt{true} or \texttt{false}

is a left\texttt{\symbol{45}}hand sided fixed\texttt{\symbol{45}}point free
inducted \mbox{\texttt{\mdseries\slshape m}}.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@Display( MultiplicationTable( SmallAntimagma(2, 2) ) );|
  [ [  2,  2 ],
    [  1,  1 ] ]
  !gapprompt@gap>| !gapinput@IsLeftFPFInducted( SmallAntimagma(2, 2) ); |
  true
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{IsRightFPFInducted (for IsMagma)}}
\logpage{[ 1, 1, 10 ]}\nobreak
\hyperdef{L}{X866768408460130D}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsRightFPFInducted({\mdseries\slshape M})\index{IsRightFPFInducted@\texttt{IsRightFPFInducted}!for IsMagma}
\label{IsRightFPFInducted:for IsMagma}
}\hfill{\scriptsize (property)}}\\
\textbf{\indent Returns:\ }
\texttt{true} or \texttt{false}

is a right\texttt{\symbol{45}}hand sided fixed\texttt{\symbol{45}}point free
inducted \mbox{\texttt{\mdseries\slshape m}}.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@Display( MultiplicationTable( SmallAntimagma(2, 1) ) );|
  [ [  2,  1 ],
    [  2,  1 ] ]
  !gapprompt@gap>| !gapinput@IsRightFPFInducted( SmallAntimagma(2, 1) );            |
  true
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{IsLeftDerangementInducted (for IsMagma)}}
\logpage{[ 1, 1, 11 ]}\nobreak
\hyperdef{L}{X7B55200A7A734759}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsLeftDerangementInducted({\mdseries\slshape M})\index{IsLeftDerangementInducted@\texttt{IsLeftDerangementInducted}!for IsMagma}
\label{IsLeftDerangementInducted:for IsMagma}
}\hfill{\scriptsize (property)}}\\
\textbf{\indent Returns:\ }
\texttt{true} or \texttt{false}

is a left\texttt{\symbol{45}}hand sided derangment inducted \mbox{\texttt{\mdseries\slshape m}}.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@M := SmallAntimagma(2, 2);|
  <magma with 2 generators>
  !gapprompt@gap>| !gapinput@IsLeftFPFInducted(M);|
  true
  !gapprompt@gap>| !gapinput@IsRightFPFInducted(M);|
  false
  !gapprompt@gap>| !gapinput@IsRightDerangementInducted(M);|
  false
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{IsRightDerangementInducted (for IsMagma)}}
\logpage{[ 1, 1, 12 ]}\nobreak
\hyperdef{L}{X7E0D0CD780550E81}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsRightDerangementInducted({\mdseries\slshape M})\index{IsRightDerangementInducted@\texttt{IsRightDerangementInducted}!for IsMagma}
\label{IsRightDerangementInducted:for IsMagma}
}\hfill{\scriptsize (property)}}\\
\textbf{\indent Returns:\ }
\texttt{true} or \texttt{false}

is a right\texttt{\symbol{45}}hand sided derangment inducted \mbox{\texttt{\mdseries\slshape m}}.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@M := SmallAntimagma(2, 1);|
  <magma with 2 generators>
  !gapprompt@gap>| !gapinput@IsLeftFPFInducted(M);|
  false
  !gapprompt@gap>| !gapinput@IsRightFPFInducted(M);|
  true
  !gapprompt@gap>| !gapinput@IsRightDerangementInducted(M);|
  true
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{IsLeftAlternative (for IsMagma)}}
\logpage{[ 1, 1, 13 ]}\nobreak
\hyperdef{L}{X83147660820830E8}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsLeftAlternative({\mdseries\slshape M})\index{IsLeftAlternative@\texttt{IsLeftAlternative}!for IsMagma}
\label{IsLeftAlternative:for IsMagma}
}\hfill{\scriptsize (property)}}\\
\textbf{\indent Returns:\ }
\texttt{true} or \texttt{false}

is a left\texttt{\symbol{45}}alternatve magma \mbox{\texttt{\mdseries\slshape M}}.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{IsRightAlternative (for IsMagma)}}
\logpage{[ 1, 1, 14 ]}\nobreak
\hyperdef{L}{X7C16C7657F8B5A61}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsRightAlternative({\mdseries\slshape M})\index{IsRightAlternative@\texttt{IsRightAlternative}!for IsMagma}
\label{IsRightAlternative:for IsMagma}
}\hfill{\scriptsize (property)}}\\
\textbf{\indent Returns:\ }
\texttt{true} or \texttt{false}

is a right\texttt{\symbol{45}}alternatve magma \mbox{\texttt{\mdseries\slshape M}}.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
\end{Verbatim}

}

}

\section{\textcolor{Chapter }{smallantimagmas automatic generated documentation of attributes}}\label{Chapter_smallantimagmas_automatic_generated_documentation_Section_smallantimagmas_automatic_generated_documentation_of_attributes}
\logpage{[ 1, 2, 0 ]}
\hyperdef{L}{X839C88FC81394B14}{}
{


\subsection{\textcolor{Chapter }{AssociativityIndex (for IsMagma)}}
\logpage{[ 1, 2, 1 ]}\nobreak
\hyperdef{L}{X809A477180F2A0F6}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{AssociativityIndex({\mdseries\slshape M})\index{AssociativityIndex@\texttt{AssociativityIndex}!for IsMagma}
\label{AssociativityIndex:for IsMagma}
}\hfill{\scriptsize (attribute)}}\\

identifies associativity index of \mbox{\texttt{\mdseries\slshape M}}.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@OneSmallAntimagma(2);|
  <magma with 2 generators>
  !gapprompt@gap>| !gapinput@AssociativityIndex(OneSmallAntimagma(2));|
  0
  !gapprompt@gap>| !gapinput@OneSmallGroup(4);|
  <pc group of size 4 with 2 generators>
  !gapprompt@gap>| !gapinput@AssociativityIndex(OneSmallGroup(4));|
  64
  !gapprompt@gap>| !gapinput@AssociativityIndex(OneSmallGroup(4)) = 4 ^ 3;|
  true
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{DiagonalOfMultiplicationTable (for IsMagma)}}
\logpage{[ 1, 2, 2 ]}\nobreak
\hyperdef{L}{X802DA9B083E7F4A3}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{DiagonalOfMultiplicationTable({\mdseries\slshape M})\index{DiagonalOfMultiplicationTable@\texttt{DiagonalOfMultiplicationTable}!for IsMagma}
\label{DiagonalOfMultiplicationTable:for IsMagma}
}\hfill{\scriptsize (attribute)}}\\

computes diaognal of multiplication table of \mbox{\texttt{\mdseries\slshape M}}.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@List(AllSmallAntimagmas(3), M -> DiagonalOfMultiplicationTable((M)));                |
  [ [ 2, 1, 1 ], [ 2, 1, 1 ],
    [ 2, 3, 2 ], [ 2, 1, 1 ],
    [ 2, 1, 1 ], [ 2, 1, 2 ],
    [ 2, 3, 2 ], [ 2, 1, 2 ],
    [ 2, 3, 1 ], [ 2, 3, 1 ]
  ]
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{CommutativityIndex (for IsMagma)}}
\logpage{[ 1, 2, 3 ]}\nobreak
\hyperdef{L}{X87FED7267EB0A068}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{CommutativityIndex({\mdseries\slshape M})\index{CommutativityIndex@\texttt{CommutativityIndex}!for IsMagma}
\label{CommutativityIndex:for IsMagma}
}\hfill{\scriptsize (attribute)}}\\

identifies commutativity index of \mbox{\texttt{\mdseries\slshape M}}.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]

\end{Verbatim}

}

\subsection{\textcolor{Chapter }{AnticommutativityIndex (for IsMagma)}}
\logpage{[ 1, 2, 4 ]}\nobreak
\hyperdef{L}{X7D32B8F780294AEC}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{AnticommutativityIndex({\mdseries\slshape M})\index{AnticommutativityIndex@\texttt{AnticommutativityIndex}!for IsMagma}
\label{AnticommutativityIndex:for IsMagma}
}\hfill{\scriptsize (attribute)}}\\

calculates anticommutativity index of \mbox{\texttt{\mdseries\slshape M}}.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{SquaresIndex (for IsMagma)}}
\logpage{[ 1, 2, 5 ]}\nobreak
\hyperdef{L}{X85904967802C27B4}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{SquaresIndex({\mdseries\slshape M})\index{SquaresIndex@\texttt{SquaresIndex}!for IsMagma}
\label{SquaresIndex:for IsMagma}
}\hfill{\scriptsize (attribute)}}\\

computes squares index of \mbox{\texttt{\mdseries\slshape M}} so the order of $\left\{ m^2 | m \in M \right\}$.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@List(AllSmallAntimagmas(2), M -> List(M, m -> m * m) );                |
  [ [ m2, m1 ], [ m2, m1 ] ]
  !gapprompt@gap>| !gapinput@List(AllSmallAntimagmas(2), M -> SquaresIndex(M ));|
  [ 2, 2 ]
  !gapprompt@gap>| !gapinput@List(AllSmallAntimagmas(3), M -> SquaresIndex(M ));|
  [ 2, 2, 2, 2, 2, 2, 2, 2, 3, 3 ]
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{IdSmallAntimagma (for IsMagma)}}
\logpage{[ 1, 2, 6 ]}\nobreak
\hyperdef{L}{X783C01BD7B931A12}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IdSmallAntimagma({\mdseries\slshape M})\index{IdSmallAntimagma@\texttt{IdSmallAntimagma}!for IsMagma}
\label{IdSmallAntimagma:for IsMagma}
}\hfill{\scriptsize (attribute)}}\\

identifies class of antiassociative magma \mbox{\texttt{\mdseries\slshape M}}.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@IsAntiassociative(OneSmallGroup(16));|
  false
  !gapprompt@gap>| !gapinput@IsAntiassociative(OneSmallAntimagma(2));|
  true
  !gapprompt@gap>| !gapinput@IsAntiassociative(OneSmallAntimagma(3));|
  true
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{LeftOrder (for IsExtLElement)}}
\logpage{[ 1, 2, 7 ]}\nobreak
\hyperdef{L}{X78BF96667D805695}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{LeftOrder({\mdseries\slshape [m]})\index{LeftOrder@\texttt{LeftOrder}!for IsExtLElement}
\label{LeftOrder:for IsExtLElement}
}\hfill{\scriptsize (attribute)}}\\

returns a left order of element \mbox{\texttt{\mdseries\slshape m}}. }

\subsection{\textcolor{Chapter }{RightOrder (for IsExtRElement)}}
\logpage{[ 1, 2, 8 ]}\nobreak
\hyperdef{L}{X850894387AD0E717}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{RightOrder({\mdseries\slshape [m]})\index{RightOrder@\texttt{RightOrder}!for IsExtRElement}
\label{RightOrder:for IsExtRElement}
}\hfill{\scriptsize (attribute)}}\\

returns a right order of element \mbox{\texttt{\mdseries\slshape m}}. }

\subsection{\textcolor{Chapter }{LeftOrdersOfElements (for IsMagma)}}
\logpage{[ 1, 2, 9 ]}\nobreak
\hyperdef{L}{X7CDCE61679BFD1F7}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{LeftOrdersOfElements({\mdseries\slshape [m]})\index{LeftOrdersOfElements@\texttt{LeftOrdersOfElements}!for IsMagma}
\label{LeftOrdersOfElements:for IsMagma}
}\hfill{\scriptsize (attribute)}}\\

returns a left order of element \mbox{\texttt{\mdseries\slshape m}}. }

\subsection{\textcolor{Chapter }{RightOrdersOfElements (for IsMagma)}}
\logpage{[ 1, 2, 10 ]}\nobreak
\hyperdef{L}{X84D8A5D88790F870}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{RightOrdersOfElements({\mdseries\slshape [m]})\index{RightOrdersOfElements@\texttt{RightOrdersOfElements}!for IsMagma}
\label{RightOrdersOfElements:for IsMagma}
}\hfill{\scriptsize (attribute)}}\\

returns a left order of element \mbox{\texttt{\mdseries\slshape m}}. }

}

\section{\textcolor{Chapter }{smallantimagmas automatic generated documentation of global functions}}\label{Chapter_smallantimagmas_automatic_generated_documentation_Section_smallantimagmas_automatic_generated_documentation_of_global_functions}
\logpage{[ 1, 3, 0 ]}
\hyperdef{L}{X7A4C95527CCD8524}{}
{


\subsection{\textcolor{Chapter }{AllSubmagmas}}
\logpage{[ 1, 3, 1 ]}\nobreak
\hyperdef{L}{X8011AFA47C662D53}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{AllSubmagmas({\mdseries\slshape M})\index{AllSubmagmas@\texttt{AllSubmagmas}}
\label{AllSubmagmas}
}\hfill{\scriptsize (function)}}\\

builds a collection of non\texttt{\symbol{45}}isomorphic submagmas of \mbox{\texttt{\mdseries\slshape M}}.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@AllSmallAntimagmas(2);|
  [ <magma with 2 generators>, <magma with 2 generators> ]
  !gapprompt@gap>| !gapinput@List(AllSmallAntimagmas(2), M -> AllSubmagmas(M));|
  [ [ <magma with 1 generator> ], [ <magma with 1 generator> ] ]
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{MagmaIsomorphismInvariantsMatch}}
\logpage{[ 1, 3, 2 ]}\nobreak
\hyperdef{L}{X7BA7BE44861A8C51}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{MagmaIsomorphismInvariantsMatch({\mdseries\slshape M})\index{MagmaIsomorphismInvariantsMatch@\texttt{MagmaIsomorphismInvariantsMatch}}
\label{MagmaIsomorphismInvariantsMatch}
}\hfill{\scriptsize (function)}}\\

computes isomorphism invariants of \mbox{\texttt{\mdseries\slshape M}}. }

\subsection{\textcolor{Chapter }{IsMagmaIsomorphic}}
\logpage{[ 1, 3, 3 ]}\nobreak
\hyperdef{L}{X7F7EAF23807EE984}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsMagmaIsomorphic({\mdseries\slshape M, N})\index{IsMagmaIsomorphic@\texttt{IsMagmaIsomorphic}}
\label{IsMagmaIsomorphic}
}\hfill{\scriptsize (function)}}\\

identifies whether magmas \mbox{\texttt{\mdseries\slshape M}}, \mbox{\texttt{\mdseries\slshape N}} are isomorphic.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@M := SmallAntimagma(2, 1);|
  <magma with 2 generators>
  !gapprompt@gap>| !gapinput@N := SmallAntimagma(2, 2);|
  <magma with 2 generators>
  !gapprompt@gap>| !gapinput@T := MagmaByMultiplicationTable([ [2, 1], [2, 1] ]);|
  <magma with 2 generators>
  !gapprompt@gap>| !gapinput@IsMagmaIsomorphic(M, M);|
  true
  !gapprompt@gap>| !gapinput@IsMagmaIsomorphic(M, T);|
  true
  !gapprompt@gap>| !gapinput@IsMagmaIsomorphic(M, N);|
  false
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{IsMagmaAntiisomorphic}}
\logpage{[ 1, 3, 4 ]}\nobreak
\hyperdef{L}{X807BD59180325341}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsMagmaAntiisomorphic({\mdseries\slshape [M, N]})\index{IsMagmaAntiisomorphic@\texttt{IsMagmaAntiisomorphic}}
\label{IsMagmaAntiisomorphic}
}\hfill{\scriptsize (function)}}\\

identifies whether magmas \mbox{\texttt{\mdseries\slshape M}}, \mbox{\texttt{\mdseries\slshape N}} are antiisomorphic.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@N := SmallAntimagma(2, 1);|
  <magma with 2 generators>
  !gapprompt@gap>| !gapinput@M := SmallAntimagma(2, 1);|
  <magma with 2 generators>
  !gapprompt@gap>| !gapinput@N := SmallAntimagma(2, 2);|
  <magma with 2 generators>
  !gapprompt@gap>| !gapinput@IsMagmaAntiisomorphic(M, M);|
  false
  !gapprompt@gap>| !gapinput@IsMagmaAntiisomorphic(M, N);|
  true
  !gapprompt@gap>| !gapinput@IsMagmaAntiisomorphic(M, TransposedMagma(M));|
  true
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{TransposedMagma}}
\logpage{[ 1, 3, 5 ]}\nobreak
\hyperdef{L}{X79BF9EB48562EF6C}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{TransposedMagma({\mdseries\slshape [M]})\index{TransposedMagma@\texttt{TransposedMagma}}
\label{TransposedMagma}
}\hfill{\scriptsize (function)}}\\

generates transposed magma \mbox{\texttt{\mdseries\slshape M}}.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@M := SmallAntimagma(2, 1);|
  <magma with 2 generators>
  !gapprompt@gap>| !gapinput@IsMagmaAntiisomorphic(M, TransposedMagma(M));|
  true
  !gapprompt@gap>| !gapinput@IsMagmaIsomorphic(M, TransposedMagma(TransposedMagma(M)));|
  true
  !gapprompt@gap>| !gapinput@M := SmallAntimagma(2, 1);|
  <magma with 2 generators>
  !gapprompt@gap>| !gapinput@Display(MultiplicationTable(M));|
  [ [  2,  1 ],
    [  2,  1 ] ]
  !gapprompt@gap>| !gapinput@Display(MultiplicationTable(TransposedMagma(M)));|
  [ [  2,  2 ],
    [  1,  1 ] ]
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{LeftPower}}
\logpage{[ 1, 3, 6 ]}\nobreak
\hyperdef{L}{X7FD21A58851687DF}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{LeftPower({\mdseries\slshape [m, k]})\index{LeftPower@\texttt{LeftPower}}
\label{LeftPower}
}\hfill{\scriptsize (function)}}\\

returns a left $k$\texttt{\symbol{45}}power of element \mbox{\texttt{\mdseries\slshape m}}. }

\subsection{\textcolor{Chapter }{RightPower}}
\logpage{[ 1, 3, 7 ]}\nobreak
\hyperdef{L}{X788005DE7947D914}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{RightPower({\mdseries\slshape [m, k]})\index{RightPower@\texttt{RightPower}}
\label{RightPower}
}\hfill{\scriptsize (function)}}\\

returns a right $k$\texttt{\symbol{45}}power of element \mbox{\texttt{\mdseries\slshape m}}. }

\subsection{\textcolor{Chapter }{AllSmallAntimagmas}}
\logpage{[ 1, 3, 8 ]}\nobreak
\hyperdef{L}{X7DDB92C284197B0B}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{AllSmallAntimagmas({\mdseries\slshape n})\index{AllSmallAntimagmas@\texttt{AllSmallAntimagmas}}
\label{AllSmallAntimagmas}
}\hfill{\scriptsize (function)}}\\

returns all antiassociative magmas of specified size \mbox{\texttt{\mdseries\slshape n}} (a number)

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@AllSmallAntimagmas(2);|
  [ <magma with 2 generators>, <magma with 2 generators> ]
  !gapprompt@gap>| !gapinput@AllSmallAntimagmas(3);|
  [
    <magma with 3 generators>, <magma with 3 generators>, <magma with 3 generators>,
    <magma with 3 generators>, <magma with 3 generators>, <magma with 3 generators>,
    <magma with 3 generators>, <magma with 3 generators>,
    <magma with 3 generators>, <magma with 3 generators>
  ]
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{NrSmallAntimagmas}}
\logpage{[ 1, 3, 9 ]}\nobreak
\hyperdef{L}{X7E5054007A985873}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{NrSmallAntimagmas({\mdseries\slshape n})\index{NrSmallAntimagmas@\texttt{NrSmallAntimagmas}}
\label{NrSmallAntimagmas}
}\hfill{\scriptsize (function)}}\\

counts number of antiassociative magmas of specified size \mbox{\texttt{\mdseries\slshape n}} (a number).

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@NrSmallAntimagmas(2);|
  2
  !gapprompt@gap>| !gapinput@NrSmallAntimagmas(3);|
  10
  !gapprompt@gap>| !gapinput@NrSmallAntimagmas(4);|
  17780
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{SmallAntimagma}}
\logpage{[ 1, 3, 10 ]}\nobreak
\hyperdef{L}{X7903CEEA79DEBA73}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{SmallAntimagma({\mdseries\slshape n, i})\index{SmallAntimagma@\texttt{SmallAntimagma}}
\label{SmallAntimagma}
}\hfill{\scriptsize (function)}}\\

returns antiassociative magma of id \mbox{\texttt{\mdseries\slshape [n, i]}}.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@SmallAntimagma(2, 1);|
  <magma with 2 generators>
  !gapprompt@gap>| !gapinput@SmallAntimagma(4, 5);|
  <magma with 4 generators>
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{OneSmallAntimagma}}
\logpage{[ 1, 3, 11 ]}\nobreak
\hyperdef{L}{X804A623879A9AC7F}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{OneSmallAntimagma({\mdseries\slshape n})\index{OneSmallAntimagma@\texttt{OneSmallAntimagma}}
\label{OneSmallAntimagma}
}\hfill{\scriptsize (function)}}\\

returns a random antiassociative magma of size \mbox{\texttt{\mdseries\slshape n}}.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@OneSmallAntimagma(2);|
  <magma with 2 generators>

  !gapprompt@gap>| !gapinput@OneSmallAntimagma(3);|
  <magma with 3 generators>
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{ReallyAllSmallAntimagmas}}
\logpage{[ 1, 3, 12 ]}\nobreak
\hyperdef{L}{X7D0A720279246D45}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{ReallyAllSmallAntimagmas({\mdseries\slshape n})\index{ReallyAllSmallAntimagmas@\texttt{ReallyAllSmallAntimagmas}}
\label{ReallyAllSmallAntimagmas}
}\hfill{\scriptsize (function)}}\\

returns really\texttt{\symbol{45}}all antiassociative magmas, isomorphic, of
specified size \mbox{\texttt{\mdseries\slshape n}} (a number)

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@ReallyAllSmallAntimagmas(2);|
  [ <magma with 2 generators>, <magma with 2 generators> ]
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{ReallyNrSmallAntimagmas}}
\logpage{[ 1, 3, 13 ]}\nobreak
\hyperdef{L}{X796637787C216BA1}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{ReallyNrSmallAntimagmas({\mdseries\slshape n})\index{ReallyNrSmallAntimagmas@\texttt{ReallyNrSmallAntimagmas}}
\label{ReallyNrSmallAntimagmas}
}\hfill{\scriptsize (function)}}\\

counts number of antiassociative magmas of specified size \mbox{\texttt{\mdseries\slshape n}} (a number)

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@ReallyNrSmallAntimagmas(3);|
  52
\end{Verbatim}

}

}

\section{\textcolor{Chapter }{smallantimagmas automatic generated documentation of methods}}\label{Chapter_smallantimagmas_automatic_generated_documentation_Section_smallantimagmas_automatic_generated_documentation_of_methods}
\logpage{[ 1, 4, 0 ]}
\hyperdef{L}{X8553DA53796C81EA}{}
{


\subsection{\textcolor{Chapter }{MagmaIsomorphism (for IsMagma, IsMagma)}}
\logpage{[ 1, 4, 1 ]}\nobreak
\hyperdef{L}{X87F3EA827FF3D302}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{MagmaIsomorphism({\mdseries\slshape M, N})\index{MagmaIsomorphism@\texttt{MagmaIsomorphism}!for IsMagma, IsMagma}
\label{MagmaIsomorphism:for IsMagma, IsMagma}
}\hfill{\scriptsize (operation)}}\\

computes an isomoprhism between magmas \mbox{\texttt{\mdseries\slshape M}}, \mbox{\texttt{\mdseries\slshape N}}.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@M := SmallAntimagma(2, 1);|
  <magma with 2 generators>
  !gapprompt@gap>| !gapinput@N := MagmaByMultiplicationTable([ [2, 1], [2, 1] ]);|
  <magma with 2 generators>
  !gapprompt@gap>| !gapinput@MagmaIsomorphism(M, N);|
  <general mapping: Domain([ m1, m2 ]) -> Domain([ m1, m2 ]) >
\end{Verbatim}

}

\subsection{\textcolor{Chapter }{MagmaAntiisomorphism (for IsMagma, IsMagma)}}
\logpage{[ 1, 4, 2 ]}\nobreak
\hyperdef{L}{X7AB3F3E37D8A5687}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{MagmaAntiisomorphism({\mdseries\slshape M, N})\index{MagmaAntiisomorphism@\texttt{MagmaAntiisomorphism}!for IsMagma, IsMagma}
\label{MagmaAntiisomorphism:for IsMagma, IsMagma}
}\hfill{\scriptsize (operation)}}\\

creates an antiisomoprhism between magmas \mbox{\texttt{\mdseries\slshape M}}, \mbox{\texttt{\mdseries\slshape N}}.

\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example]
  !gapprompt@gap>| !gapinput@M := SmallAntimagma(2, 1);|
  <magma with 2 generators>
  !gapprompt@gap>| !gapinput@N := SmallAntimagma(2, 2);|
  <magma with 2 generators>
  !gapprompt@gap>| !gapinput@MagmaAntiisomorphism(M, N);|
  <mapping: Domain([ m1, m2 ]) -> Domain([ m1, m2 ]) >
\end{Verbatim}

}

}

}

\def\indexname{Index\logpage{[ "Ind", 0, 0 ]}
\hyperdef{L}{X83A0356F839C696F}{}
}

\cleardoublepage
\phantomsection
\addcontentsline{toc}{chapter}{Index}

\printindex

\immediate\write\pagenrlog{["Ind", 0, 0], \arabic{page},}
\newpage
\immediate\write\pagenrlog{["End"], \arabic{page}];}
\immediate\closeout\pagenrlog
\end{document}

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