path: root/macros/latex/contrib/galois/galois.dtx

\def\fileversion{1.05}
\def\filedate{2005/03/19}
% \iffalse meta-comment
%
% This file can be redistributed and/or modified under the terms of 
% the LaTeX Project Public License from CTAN archives, as described at
% http://www.latex-project.org/lppl.txt.  Either LPPL Version 1.3 or, 
% at your option, any later version.
% 
% \fi
%
% \CheckSum{423}
%% \CharacterTable
%%  {Upper-case    \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
%%   Lower-case    \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z
%%   Digits        \0\1\2\3\4\5\6\7\8\9
%%   Exclamation   \!     Double quote  \"     Hash (number) \#
%%   Dollar        \$     Percent       \%     Ampersand     \&
%%   Acute accent  \'     Left paren    \(     Right paren   \)
%%   Asterisk      \*     Plus          \+     Comma         \,
%%   Minus         \-     Point         \.     Solidus       \/
%%   Colon         \:     Semicolon     \;     Less than     \<
%%   Equals        \=     Greater than  \>     Question mark \?
%%   Commercial at \@     Left bracket  \[     Backslash     \\
%%   Right bracket \]     Circumflex    \^     Underscore    \_
%%   Grave accent  \`     Left brace    \{     Vertical bar  \|
%%   Right brace   \}     Tilde         \~}
%%
%
% \iffalse
%%
%% Source File: galois.dtx 
%% Copyright 1994 1998 1999 2006 Patrick.Cousot@ens.fr
%%
%
%<*dtx>
          \ProvidesFile{galois.dtx}
%</dtx>
%<package>\NeedsTeXFormat{LaTeX2e}
%<package>\ProvidesPackage{galois}
%    \begin{macrocode}
%<*driver>
\documentclass{ltxdoc}
%    \end{macrocode}
%
%    We don't want everything to appear in the index.
%    \begin{macrocode}
\DoNotIndex{\,,\:,\ ,\hfill,\hskip,\hspace}
\DoNotIndex{\@ifundefined,\@tempdima,\@tempdimb,\@tempdimc}
\DoNotIndex{\DoNotIndex,\NeedsTeXFormat,\ifx,\else,\fi,\ifdim,\fi,\endtrivlist}
\DoNotIndex{\addtolength,\divide,\circ,\cleaders,\ensuremath}
\DoNotIndex{\begin,\end,\bgroup,\egroup,\begingroup,\endgroup}
\DoNotIndex{\catcode,\noexpand,\protect,\string,\space,\put,\equiv}
\DoNotIndex{\filedate,\filename,\fileversion,\ProcessOptions}
\DoNotIndex{\hbox,\raisebox,\rlap,\llap,\relax,\rule,\smash}
\DoNotIndex{\let,\def,\newcommand,\newenvironment,\DeclareOption}
\DoNotIndex{\m@th,\mathchoice,\mathord,\mathrel,\max,\mkern,\newlength}
\DoNotIndex{\setlength,\settodepth,\settoheight,\settowidth}
\DoNotIndex{\tiny,\scriptstyle,\scriptscriptstyle,\undefined}
\DoNotIndex{\leftarrow,\rightarrow,\longleftarrow,\longrightarrow}
\makeatother
%
\CodelineNumbered
\CodelineIndex
\EnableCrossrefs
\RecordChanges
%
\usepackage[color]{galois}
\begin{document}
\DocInput{galois.dtx}
\PrintIndex
\PrintChanges
\end{document}
%</driver>
%    \end{macrocode}
% \fi
%         \ProvidesFile{galois.dtx}
       [\filedate\space v\fileversion, Galois connections, Patrick.Cousot@ens.fr]
%
%
% \changes{v0.00}{1996/11/25}%
%    {Initial version for LaTeX 2.09}
% \changes{v1.00}{1997/05/19}%
%    {Adapted to LaTeX2e}
% \changes{v1.01}{1998/10/19}%
%    {Check "\textbackslash comp" is not already defined (as in "mathtime.sty")}
% \changes{v1.02}{1998/11/13}%
%    {Rewritting of extendable arrows using the TeX book left/right arrowfill 
%     p. 357. Added and renamed style parameters ("\textbackslash GaloisSep" now 
%     "\textbackslash GaloisArrowTagSep")}
% \changes{v1.03}{1999/04/02}%
%    {Renamed internal macros to avoid interferences with other packages
%     "\textbackslash rightarrowfill" $\rightarrow$ "\textbackslash @GALOISrightarrowfill"
%     "\textbackslash leftarrowfill"  $\rightarrow$ "\textbackslash @GALOISleftarrowfill"}
% \changes{v1.04}{1999/05/01}%
%    {\LaTeX{} package file documentation}
% \changes{v1.05}{2005/03/19}%
%    {Added color option}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \title{The \textsf{galois} package\thanks{This file
%        has version number \fileversion, last
%        revised \filedate.}}
% \author{Patrick Cousot\\
% \texttt{Patrick.Cousot@ens.fr}}
% 
% \date{\filedate}
%
% \maketitle
%
% ^^A                 x  
% ^^A \mor{x} =   |------->   total morphism 
% ^^A
% \newcommand{\mor}[1]{\mathrel{\raisebox{-0.1ex}^^A fix heigt of 2nd parameter of  stackrel
% {$\stackrel{\hbox{$\scriptscriptstyle #1$}}{\raisebox{0pt}[0.5ex][0pt]^^A fix height of 1st parameter of stackrel
% {$\longmapsto$}}$}}}
% 
% \section{Introduction}
% 
% This |galois| package introduces two-dimensional notations for 
% Galois connections.
%
% \section{Detailled explanations on Galois connections}
%
% If $(L$, ${\leq})$ and $(M$, ${\sqsubseteq})$ are posets, $\alpha\in 
% L\mapsto M$, $\gamma\in M\mapsto L$ and $\forall x\in L, y\in M$: 
% $\alpha(x)\sqsubseteq y$ $\iff$ $x\leq\gamma(y)$ then the pair 
% $\langle \alpha$, $\gamma\rangle$ is a {\em Galois connection}, 
% written |\galois{\alpha}{\gamma}|:
% \DescribeMacro{\galois}
% \begin{displaymath}
% (L,\,{\leq})\galois{\alpha}{\gamma}(M,\,{\sqsubseteq})
% \end{displaymath}
% In a Galois connection, $\alpha$ is onto if and only if $\gamma$ is 
% one-to-one if and only if $\alpha \comp \gamma$ = 1 (where ${\comp}$ 
% is the functional composition and 1 the identity), written
% |\galoiS{\alpha}{\gamma}|:
% \DescribeMacro{\galoiS}
% \begin{displaymath}
% (L,\,{\leq})\galoiS{\alpha}{\gamma}(M,\,{\sqsubseteq})
% \end{displaymath}
% $\alpha$ is one-to-one if and only if $\gamma$ is onto
% if and only if $\gamma\comp\alpha$ = 1, written
% |\Galois{\alpha}{\gamma}|:
% \DescribeMacro{\Galois}
% \begin{displaymath}
% (L,\,{\leq})\Galois{\alpha}{\gamma}(M,\,{\sqsubseteq})
% \end{displaymath}
% For a bijection, we write
% |\GaloiS{\alpha}{\gamma}|:
% \DescribeMacro{\GaloiS}
% \begin{displaymath}
% (L,\,{\leq})\GaloiS{\alpha}{\gamma}(M,\,{\sqsubseteq})
% \end{displaymath}
% The surjection on the quotient of $M$ by the equivalence relation 
% $x\equiv y$ defined by $\gamma(x)=\gamma(y)$ is denoted
% |\galoiSr{\alpha}{\gamma}|:
% \DescribeMacro{\galoiSr}
% \begin{displaymath}
% (L,\,{\leq})\galoiSr{\alpha}{\gamma}(M,\,{\sqsubseteq})
% \end{displaymath}
% The composition of Galois connections:
% \begin{displaymath}
% (L,\,{\leq})\galois{\alpha_1}{\gamma_1}(M,\,{\sqsubseteq})
% \quad{\rm and}\quad
% (M,\,{\sqsubseteq})\galois{\alpha_2}{\gamma_2}(N,\,{\preceq})
% \end{displaymath}
% is a Galois connection (the composition $\comp$ of functions is
% |\comp|):
% \DescribeMacro{\comp}
% \begin{displaymath}
% (L,\,{\leq})\galois{\alpha_2\comp\alpha_1}{\gamma_1\comp\gamma_2}(N,\,{\preceq})
% \end{displaymath}
% Galois connections 
% $(L,\,{\leq})\galois{\alpha}{\gamma}(M,\,\sqsubseteq)$ can be lifted 
% from sets of properties to sets of monotone functions:
% \begin{displaymath}
% (L\mor{m}L,\,{\leq})\galois{\lambda{\varphi}\cdot{\alpha\comp\varphi\comp\gamma}}{\lambda{\phi}\cdot{\gamma\comp\phi\comp\alpha}}(M\mor{m}M,\,{\sqsubseteq})
% \end{displaymath}
% where the ordering on functions is pointwise that is 
% $\varphi\preceq\phi$ if and only if $\forall 
% x:\varphi(x)\preceq\phi(x)$.  Observe that the length of the arrows
% stretches automatically to the appropriate width.
%
% \section{Package options}
%
%  \begin{macro}{color}
%  The \texttt{color} option is required for colored Galois
%  connections is in 
%     \begin{center}
%        \begin{tabular}{cc}
%          |\galois[red]{\alpha}{\gamma}| &
%          $\galois[red]{\alpha}{\gamma}$,\\[1ex]
%          |\Galois{\alpha}[blue]{\gamma}| &
%          $\Galois{\alpha}[blue]{\gamma}$,\\[1ex]
%          |\GaloiS[red]{\alpha}[blue]{\gamma}| &
%          $\GaloiS[red]{\alpha}[blue]{\gamma}$,\\[1ex]
%          |\galoiSr[red]{\alpha}[blue]{\gamma}| &
%          $\galoiSr[red]{\alpha}[blue]{\gamma}$, or\\[1ex]
%         |\comp[red]| & $\comp[red]$\ .
%       \end{tabular}
%    \end{center}
%  Without '|color|' option, these colors are ignored.
%  \begin{macro}{\@GALOIScolor}
% |\@GALOIScolor| is |\color| with the |color| option and later
% defined as |\relax| in absence of |color| option.
%    \begin{macrocode}
\DeclareOption{color}{%
  \def\@GALOIScolor{\color}}
%    \end{macrocode}
%
%    \begin{macrocode}
\ProcessOptions
%    \end{macrocode}
%  \end{macro}
%  \end{macro}
%
% \section{Style parameters}
% 
% You can use Galois connections in any size (footnotes, transparencies, 
% etc.) :
% {\tiny tiny $L\galois{\alpha}{\gamma}M$},
% {\scriptsize scriptsize $L\galois{\alpha}{\gamma}M$},
% {\footnotesize footnotesize $L\galois{\alpha}{\gamma}M$},
% {\small small $L\galois{\alpha}{\gamma}M$},
% {\normalsize normalsize $L\galois{\alpha}{\gamma}M$},
% {\large large $L\galois{\alpha}{\gamma}M$},
% {\Large Large $L\galois{\alpha}{\gamma}M$},
% {\LARGE LARGE $L\galois{\alpha}{\gamma}M$},
% {\huge huge $L\galois{\alpha}{\gamma}M$},
% {\Huge Huge $L\galois{\alpha}{\gamma}M$}.
% Observe that in $\GaloiS{\rule{3cm}{1cm}}{\rule{0.5cm}{1cm}}$ the 
% width of arrows and height of enclosing box are automatically adjusted 
% according to the size of $\alpha$ and $\gamma$.  You can adjust the 
% following parameters:
% \DescribeMacro{\GaloisStyle}
% \begin{description}
% \item[]|\GaloisStyle|          : style of upper and lower tags 
% (|\scripstyle | by default);
% \end{description}
% \DescribeMacro{\GaloisArrowThickness}
% \begin{description}
% \item[]|\GaloisArrowThickness| : thickness of the arrow stems 
% \footnote{stem is ``tige'' in french.}; 
% (|0.1ex| by default);
% \end{description}
% \DescribeMacro{\GaloisArrowsSep}
% \begin{description}
% \item[]|\GaloisArrowsSep|      : distance between the arrows  
% (|0.2ex| by default);
% \end{description}
% \DescribeMacro{\GaloisArrowTagSep}
% \begin{description}
% \item[]|\GaloisArrowTagSep|    : distance between arrows and tags 
% (|0.5ex| by default).
% \end{description}
% For example with:
% \begin{quote}
% |\renewcommand{\GaloisArrowsSep}{1cm}|\\
% |\renewcommand{\GaloisArrowTagSep}{0pt}|
% \end{quote}
% we get
% \renewcommand{\GaloisArrowsSep}{1cm}\renewcommand{\GaloisArrowTagSep}{0pt}
% $\GaloiS{\rule{3cm}{1cm}}{\rule{0.5cm}{1cm}}$ while with:
% \begin{quote}
% |\renewcommand{\GaloisArrowsSep}{0pt}|\\
% |\renewcommand{\GaloisArrowTagSep}{5mm}|
% \end{quote}
% we get \renewcommand{\GaloisArrowsSep}{0pt}\renewcommand{\GaloisArrowTagSep}{5mm}
% $\GaloiS{\rule{3cm}{1cm}}{\rule{0.5cm}{1cm}}$ and with
% \begin{quote}
% |\renewcommand{\GaloisArrowsSep}{0pt}|\\
% |\renewcommand{\GaloisArrowTagSep}{0pt}|
% \end{quote}
% we get 
% \renewcommand{\GaloisArrowsSep}{0pt}\renewcommand{\GaloisArrowTagSep}{0pt} 
% $\GaloiS{\rule{3cm}{1cm}}{\rule{0.5cm}{1cm}}$.
%
% \StopEventually{}
%
% \section{Implementation}
%    \begin{macrocode}
%<*package>
%    \end{macrocode}
%
% Require |color| package for '|color|' option else coloring is ignored.
%    \begin{macrocode}
\ifx\@GALOIScolor\undefined
\def\@GALOIScolor#1{\relax}%
\else
\RequirePackage{color}%
\fi
%    \end{macrocode}
%
% ^^A Reset style parameters to default
% \renewcommand{\GaloisStyle}{\scriptstyle}%
% \renewcommand{\GaloisArrowThickness}{0.1ex}%
% \renewcommand{\GaloisArrowsSep}{0.2ex}%
% \renewcommand{\GaloisArrowTagSep}{0.5ex}% 
% \begin{macro}{\comp}
% \begin{macro}{\@GALOIScomp}
% Define functional composition $f\comp g(x)$ is $f(g(x))$ (if not 
% already defined e.g.\ as in |mathtime.sty|). |\comp[color]| will
% draw in color (black by default).
%    \begin{macrocode}
\@ifundefined{comp}{%
% Scan the optional color argument
\newcommand{\comp}{\@ifnextchar[{\@GALOIScomp}{\@GALOIScomp[black]}}%
% Defined the colored functional composition \@GALOIScomp[color]
\def\@GALOIScomp[#1]{\mathchoice
{\mathrel{\raisebox{0.2ex}{$\@GALOIScolor{#1}\scriptstyle\circ$}}}%
{\mathrel{\raisebox{0.2ex}{$\@GALOIScolor{#1}\scriptstyle\circ$}}}%
{\mathrel{\raisebox{0.1ex}{$\@GALOIScolor{#1}\scriptscriptstyle\circ$}}}%
{\mathrel{\raisebox{0.1ex}{$\@GALOIScolor{#1}\scriptscriptstyle\circ$}}}}%
}{}%
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% Style commands:
% \begin{macro}{\GaloisStyle}
% Style of $a$ and $b$ in $\galois{a}{b}$, $\Galois{a}{b}$, $\galoiS{a}{b}$ or 
% $\GaloiS{a}{b}$:
%    \begin{macrocode}
\newcommand{\GaloisStyle}{\scriptstyle}%
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\GaloisArrowThickness}
% Thickness of the arrow stems (0.1ex by default):
%    \begin{macrocode}
\newcommand{\GaloisArrowThickness}{0.1ex}%
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\GaloisArrowsSep}
% Distance between the lower and upper arrows (0.2ex by default):
%    \begin{macrocode}
\newcommand{\GaloisArrowsSep}{0.2ex}%
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\GaloisArrowTagSep}
% Distance between the lower arrow and the top of $a$ and the 
% top-arrow and the bottom of $b$ (0.5ex by default)
%    \begin{macrocode}
\newcommand{\GaloisArrowTagSep}{0.5ex}% 
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@GALOISalphadepth}
% \begin{macro}{\@GALOISalphaheight}
% \begin{macro}{\@GALOISgammadepth}
% \begin{macro}{\@GALOISwidth}
% \begin{macro}{\@GALOISheight}
% \begin{macro}{\@GALOISdepth}
% \begin{macro}{\@GALOIStotalheight}
% \begin{macro}{\@GALOISGap}
% \begin{macro}{\@GALOISalphaarrowwidth}
% \begin{macro}{\@GALOISalphaarrowhalfheight}
% \begin{macro}{\@GALOISgammaarrowwidth}
% \begin{macro}{\@GALOISgammaarrowhalfheight}
% Auxiliary lengths:
%    \begin{macrocode}{}
\newlength{\@GALOISalphadepth}%
\newlength{\@GALOISalphaheight}%
\newlength{\@GALOISgammadepth}%
\newlength{\@GALOISwidth}%
\newlength{\@GALOISheight}%
\newlength{\@GALOISdepth}%
\newlength{\@GALOIStotalheight}%
\newlength{\@GALOISGap}%
\newlength{\@GALOISalphaarrowwidth}%
\newlength{\@GALOISalphaarrowhalfheight}%
\newlength{\@GALOISgammaarrowwidth}%
\newlength{\@GALOISgammaarrowhalfheight}%
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\Galois@put}
% |\Galois@put(x,y-d){text}| puts text at coordinates $(x,y-d)$, in
% a box of size 0\texttt{pt} $\times$ 0\texttt{pt}:
%    \begin{macrocode}
\def\Galois@put(#1,#2-#3)#4{\rlap{\smash{\hskip#1\setlength{\@tempdimc}{#2}%
\addtolength{\@tempdimc}{-#3}\raisebox{\@tempdimc}{#4}}}}%
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@GALOISrightarrowfill}
%  |\@GALOISrightarrowfill{\rightarrow}|, see \TeX book p.\ 357.
%    \begin{macrocode}
\def\@GALOISrightarrowfill#1{$\m@th \smash- \mkern-7mu%
 \cleaders\hbox{$\mkern-2mu \smash- \mkern-2mu$}\hfill%
 \mkern-7mu \mathord{#1}$}%
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@GALOISleftarrowfill}
% |\@GALOISleftarrowfill{\leftarrow}|, see \TeX book p.\ 357.
%    \begin{macrocode}
\def\@GALOISleftarrowfill#1{$\m@th \mathord{#1} \mkern-7mu%
  \cleaders\hbox{$\mkern-2mu \smash- \mkern-2mu$}\hfill%
  \mkern-7mu \smash-$}%
%    \end{macrocode}
% \end{macro}
%
% Stacking $a$, the arrows and $g$ in $\galois{a}{g}$:
% \begin{macro}{\@GALOIS}
% \begin{macro}{\@GALOISca}
% \begin{macro}{\@GALOISca}
% |\@GALOIS{-->}{<--}{a}{g}| constructs $\galois{a}{g}$.
% |\@GALOIS{-->}{<--}[colora]{a}{g}|,
% |\@GALOIS{-->}{<--}{a}[colorg]{g}| and
% |\@GALOIS{-->}{<--}[colora]{a}[colorg]{g}|
% add colors |colora| for the $a$-arrow and |colorg| for
% the$ g$ arrow.
%    \begin{macrocode}
%    First, scan the alpha color optional argument (black
%    otherwise)
\def\@GALOIS#1#2{\@ifnextchar[{\@GALOISca{#1}{#2}}{\@GALOISca{#1}{#2}[black]}}%
%    Second scan the gamma color optional argument (black
%    otherwise)
\def\@GALOISca#1#2[#3]#4{\@ifnextchar[{\@GALOIScacg{#1}{#2}[#3]{#4}}%
                                      {\@GALOIScacg{#1}{#2}[#3]{#4}[black]}}%
%    Finally \@GALOIScacg{-->}{<--}[colora]{a}[colorg]{g} stacks $a$,
%    the arrows and $g$ in $\galois{a}{g}$, using colors with the
%    'color' option.
\def\@GALOIScacg#1#2[#3]#4[#5]#6{%
\ensuremath{\mathrel{%
\def\@GALOISalphatag{\ $\@GALOIScolor{#3}\GaloisStyle#4$\ }%
\def\@GALOISgammatag{\ $\@GALOIScolor{#5}\GaloisStyle#6$\ }%
% compute width of alpha/lower and gamma/upper arrows
\settowidth{\@GALOISalphaarrowwidth}{$\mathord{#1}$}%
\settowidth{\@GALOISgammaarrowwidth}{$\mathord{#2}$}%
% compute width of the picture \@GALOISwidth
\ifdim\@GALOISalphaarrowwidth>\@GALOISgammaarrowwidth%
\settowidth{\@tempdima}{\hbox{\hspace*{\@GALOISalphaarrowwidth}\@GALOISalphatag}}%
\settowidth{\@tempdimb}{\hbox{\hspace*{\@GALOISalphaarrowwidth}\@GALOISgammatag}}%
\else%
\settowidth{\@tempdima}{\hbox{\hspace*{\@GALOISgammaarrowwidth}\@GALOISalphatag}}%
\settowidth{\@tempdimb}{\hbox{\hspace*{\@GALOISgammaarrowwidth}\@GALOISgammatag}}%
\fi%
\ifdim\@tempdima>\@tempdimb%
\setlength{\@GALOISwidth}{\@tempdima}%
\else%
\setlength{\@GALOISwidth}{\@tempdimb}%
\fi%
\def\@GALOISrightarrow{\hbox to\@GALOISwidth
{\@GALOIScolor{#3}\@GALOISrightarrowfill{#1}}}%
\def\@GALOISleftarrow{\hbox to\@GALOISwidth
{\@GALOIScolor{#5}\@GALOISleftarrowfill{#2}}}%
% compute half height of alpha/lower arrow
\settodepth{\@GALOISalphaarrowhalfheight}{$\mathord{#1}$}%
\settoheight{\@tempdima}{$\mathord{#1}$}%
\addtolength{\@GALOISalphaarrowhalfheight}{\@tempdima}%
\divide \@GALOISalphaarrowhalfheight by 2%
% compute half height of gamma/upper arrow
\settodepth{\@GALOISgammaarrowhalfheight}{$\mathord{#2}$}%
\settoheight{\@tempdima}{$\mathord{#2}$}%
\addtolength{\@GALOISgammaarrowhalfheight}{\@tempdima}%
\divide \@GALOISgammaarrowhalfheight by 2%
% compute the distance between the two arrows \@GALOISGap = 
%   \max(\@GALOISalphaarrowhalfheight,
%        \@GALOISgammaarrowhalfheight)+\GaloisArrowsSep
\ifdim\@GALOISalphaarrowhalfheight>\@GALOISgammaarrowhalfheight%
\setlength{\@GALOISGap}{\@GALOISalphaarrowhalfheight}%
\else%
\addtolength{\@GALOISGap}{\@GALOISgammaarrowhalfheight}%
\fi%
\addtolength{\@GALOISGap}{\GaloisArrowsSep}%
% lift from the stems thickness
\addtolength{\@GALOISGap}{\GaloisArrowThickness }%
\addtolength{\@GALOISGap}{\GaloisArrowThickness }%
% compute height \@GALOISheight depth \@GALOISdepth
% and total height \@GALOIStotalheight  of the picture
\settodepth{\@GALOISalphadepth}{\@GALOISalphatag}%
\settoheight{\@GALOISalphaheight}{\@GALOISalphatag}%
\settodepth{\@GALOISgammadepth}{\@GALOISgammatag}%
% compute depth \@GALOISdepth of the picture
% \@GALOISdepth = \@GALOISalphadepth
%               + \@GALOISalphaheight % vertical size of alpha tag
%               + \GaloisArrowTagSep  % between top of tag and arrow 
\setlength{\@GALOISdepth}{\@GALOISalphadepth}%
\addtolength{\@GALOISdepth}{\@GALOISalphaheight}%
\addtolength{\@GALOISdepth}{\GaloisArrowTagSep}%
%  lift from the stem thickness
\addtolength{\@GALOISdepth}{-\GaloisArrowThickness }%
% compute height \@GALOISheight of the picture
\setlength{\@GALOISheight}{\@GALOISGap}%
\addtolength{\@GALOISheight}{\GaloisArrowTagSep}%
\addtolength{\@GALOISheight}{\@GALOISgammadepth}%
\settoheight{\@tempdima}{\@GALOISgammatag}%
\addtolength{\@GALOISheight}{\@tempdima}%
% compute  total height \@GALOIStotalheight of the picture
% \@GALOIStotalheight = \@GALOISdepth + \@GALOISheight
\setlength{\@GALOIStotalheight}{\@GALOISdepth}%
\addtolength{\@GALOIStotalheight}{\@GALOISheight}%
% put alpha arrow
\Galois@put(0pt,0pt-\@GALOISalphaarrowhalfheight){\@GALOISrightarrow}%
% put gamma arrow
\Galois@put(0pt,\@GALOISGap-\@GALOISalphaarrowhalfheight){\@GALOISleftarrow}%
% put alpha
\setlength{\@tempdima}{\@GALOISwidth}%
\settowidth{\@tempdimb}{\@GALOISalphatag}%
\addtolength{\@tempdima}{-\@tempdimb}%
\divide\@tempdima by 2%
\Galois@put(\@tempdima,\@GALOISalphadepth-\@GALOISdepth){\@GALOISalphatag}%
% put gamma
\setlength{\@tempdima}{\@GALOISwidth}%
\settowidth{\@tempdimb}{\@GALOISgammatag}%
\addtolength{\@tempdima}{-\@tempdimb}%
\divide\@tempdima by 2%
\setlength{\@tempdimb}{\@GALOISalphadepth}%
\addtolength{\@tempdimb}{\@GALOISalphaheight}%
\addtolength{\@tempdimb}{\GaloisArrowTagSep}%
\addtolength{\@tempdimb}{\GaloisArrowTagSep}%
\addtolength{\@tempdimb}{\@GALOISGap}%
\addtolength{\@tempdimb}{\@GALOISgammadepth}%
\Galois@put(\@tempdima,\@tempdimb-\@GALOISdepth){\@GALOISgammatag}%
\rule[-\@GALOISdepth]{0pt}{\@GALOIStotalheight}% set depth and height
\hspace*{\@GALOISwidth}% set width
}}}%
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\galois}
% |\galois{a}{g}| is $\galois{a}{g}$.
%    \begin{macrocode}
\newcommand{\galois}{\@GALOIS{\rightarrow}{\leftarrow}}%
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\galoiS}
% |\galoiS{a}{g}| is $\galoiS{a}{g}$ ($a$ onto, $g$ one-to-one, $a 
% \comp g = 1$):
%    \begin{macrocode}
\def\@GALOISmytwoheadrightarrow{\rlap{$\:\,{\rightarrow}$}{\longrightarrow}}%
\def\@GALOIStwoheadrightarrow{\protect\@GALOISmytwoheadrightarrow}%
\newcommand{\galoiS}{\@GALOIS{\@GALOIStwoheadrightarrow}{\leftarrow}}%
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\galoiSr}
% |\galoiSr{a}{g}| is $\galoiSr{a}{g}$.
%    \begin{macrocode}
\def\@GALOISmytwoheadrightarrowreduc{\rlap{\smash{\hskip1ex\raisebox{0.815ex}%
{\tiny$\equiv$}}}\rlap{$\:\,{\rightarrow}$}{\longrightarrow}}%
\def\@GALOIStwoheadrightarrowreduc{\protect\@GALOISmytwoheadrightarrowreduc}%
\newcommand{\galoiSr}{\@GALOIS{\@GALOIStwoheadrightarrowreduc}{\leftarrow}}%
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\Galois}
% |\Galois{a}{g}| is $\Galois{a}{g}$ ($a$ one-to-one, $g$ onto, $g 
% \comp a = 1$):
%    \begin{macrocode}
\def\@GALOISmytwoheadleftarrow{\rlap{$\:{\leftarrow}$}{\longleftarrow}}%
\def\@GALOIStwoheadleftarrow{\protect\@GALOISmytwoheadleftarrow}%
\newcommand{\Galois}{\@GALOIS{\rightarrow}{\@GALOIStwoheadleftarrow}}%
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\GaloiS}
% |\GaloiS{a}{g}| is $\GaloiS{a}{g}$ ($a$ bijective with inverse $g$).
%    \begin{macrocode}
\newcommand{\GaloiS}{\@GALOIS%
{\@GALOIStwoheadrightarrow}{\@GALOISmytwoheadleftarrow}}%
%    \end{macrocode}
% \end{macro}
%
%    \begin{macrocode}
%</package>
%    \end{macrocode}
%
% \Finale
%
\endinput