File:Integer examples for groups and non-groups.pdf

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Description
English: Shows a Karnaugh diagram of all possibilities of the Commutative, Associative, Inverse, and Identity law to hold (y) or not to hold (n), and the definition of an according binary operation ⊕ on the integers (ℤ) for almost each possibility. The rightmost column is omitted since Inverse elements can't be defined in absence of an Identity element. Where applicable, the Identity element always happens to be 0, and the Inverse element of x always happens to be -x.
Date
Source Own work
Author Jochen Burghardt
Other versions File:Integer examples for groups and non-groups svg.svg
LaTeX source Code 
\documentclass[12pt]{article}
\usepackage[pdftex]{color}
\usepackage[paperwidth=250mm,paperheight=230mm]{geometry}
\setlength{\unitlength}{1mm}
\setlength{\topmargin}{-45mm}
\setlength{\textwidth}{250mm}
\setlength{\textheight}{230mm}
\setlength{\oddsidemargin}{-25mm}
\pagestyle{empty}

\begin{document}

% foregrounds
\definecolor{fLin}      {rgb}{0.50,0.50,0.50}   % line
\definecolor{fALn}      {rgb}{0.80,0.80,0.80}   % auxiliary line
\definecolor{fCap}      {rgb}{0.30,0.30,0.30}   % karnaugh caption
\definecolor{fACp}      {rgb}{0.80,0.80,0.80}   % auxiliary karnaugh caption
\definecolor{fExm}      {rgb}{0.00,0.00,0.00}   % examples

\definecolor{fMag}      {rgb}{0.00,0.00,0.50}   % magma
\definecolor{fSGp}      {rgb}{0.00,0.50,0.00}   % monoid
\definecolor{fMon}      {rgb}{0.50,0.00,0.00}   % semigroup
\definecolor{fGrp}      {rgb}{0.50,0.50,0.00}   % group
\definecolor{fAGp}      {rgb}{0.00,0.50,0.50}   % abelian group

% backgrounds
\definecolor{bMag}      {rgb}{0.90,0.90,0.99}   % magma
\definecolor{bSGp}      {rgb}{0.90,0.99,0.90}   % monoid
\definecolor{bMon}      {rgb}{0.99,0.90,0.90}   % semigroup
\definecolor{bGrp}      {rgb}{0.99,0.99,0.80}   % group
\definecolor{bAGp}      {rgb}{0.80,0.99,0.99}   % abelian group

\begin{picture}(240,230)

\thicklines
% magma
\textcolor{bMag}{\put(37,2){\makebox(0,0)[bl]{\rule{146mm}{196mm}}}}%
\textcolor{fMag}{\put(37,2){\framebox(146,196)[bl]{}}}%
\textcolor{fMag}{\put(39,196){\makebox(0,0)[tl]{\large\bf Magma}}}%
% semigroup
\textcolor{bSGp}{\put(39,52){\makebox(0,0)[bl]{\rule{142mm}{96mm}}}}%
\textcolor{fSGp}{\put(39,52){\framebox(142,96)[bl]{}}}%
\textcolor{fSGp}{\put(41,146){\makebox(0,0)[tl]{\large\bf Semigroup}}}%
% monoid
\textcolor{bMon}{\put(87,54){\makebox(0,0)[bl]{\rule{92mm}{92mm}}}}%
\textcolor{fMon}{\put(87,54){\framebox(92,92)[bl]{}}}%
\textcolor{fMon}{\put(89,144){\makebox(0,0)[tl]{\large\bf Monoid}}}%
% group
\textcolor{bGrp}{\put(137,56){\makebox(0,0)[bl]{\rule{40mm}{88mm}}}}%
\textcolor{fGrp}{\put(137,56){\framebox(40,88)[bl]{}}}%
\textcolor{fGrp}{\put(139,142){\makebox(0,0)[tl]{\large\bf Group}}}%
% abelian group
\textcolor{bAGp}{\put(139,58){\makebox(0,0)[bl]{\rule{36mm}{40mm}}}}%
\textcolor{fAGp}{\put(139,58){\framebox(36,40)[bl]{}}}%
\textcolor{fAGp}{\put(141,96){\makebox(0,0)[tl]{\large\bf Abel.\ Group}}}%

\thinlines
% diagram hor lines
\textcolor{fLin}{\put(35,0){\line(1,0){150}}}%
\textcolor{fLin}{\put(35,50){\line(1,0){150}}}%
\textcolor{fLin}{\put(35,100){\line(1,0){150}}}%
\textcolor{fLin}{\put(35,150){\line(1,0){150}}}%
\textcolor{fLin}{\put(35,200){\line(1,0){150}}}%
\textcolor{fALn}{\put(185,0){\line(1,0){50}}}%
\textcolor{fALn}{\put(185,50){\line(1,0){50}}}%
\textcolor{fALn}{\put(185,100){\line(1,0){50}}}%
\textcolor{fALn}{\put(185,150){\line(1,0){50}}}%
\textcolor{fALn}{\put(185,200){\line(1,0){50}}}%
% diagram vert lines
\textcolor{fLin}{\put(35,0){\line(0,1){200}}}%
\textcolor{fLin}{\put(85,0){\line(0,1){200}}}%
\textcolor{fLin}{\put(135,0){\line(0,1){200}}}%
\textcolor{fLin}{\put(185,0){\line(0,1){200}}}%
\textcolor{fALn}{\put(235,0){\line(0,1){200}}}%
% diagram hor caption
\textcolor{fCap}{\put(39,220){\makebox(0,0)[l]{\large\bf Inv,Ide}}}%
\textcolor{fCap}{\put(60,210){\makebox(0,0){\large\bf nn}}}%
\textcolor{fCap}{\put(110,210){\makebox(0,0){\large\bf ny}}}%
\textcolor{fCap}{\put(160,210){\makebox(0,0){\large\bf yy}}}%
\textcolor{fACp}{\put(210,210){\makebox(0,0){\large\bf yn}}}%
% diagram vert caption
\textcolor{fCap}{\put(25,196){\makebox(0,0)[tr]{\large\bf Com,Ass}}}%
\textcolor{fCap}{\put(25,175){\makebox(0,0)[r]{\large\bf nn}}}%
\textcolor{fCap}{\put(25,125){\makebox(0,0)[r]{\large\bf ny}}}%
\textcolor{fCap}{\put(25,75){\makebox(0,0)[r]{\large\bf yy}}}%
\textcolor{fCap}{\put(25,25){\makebox(0,0)[r]{\large\bf yn}}}%

% example nnnn
\textcolor{fExm}{\put(60,175){\makebox(0,0){%
        $\begin{array}{r@{}c@{}l}%
        x \oplus y & = & 2 x - y        \\%
        \end{array}$%
        }}}%

% example nnny
\textcolor{fExm}{\put(110,175){\makebox(0,0){%
        $\begin{array}{r@{}c@{}l}%
        x \oplus y & = & x + y + x^2 y  \\%
        \end{array}$%
        }}}%

% example nnyy
\textcolor{fExm}{\put(160,175){\makebox(0,0){%
        $\begin{array}{r@{}c@{}l}%
        x \oplus y & = & x \!+\! y \!+\! x^2 y^2 \!+\! x y^3    \\%
        \end{array}$%
        }}}%

% example nynn
\textcolor{fExm}{\put(60,125){\makebox(0,0){%
        $\begin{array}{r@{}c@{}l}%
        x \oplus y & = & x      \\%
        \end{array}$%
        }}}%

% example yynn
\textcolor{fExm}{\put(60,75){\makebox(0,0){%
        $\begin{array}{r@{}c@{}l}%
        x \oplus y & = & 2 x y  \\%
        \end{array}$%
        }}}%

% example yyny
\textcolor{fExm}{\put(110,75){\makebox(0,0){%
        $\begin{array}{r@{}c@{}l}%
        x \oplus y & = & x + y + x y    \\%
        \end{array}$%
        }}}%

% example yyyy
\textcolor{fExm}{\put(158,75){\makebox(0,0){%
        $\begin{array}{r@{}c@{}l}%
        x \oplus y & = & x + y  \\%
        \end{array}$%
        }}}%

% example ynnn
\textcolor{fExm}{\put(60,25){\makebox(0,0){%
        $\begin{array}{r@{}c@{}l}%
        x \oplus y & = & x y - 1        \\%
        \end{array}$%
        }}}%

% example ynny
\textcolor{fExm}{\put(110,25){\makebox(0,0){%
        $\begin{array}{r@{}c@{}l}%
        x \oplus y & = & x + y + x^2 y^2        \\%
        \end{array}$%
        }}}%

% example ynyy
\textcolor{fExm}{\put(160,25){\makebox(0,0){%
        $\begin{array}{r@{}c@{}l}%
        x \oplus y & = & x \!+\! y \!+\! x^2 y \!+\! x y^2      \\%
        \end{array}$%
        }}}%

\end{picture}
\end{document}

Licensing[edit]

I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
attribution share alike
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current11:39, 15 November 2013Thumbnail for version as of 11:39, 15 November 20131,475 × 1,356 (18 KB)Jochen Burghardt (talk | contribs)User created page with UploadWizard

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