File:Integer examples for groups and non-groups.pdf
Summary
| 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}
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