File:Exponentiation as monoid homomorphism.pdf
Size of this JPG preview of this PDF file: 646 × 600 pixels. Other resolutions: 259 × 240 pixels | 517 × 480 pixels | 825 × 766 pixels.
Original file (825 × 766 pixels, file size: 16 KB, MIME type: application/pdf)
 | This is a file from the Wikimedia Commons. Information from its description page there is shown below. Commons is a freely licensed media file repository. You can help. |
Summary
| Description |
English: Sketches an (injective, but not surjective) monoid homomorphism f from the monoid (ℕ,+,0) to the monoid (ℕ,*,1), defined by f(x)=2x |
| Date | |
| Source | own drawing using LaTeX) |
| Author | Jochen Burghardt |
| Other versions | File:Exponentiation as monoid homomorphism svg.svg |
Latex source code
|
|---|
\documentclass[12pt]{article}
\usepackage[pdftex]{color}
\usepackage[paperwidth=140mm,paperheight=130mm]{geometry}
\setlength{\topmargin}{-36mm}
\setlength{\textwidth}{30cm}
\setlength{\textheight}{26cm}
\setlength{\oddsidemargin}{-2.4cm}
\setlength{\unitlength}{1mm}
\newcommand{\hd}[1]{{\bf #1}}
\begin{document}
% foreground
\definecolor{fAH} {rgb}{0.00,0.80,0.00} % add head
\definecolor{fAC} {rgb}{0.00,0.80,0.00} % add cont
\definecolor{fMh} {rgb}{0.80,0.50,0.50} % mult unreached head
\definecolor{fMc} {rgb}{0.80,0.50,0.50} % mult unreached cont
\definecolor{fMH} {rgb}{0.80,0.00,0.00} % mult reached head
\definecolor{fMC} {rgb}{0.80,0.00,0.00} % mult reached cont
\definecolor{fH} {rgb}{0.00,0.00,0.99} % homomorphism
% background
\definecolor{bA} {rgb}{0.91,0.99,0.91} % add
\definecolor{bAM} {rgb}{0.99,0.99,0.83} % add+mult
\definecolor{bM} {rgb}{0.99,0.91,0.91} % mult
\begin{picture}(140,130)(10,70)
\textcolor{bM}{\put(10,200){\makebox(0,0)[tl]{\rule{118mm}{118mm}}}}%
\textcolor{bA}{\put(75,145){\makebox(0,0)[tl]{\rule{68mm}{68mm}}}}%
\textcolor{bAM}{\put(75,145){\makebox(0,0)[tl]{\rule{53mm}{63mm}}}}%
\thicklines
%%%%% multiplication
\textcolor{fMH}{\put( 10,200){\line(1,0){120}}}%
\textcolor{fMH}{\put( 10,190){\line(1,0){120}}}%
\textcolor{fMH}{\put( 10,200){\line(0,-1){120}}}%
\textcolor{fMH}{\put( 20,200){\line(0,-1){120}}}%
\textcolor{fMH}{\put( 15,195){\makebox(0,0){\hd{$\times$}}}}%
% mult hor head
\textcolor{fMh}{\put( 25,195){\makebox(0,0){\hd{0}}}}%
\textcolor{fMH}{\put( 35,195){\makebox(0,0){\hd{1}}}}%
\textcolor{fMH}{\put( 45,195){\makebox(0,0){\hd{2}}}}%
\textcolor{fMh}{\put( 55,195){\makebox(0,0){\hd{3}}}}%
\textcolor{fMH}{\put( 65,195){\makebox(0,0){\hd{4}}}}%
\textcolor{fMh}{\put( 75,195){\makebox(0,0){\hd{5}}}}%
\textcolor{fMh}{\put( 85,195){\makebox(0,0){\hd{6}}}}%
\textcolor{fMh}{\put( 95,195){\makebox(0,0){\hd{7}}}}%
\textcolor{fMH}{\put(105,195){\makebox(0,0){\hd{8}}}}%
\textcolor{fMh}{\put(115,195){\makebox(0,0){\hd{9}}}}%
\textcolor{fMh}{\put(125,195){\makebox(0,0){\hd{\ldots}}}}%
% mult vert head
\textcolor{fMh}{\put( 15,185){\makebox(0,0){\hd{0}}}}%
\textcolor{fMH}{\put( 15,175){\makebox(0,0){\hd{1}}}}%
\textcolor{fMH}{\put( 15,165){\makebox(0,0){\hd{2}}}}%
\textcolor{fMh}{\put( 15,155){\makebox(0,0){\hd{3}}}}%
\textcolor{fMH}{\put( 15,145){\makebox(0,0){\hd{4}}}}%
\textcolor{fMh}{\put( 15,135){\makebox(0,0){\hd{5}}}}%
\textcolor{fMh}{\put( 15,125){\makebox(0,0){\hd{6}}}}%
\textcolor{fMh}{\put( 15,115){\makebox(0,0){\hd{7}}}}%
\textcolor{fMH}{\put( 15,105){\makebox(0,0){\hd{8}}}}%
\textcolor{fMh}{\put( 15, 95){\makebox(0,0){\hd{9}}}}%
\textcolor{fMh}{\put( 15, 85){\makebox(0,0){\hd{\vdots}}}}%
% mult contents 0
\textcolor{fMc}{\put( 25,185){\makebox(0,0){$0$}}}%
\textcolor{fMc}{\put( 35,185){\makebox(0,0){$0$}}}%
\textcolor{fMc}{\put( 45,185){\makebox(0,0){$0$}}}%
\textcolor{fMc}{\put( 55,185){\makebox(0,0){$0$}}}%
\textcolor{fMc}{\put( 65,185){\makebox(0,0){$0$}}}%
\textcolor{fMc}{\put( 75,185){\makebox(0,0){$0$}}}%
\textcolor{fMc}{\put( 85,185){\makebox(0,0){$0$}}}%
\textcolor{fMc}{\put( 95,185){\makebox(0,0){$0$}}}%
\textcolor{fMc}{\put(105,185){\makebox(0,0){$0$}}}%
\textcolor{fMc}{\put(115,185){\makebox(0,0){$0$}}}%
\textcolor{fMc}{\put(125,185){\makebox(0,0){$\ldots$}}}%
% mult 1
\textcolor{fMc}{\put( 25,175){\makebox(0,0){$0$}}}%
\textcolor{fMC}{\put( 35,175){\makebox(0,0){$1$}}}%
\textcolor{fMC}{\put( 45,175){\makebox(0,0){$2$}}}%
\textcolor{fMc}{\put( 55,175){\makebox(0,0){$3$}}}%
\textcolor{fMC}{\put( 65,175){\makebox(0,0){$4$}}}%
\textcolor{fMc}{\put( 75,175){\makebox(0,0){$5$}}}%
\textcolor{fMc}{\put( 85,175){\makebox(0,0){$6$}}}%
\textcolor{fMc}{\put( 95,175){\makebox(0,0){$7$}}}%
\textcolor{fMC}{\put(105,175){\makebox(0,0){$8$}}}%
\textcolor{fMc}{\put(115,175){\makebox(0,0){$9$}}}%
\textcolor{fMc}{\put(125,175){\makebox(0,0){$\ldots$}}}%
% mult 2
\textcolor{fMc}{\put( 25,165){\makebox(0,0){$0$}}}%
\textcolor{fMC}{\put( 35,165){\makebox(0,0){$2$}}}%
\textcolor{fMC}{\put( 45,165){\makebox(0,0){$4$}}}%
\textcolor{fMc}{\put( 55,165){\makebox(0,0){$6$}}}%
\textcolor{fMC}{\put( 65,165){\makebox(0,0){$8$}}}%
\textcolor{fMc}{\put( 75,165){\makebox(0,0){$10$}}}%
\textcolor{fMc}{\put( 85,165){\makebox(0,0){$12$}}}%
\textcolor{fMc}{\put( 95,165){\makebox(0,0){$14$}}}%
\textcolor{fMC}{\put(105,165){\makebox(0,0){$16$}}}%
\textcolor{fMc}{\put(115,165){\makebox(0,0){$18$}}}%
\textcolor{fMc}{\put(125,165){\makebox(0,0){$\ldots$}}}%
% mult 3
\textcolor{fMc}{\put( 25,155){\makebox(0,0){$0$}}}%
\textcolor{fMc}{\put( 35,155){\makebox(0,0){$3$}}}%
\textcolor{fMc}{\put( 45,155){\makebox(0,0){$6$}}}%
\textcolor{fMc}{\put( 55,155){\makebox(0,0){$9$}}}%
\textcolor{fMc}{\put( 65,155){\makebox(0,0){$12$}}}%
\textcolor{fMc}{\put( 75,155){\makebox(0,0){$15$}}}%
\textcolor{fMc}{\put( 85,155){\makebox(0,0){$18$}}}%
\textcolor{fMc}{\put( 95,155){\makebox(0,0){$21$}}}%
\textcolor{fMc}{\put(105,155){\makebox(0,0){$24$}}}%
\textcolor{fMc}{\put(115,155){\makebox(0,0){$27$}}}%
\textcolor{fMc}{\put(125,155){\makebox(0,0){$\ldots$}}}%
% mult 4
\textcolor{fMc}{\put( 25,145){\makebox(0,0){$0$}}}%
\textcolor{fMC}{\put( 35,145){\makebox(0,0){$4$}}}%
\textcolor{fMC}{\put( 45,145){\makebox(0,0){$8$}}}%
\textcolor{fMc}{\put( 55,145){\makebox(0,0){$12$}}}%
\textcolor{fMC}{\put( 65,145){\makebox(0,0){$16$}}}%
\textcolor{fMc}{\put( 75,145){\makebox(0,0){$20$}}}%
\textcolor{fMc}{\put( 85,145){\makebox(0,0){$24$}}}%
\textcolor{fMc}{\put( 95,145){\makebox(0,0){$28$}}}%
\textcolor{fMC}{\put(105,145){\makebox(0,0){$32$}}}%
\textcolor{fMc}{\put(115,145){\makebox(0,0){$36$}}}%
\textcolor{fMc}{\put(125,145){\makebox(0,0){$\ldots$}}}%
% mult 5
\textcolor{fMc}{\put( 25,135){\makebox(0,0){$0$}}}%
\textcolor{fMc}{\put( 35,135){\makebox(0,0){$5$}}}%
\textcolor{fMc}{\put( 45,135){\makebox(0,0){$10$}}}%
\textcolor{fMc}{\put( 55,135){\makebox(0,0){$15$}}}%
\textcolor{fMc}{\put( 65,135){\makebox(0,0){$20$}}}%
\textcolor{fMc}{\put( 75,135){\makebox(0,0){$25$}}}%
\textcolor{fMc}{\put( 85,135){\makebox(0,0){$30$}}}%
\textcolor{fMc}{\put( 95,135){\makebox(0,0){$35$}}}%
\textcolor{fMc}{\put(105,135){\makebox(0,0){$40$}}}%
\textcolor{fMc}{\put(115,135){\makebox(0,0){$45$}}}%
\textcolor{fMc}{\put(125,135){\makebox(0,0){$\ldots$}}}%
% mult 6
\textcolor{fMc}{\put( 25,125){\makebox(0,0){$0$}}}%
\textcolor{fMc}{\put( 35,125){\makebox(0,0){$6$}}}%
\textcolor{fMc}{\put( 45,125){\makebox(0,0){$12$}}}%
\textcolor{fMc}{\put( 55,125){\makebox(0,0){$18$}}}%
\textcolor{fMc}{\put( 65,125){\makebox(0,0){$24$}}}%
\textcolor{fMc}{\put( 75,125){\makebox(0,0){$30$}}}%
\textcolor{fMc}{\put( 85,125){\makebox(0,0){$36$}}}%
\textcolor{fMc}{\put( 95,125){\makebox(0,0){$42$}}}%
\textcolor{fMc}{\put(105,125){\makebox(0,0){$48$}}}%
\textcolor{fMc}{\put(115,125){\makebox(0,0){$54$}}}%
\textcolor{fMc}{\put(125,125){\makebox(0,0){$\ldots$}}}%
% mult 7
\textcolor{fMc}{\put( 25,115){\makebox(0,0){$0$}}}%
\textcolor{fMc}{\put( 35,115){\makebox(0,0){$7$}}}%
\textcolor{fMc}{\put( 45,115){\makebox(0,0){$14$}}}%
\textcolor{fMc}{\put( 55,115){\makebox(0,0){$21$}}}%
\textcolor{fMc}{\put( 65,115){\makebox(0,0){$28$}}}%
\textcolor{fMc}{\put( 75,115){\makebox(0,0){$35$}}}%
\textcolor{fMc}{\put( 85,115){\makebox(0,0){$42$}}}%
\textcolor{fMc}{\put( 95,115){\makebox(0,0){$49$}}}%
\textcolor{fMc}{\put(105,115){\makebox(0,0){$56$}}}%
\textcolor{fMc}{\put(115,115){\makebox(0,0){$63$}}}%
\textcolor{fMc}{\put(125,115){\makebox(0,0){$\ldots$}}}%
% mult 8
\textcolor{fMc}{\put( 25,105){\makebox(0,0){$0$}}}%
\textcolor{fMC}{\put( 35,105){\makebox(0,0){$8$}}}%
\textcolor{fMC}{\put( 45,105){\makebox(0,0){$16$}}}%
\textcolor{fMc}{\put( 55,105){\makebox(0,0){$24$}}}%
\textcolor{fMC}{\put( 65,105){\makebox(0,0){$32$}}}%
\textcolor{fMc}{\put( 75,105){\makebox(0,0){$40$}}}%
\textcolor{fMc}{\put( 85,105){\makebox(0,0){$48$}}}%
\textcolor{fMc}{\put( 95,105){\makebox(0,0){$56$}}}%
\textcolor{fMC}{\put(105,105){\makebox(0,0){$64$}}}%
\textcolor{fMc}{\put(115,105){\makebox(0,0){$72$}}}%
\textcolor{fMc}{\put(125,105){\makebox(0,0){$\ldots$}}}%
% mult 9
\textcolor{fMc}{\put( 25,95){\makebox(0,0){$0$}}}%
\textcolor{fMc}{\put( 35,95){\makebox(0,0){$9$}}}%
\textcolor{fMc}{\put( 45,95){\makebox(0,0){$18$}}}%
\textcolor{fMc}{\put( 55,95){\makebox(0,0){$27$}}}%
\textcolor{fMc}{\put( 65,95){\makebox(0,0){$36$}}}%
\textcolor{fMc}{\put( 75,95){\makebox(0,0){$45$}}}%
\textcolor{fMc}{\put( 85,95){\makebox(0,0){$54$}}}%
\textcolor{fMc}{\put( 95,95){\makebox(0,0){$63$}}}%
\textcolor{fMc}{\put(105,95){\makebox(0,0){$72$}}}%
\textcolor{fMc}{\put(115,95){\makebox(0,0){$81$}}}%
\textcolor{fMc}{\put(125,95){\makebox(0,0){$\ldots$}}}%
% mult 10...
\textcolor{fMc}{\put( 25, 85){\makebox(0,0){$\vdots$}}}%
\textcolor{fMc}{\put( 35, 85){\makebox(0,0){$\vdots$}}}%
\textcolor{fMc}{\put( 45, 85){\makebox(0,0){$\vdots$}}}%
\textcolor{fMc}{\put( 55, 85){\makebox(0,0){$\vdots$}}}%
\textcolor{fMc}{\put( 65, 85){\makebox(0,0){$\vdots$}}}%
\textcolor{fMc}{\put( 75, 85){\makebox(0,0){$\vdots$}}}%
\textcolor{fMc}{\put( 85, 85){\makebox(0,0){$\vdots$}}}%
\textcolor{fMc}{\put( 95, 85){\makebox(0,0){$\vdots$}}}%
\textcolor{fMc}{\put(105, 85){\makebox(0,0){$\vdots$}}}%
\textcolor{fMc}{\put(115, 85){\makebox(0,0){$\vdots$}}}%
\textcolor{fMc}{\put(125, 85){\makebox(0,0){$\ddots$}}}%
%%%%% addition
\textcolor{fAH}{\put( 75,145){\line(1,0){70}}}%
\textcolor{fAH}{\put( 75,135){\line(1,0){70}}}%
\textcolor{fAH}{\put( 75,145){\line(0,-1){70}}}%
\textcolor{fAH}{\put( 85,145){\line(0,-1){70}}}%
\textcolor{fAH}{\put( 80,140){\makebox(0,0){\hd{+}}}}%
% add hor head
\textcolor{fAH}{\put( 90,140){\makebox(0,0){\hd{0}}}}%
\textcolor{fAH}{\put( 100,140){\makebox(0,0){\hd{1}}}}%
\textcolor{fAH}{\put(110,140){\makebox(0,0){\hd{2}}}}%
\textcolor{fAH}{\put(120,140){\makebox(0,0){\hd{3}}}}%
\textcolor{fAH}{\put(130,140){\makebox(0,0){\hd{4}}}}%
\textcolor{fAH}{\put(140,140){\makebox(0,0){\hd{\ldots}}}}%
% add vert head
\textcolor{fAH}{\put( 80,130){\makebox(0,0){\hd{0}}}}%
\textcolor{fAH}{\put( 80,120){\makebox(0,0){\hd{1}}}}%
\textcolor{fAH}{\put( 80,110){\makebox(0,0){\hd{2}}}}%
\textcolor{fAH}{\put( 80,100){\makebox(0,0){\hd{3}}}}%
\textcolor{fAH}{\put( 80, 90){\makebox(0,0){\hd{4}}}}%
\textcolor{fAH}{\put( 80, 80){\makebox(0,0){\hd{\vdots}}}}%
% add contents 0
\textcolor{fAC}{\put( 90,130){\makebox(0,0){$0$}}}%
\textcolor{fAC}{\put( 100,130){\makebox(0,0){$1$}}}%
\textcolor{fAC}{\put(110,130){\makebox(0,0){$2$}}}%
\textcolor{fAC}{\put(120,130){\makebox(0,0){$3$}}}%
\textcolor{fAC}{\put(130,130){\makebox(0,0){$4$}}}%
\textcolor{fAC}{\put(140,130){\makebox(0,0){$\ldots$}}}%
% add 1
\textcolor{fAC}{\put( 90,120){\makebox(0,0){$1$}}}%
\textcolor{fAC}{\put( 100,120){\makebox(0,0){$2$}}}%
\textcolor{fAC}{\put(110,120){\makebox(0,0){$3$}}}%
\textcolor{fAC}{\put(120,120){\makebox(0,0){$4$}}}%
\textcolor{fAC}{\put(130,120){\makebox(0,0){$5$}}}%
\textcolor{fAC}{\put(140,120){\makebox(0,0){$\ldots$}}}%
% add 2
\textcolor{fAC}{\put( 90,110){\makebox(0,0){$2$}}}%
\textcolor{fAC}{\put( 100,110){\makebox(0,0){$3$}}}%
\textcolor{fAC}{\put(110,110){\makebox(0,0){$4$}}}%
\textcolor{fAC}{\put(120,110){\makebox(0,0){$5$}}}%
\textcolor{fAC}{\put(130,110){\makebox(0,0){$6$}}}%
\textcolor{fAC}{\put(140,110){\makebox(0,0){$\ldots$}}}%
% add 3
\textcolor{fAC}{\put( 90,100){\makebox(0,0){$3$}}}%
\textcolor{fAC}{\put( 100,100){\makebox(0,0){$4$}}}%
\textcolor{fAC}{\put(110,100){\makebox(0,0){$5$}}}%
\textcolor{fAC}{\put(120,100){\makebox(0,0){$6$}}}%
\textcolor{fAC}{\put(130,100){\makebox(0,0){$7$}}}%
\textcolor{fAC}{\put(140,100){\makebox(0,0){$\ldots$}}}%
% add 4
\textcolor{fAC}{\put( 90, 90){\makebox(0,0){$4$}}}%
\textcolor{fAC}{\put( 100, 90){\makebox(0,0){$5$}}}%
\textcolor{fAC}{\put(110, 90){\makebox(0,0){$6$}}}%
\textcolor{fAC}{\put(120, 90){\makebox(0,0){$7$}}}%
\textcolor{fAC}{\put(130, 90){\makebox(0,0){$8$}}}%
\textcolor{fAC}{\put(140, 90){\makebox(0,0){$\ldots$}}}%
% add 5...
\textcolor{fAC}{\put( 90, 80){\makebox(0,0){$\vdots$}}}%
\textcolor{fAC}{\put( 100, 80){\makebox(0,0){$\vdots$}}}%
\textcolor{fAC}{\put(110, 80){\makebox(0,0){$\vdots$}}}%
\textcolor{fAC}{\put(120, 80){\makebox(0,0){$\vdots$}}}%
\textcolor{fAC}{\put(130, 80){\makebox(0,0){$\vdots$}}}%
\textcolor{fAC}{\put(140, 80){\makebox(0,0){$\ddots$}}}%
% mapping hor
\textcolor{fH}{\put( 88,142){\vector(-1,1){51}}}%
\textcolor{fH}{\put( 98,142){\vector(-1,1){51}}}%
\textcolor{fH}{\put( 108,142){\line(-3,4){30}}}%
\textcolor{fH}{\put( 78,182){\vector(-1,1){11}}}%
\textcolor{fH}{\put(118,142){\line(0,1){7}}}%
\textcolor{fH}{\put(118,149){\vector(-1,4){11}}}%
% mapping vert
%\textcolor{fH}{\put( 68,132){\vector(-1,1){51}}}%
%\textcolor{fH}{\put( 68,122){\vector(-1,1){51}}}%
%\textcolor{fH}{\put(68,112){\line(-4,3){40}}}%
%\textcolor{fH}{\put(28,142){\vector(-1,1){11}}}%
%\textcolor{fH}{\put(68,102){\line(-1,0){7}}}%
%\textcolor{fH}{\put(61,102){\vector(-4,1){44}}}%
\end{picture}
\end{document}
|
Licensing
I, the copyright holder of this work, hereby publish it under the following licenses:
This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
|
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License. |
You may select the license of your choice.
File history
Click on a date/time to view the file as it appeared at that time.
| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 18:12, 10 October 2013 |  | 825 × 766 (16 KB) | Jochen Burghardt | adapted as suggested on Talk:Monoid#Suggested tweaks to new illustration |
| 18:11, 10 October 2013 | No thumbnail | 0 × 0 (15 KB) | Jochen Burghardt | Reverted to version as of 21:36, 9 October 2013 | |
| 18:09, 10 October 2013 |  | 825 × 766 (16 KB) | Jochen Burghardt | ||
| 21:36, 9 October 2013 | No thumbnail | 0 × 0 (15 KB) | Jochen Burghardt | {{Information |Description ={{en|1=Sketches an (injective, but not surjective) monoid homomorphism h from the monoid {{color|#008000|(ℕ<sub>0</sub>,+)}} to the monoid {{color|#800000|(ℕ<sub>1</sub>,*)}}, defined by h(x)=2<sup>x</sup>}} |Source ... |
File usage
No pages on the English Wikipedia use this file (pages on other projects are not listed).