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 Topics FrontPage SandBox SandBox Mapping SandBoxMorphism <-- You are here. Using OpenAxiom \begin{axiom} )version \end{axiom} \begin{spad} )abbrev domain MORPH Morphism T ==> SetCategory Morphism(source:T, target:T): with domain:%->T codomain:%->T coerce:(source->target) -> % coerce:%->(source->target) coerce:%->OutputForm elt:(%,source)->target == (source->target) add -- Rep == (source->target) Rep == SExpression domain(p:%):T == source codomain(p:%):T == target coerce(r:(source->target)):% == per(r pretend SExpression) coerce(p:%):(source->target) == (rep(p) pretend (source->target)) coerce(p:%):OutputForm == p pretend OutputForm elt(f,x) == (rep(f) pretend (source->target))(x) \end{spad} The naive representation of Rep == (source->target) gives the following error: Error: Caught fatal error [memory may be damaged] ... Error signalled by EVALHOOK. Backtrace: system:universal-error-handler > evalhook > lambda > lambda-closure > block > apply > APPLY But using SExpression seems to work. \begin{axiom} )show Morphism(Float,Integer) f:Morphism(Float,Integer) f:=(x:Float):Integer +->wholePart(x) f(1.1) (f::(Float->Integer))(1.1) domain f codomain f (f::(domain(f)->codomain(f)))(1.1) g:Morphism(Integer,String):=(i:Integer):String +-> i::String g(1985) \end{axiom} Next let's study how this sort of thing is accomplished in a similar domain. \begin{spad} )abbrev domain HOM Hom ++ Based on Automorphism by: Manuel Bronstein ++ Description: ++ Hom(R,S) is the domain of homomorphisms - Mapping(S,R). Hom(R:SetCategory?,S:SetCategory?): Join(SetCategory?, Eltable(R, S)) with morphism: (R -> S) -> % ++ morphism(f) returns the non-invertible morphism given by f. == add -- Why doesn't the obvious representation just (R->S) work? Rep == SExpression? morphism(f) == per(f pretend SExpression?) elt(f, r) == (rep(f) pretend (R->S))(r) --Rep == List(R -> S) --morphism(f) == per([f #1]?) --elt(f, r) == (rep(f)(1))(r) --Rep := ((R, Integer) -> S) --morphism(f) == (#2; f(#1)) --elt(f, r) == (g := f pretend ((R, Integer) -> S); g(r, 1)) f = g == EQ(f, g)$Lisp coerce(f:%):OutputForm? == message("theMap ...") \end{spad} \begin{axiom} p:=morphism((x:Integer):Float +-> x+1.0)$Hom(Integer,Float) p(2) p(-1) \end{axiom} Some or all expressions may not have rendered properly, because Axiom returned the following error:Error: export FRICAS=/usr/local/lib/open-axiom/x86_64-unknown-linux/1.4.0-2009-10-13; export ALDORROOT=/usr/local/aldor/linux/1.1.0; export PATH=$ALDORROOT/bin:$PATH; export HOME=/var/zope2/var/LatexWiki; ulimit -t 600; export LD_LIBRARY_PATH=/usr/local/lib/open-axiom/x86_64-unknown-linux/1.4.0-2009-10-13/lib; LANG=en_US.UTF-8 $FRICAS/bin/FRICASsys < /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/8560717210793468152-25px.axm /bin/sh: /usr/local/lib/open-axiom/x86_64-unknown-linux/1.4.0-2009-10-13/bin/FRICASsys: not found  Some or all expressions may not have rendered properly, because Latex returned the following error:This is pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6) \write18 enabled. %&-line parsing enabled. entering extended mode (./7813582976431298218-16.0px.tex LaTeX2e <2005/12/01> Babel and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, arabic, farsi, croatian, ukrainian, russian, bulgarian, czech, slov ak, danish, dutch, finnish, basque, french, german, ngerman, ibycus, greek, mon ogreek, ancientgreek, hungarian, italian, latin, mongolian, norsk, icelandic, i nterlingua, turkish, coptic, romanian, welsh, serbian, slovenian, estonian, esp eranto, uppersorbian, indonesian, polish, portuguese, spanish, catalan, galicia n, swedish, ukenglish, pinyin, loaded. 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(/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/setspace/setspace.sty Package: setspace 6.7 <2000/12/01> ) (/usr/share/texmf-texlive/tex/generic/xypic/xy.sty (/usr/share/texmf-texlive/tex/generic/xypic/xy.tex Bootstrap'ing: catcodes, docmode, (/usr/share/texmf-texlive/tex/generic/xypic/xyrecat.tex) (/usr/share/texmf-texlive/tex/generic/xypic/xyidioms.tex) Xy-pic version 3.7 <1999/02/16> Copyright (c) 1991-1998 by Kristoffer H. Rose Xy-pic is free software: see the User's Guide for details. 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[1] LaTeX Warning: Characters dropped after \end{axiom}' on input line 155. You can't use macro parameter character #' in horizontal mode. l.171 --morphism(f) == per([f # 1]) You can't use macro parameter character #' in horizontal mode. l.175 ... --morphism(f) == (# 2; f(#1)) You can't use macro parameter character #' in horizontal mode. l.175 ...rphism(f) == (#2; f(# 1)) (/usr/share/texmf-texlive/tex/latex/jknapltx/ursfs.fd) (/usr/share/texmf-texlive/tex/latex/amsfonts/umsa.fd) (/usr/share/texmf-texlive/tex/latex/amsfonts/umsb.fd) (/usr/share/texmf-texlive/tex/latex/base/ulasy.fd) Missing$ inserted. \$ l.180 [2] LaTeX Warning: Characters dropped after \end{axiom}' on input line 186. [3] (./7813582976431298218-16.0px.aux) ) (see the transcript file for additional information) Output written on 7813582976431298218-16.0px.dvi (3 pages, 2696 bytes). Transcript written on 7813582976431298218-16.0px.log. `

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