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Algebra

SandBoxNonZeroInteger is an attempt to define the domain of Integers without 0 as a SubDomain.

\begin{spad} )abbrev domain NZINT NonZeroInteger NonZeroInteger: Join(OrderedAbelianSemiGroup, Monoid --,CoercibleTo(Integer) ) with commutative("*") gcd: (%,%) -> % ++ gcd(a,b) computes the greatest common divisor of two ++ positive integers \spad{a} and b. _-: % -> % --retract:Integer->% --retractIfCan:Integer->Union(%,"failed") --retract:NonNegativeInteger->% --retractIfCan:NonNegativeInteger->Union(%,"failed") --convert:NonNegativeInteger->% == SubDomain(Integer,not(#1 = 0)) add Rep == Integer x:% y:% z:Integer nz:NonNegativeInteger -x == (-(x pretend Integer)) pretend % x+y == (z:=(x pretend Integer)+(y pretend Integer)) = 0 => error "zero" z pretend %

--coerce(x):Integer == x pretend Integer --retract(z)== z pretend % --convert(nz):% == retract(nz) --retractIfCan(z) == -- zero?(z) => "failed" -- retract(z) --retract(nz)== nz pretend % --retractIfCan(nz) == -- zero?(nz) => "failed" -- retract(nz) \end{spad}

\begin{axiom} )show NZINT \end{axiom}

\begin{axiom} i:NZINT := 1$NZINT i:NZINT := (1::Integer) i:NZINT := 1 \end{axiom} \begin{axiom} j:NZINT := -1 \end{axiom} \begin{axiom} j:NZINT:=-i i+1 j+1 i+j \end{axiom}

\begin{axiom} i-j retractIfCan(0)$NZINT \end{axiom}

Can we define NonZero as a functor? \begin{spad} )abbrev domain NZ NonZero NonZero(T:Ring): Join(OrderedAbelianSemiGroup, Monoid) with Commutative("*") _-: % -> % == SubDomain(T,not(#1 = 0)) add Rep == T x:% y:% z:T -x == (-(x pretend T)) pretend % x+y == (z:=(x pretend T)+(y pretend T)) = 0 => error "zero" z pretend % \end{spad}

\begin{axiom} )show NonZero(Float) \end{axiom}

\begin{axiom} f:NonZero(Float):=1 g:NonZero(Float):=-f f+(1.1::NonZero(Float)) f + -g f+g \end{axiom}


Some or all expressions may not have rendered properly, because Axiom returned the following error:
Error: export AXIOM=/usr/local/lib/open-axiom/x86_64-unknown-linux/1.3.0-2009-01-05; 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/1.3.0-2009-01-05/lib; LANG=en_US.UTF-8 $AXIOM/bin/AXIOMsys < /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3437484353503122027-25px.axm
/bin/sh: /usr/local/lib/open-axiom/x86_64-unknown-linux/1.3.0-2009-01-05/bin/AXIOMsys: 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)
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ogreek, ancientgreek, hungarian, italian, latin, mongolian, norsk, icelandic, i
nterlingua, turkish, coptic, romanian, welsh, serbian, slovenian, estonian, esp
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Package geometry Warning: `lmargin' and `rmargin' result in NEGATIVE (-108.405p t). `width' should be shortened in length.

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Xy-pic version 3.7 <1999/02/16> Copyright (c) 1991-1998 by Kristoffer H. Rose <krisrose@ens-lyon.fr> Xy-pic is free software: see the User's Guide for details.

Loading kernel: messages; fonts; allocations: state, direction, utility macros; pictures: \xy, positions, objects, decorations; kernel objects: directionals, circles, text; options; algorithms: directions, edges, connections; Xy-pic loaded)) (/usr/share/texmf-texlive/tex/generic/xypic/xyall.tex Xy-pic option: All features v.3.3 (/usr/share/texmf-texlive/tex/generic/xypic/xycurve.tex Xy-pic option: Curve and Spline extension v.3.7 curve, circles, loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xyframe.tex Xy-pic option: Frame and Bracket extension v.3.7 loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xycmtip.tex Xy-pic option: Computer Modern tip extension v.3.3 (/usr/share/texmf-texlive/tex/generic/xypic/xytips.tex Xy-pic option: More Tips extension v.3.3 loaded) loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xyline.tex Xy-pic option: Line styles extension v.3.6 loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xyrotate.tex Xy-pic option: Rotate and Scale extension v.3.3 loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xycolor.tex Xy-pic option: Colour extension v.3.3 loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xymatrix.tex Xy-pic option: Matrix feature v.3.4 loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xyarrow.tex Xy-pic option: Arrow and Path feature v.3.5 path, \ar, loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xygraph.tex Xy-pic option: Graph feature v.3.7 loaded) loaded) (/usr/share/texmf-texlive/tex/latex/tools/verbatim.sty) (/usr/share/texmf/tex/latex/graphviz/graphviz.sty (/usr/share/texmf-texlive/tex/latex/psfrag/psfrag.sty)) (/usr/share/texmf/tex/latex/sagetex.sty Writing sage input file 7851149973523920894-16.0px.sage (./7851149973523920894-16.0px.sout)) (/usr/share/texmf-texlive/tex/latex/gnuplottex/gnuplottex.sty (/usr/share/texmf-texlive/tex/latex/base/latexsym.sty) (/usr/share/texmf-texlive/tex/latex/moreverb/moreverb.sty) (/usr/share/texmf-texlive/tex/latex/base/ifthen.sty)) (./7851149973523920894-16.0px.aux) (/usr/share/texmf-texlive/tex/latex/ucs/ucsencs.def) Missing $ inserted. <inserted text> $ l.130 _ -: % -> % (/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) You can't use `macro parameter character #' in math mode. l.136 == SubDomain(Integer,not(# 1 = 0)) add Missing $ inserted. <inserted text> $ l.146

[1] (/usr/share/texmf-texlive/tex/latex/base/t1cmtt.fd)

LaTeX Warning: Characters dropped after `\end{axiom}' on input line 160.

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Missing $ inserted. <inserted text> $ l.183 _ -: % -> % You can't use `macro parameter character #' in math mode. l.184 == SubDomain(T,not(# 1 = 0)) add Missing $ inserted. <inserted text> $ l.193 \end{spad} \newpage

Overfull \hbox (19.6396pt too wide) in paragraph at lines 180--193 \T1/cmr/m/n/12 )abbrev do-main NZ NonZero NonZero(T:Ring): Join(OrderedAbelianS emiGroup, Monoid) with Com-mu-ta-tive("*") $[] \OT1/cmr/m/n/12 :== \OML/cmm/m/i t/12 SubDomain\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 T; not\OT1/cmr/m/n/12 (1 = [2]

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[3] (./7851149973523920894-16.0px.aux) ) (see the transcript file for additional information) Output written on 7851149973523920894-16.0px.dvi (3 pages, 3368 bytes). Transcript written on 7851149973523920894-16.0px.log.




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