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Edit detail for SandBoxMorphism revision 5 of 15

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Editor: Bill Page Time: 2008/06/18 16:51:51 GMT-7
Note: simplified Automorphism

changed:
-)abbrev domain AUTOMOR Automorphism
)abbrev domain AUTOM Automorph

changed:
-++       Automorphism R is the multiplicative group of automorphisms of R.
--- In fact, non-invertible endomorphism are allowed as partial functions.
--- This domain is noncanonical in that f*f^{-1} will be the identity
--- function but won't be equal to 1.
-Automorphism(R:Ring): Join(SetCategory, Eltable(R, R)) with
++       Automorph R is the domain of automorphisms on R.
Automorph(R:Ring): Join(SetCategory, Eltable(R, R)) with

changed:
-      iter:  (R -> R, NonNegativeInteger, R) -> R
-      apply: (%, R, Integer) -> R
-
-      Rep := ((R, Integer) -> R)

      Rep := (R -> R)

changed:
-      apply(f, r, n) == (g := f pretend ((R, Integer) -> R); g(r, n))
-      elt(f, r)                       == apply(f, r, 1)
      elt(f, r) == (g := f pretend (R -> R); g(r))

changed:
-      morphism(f:R -> R):%            == iter(f, #2, #1)
-      iter(f, n, r) ==
-          for i in 1..n repeat r := f r
-          r
      morphism(f:R -> R):%            == f

fricas

(1) -> <spad>

fricas

)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)
  domain(p:%):T == source
  codomain(p:%):T == target
  coerce(r:(source->target)):% == per(r)
  coerce(p:%):(source->target) == p pretend (source->target)
  coerce(p:%):OutputForm == p pretend OutputForm
  --elt(f,x) == (f::(source->target))(x)</spad>

fricas

Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/2396666756253990559-25px001.spad
      using system compiler.
   MORPH abbreviates domain Morphism 
------------------------------------------------------------------------
   initializing NRLIB MORPH for Morphism 
   compiling into NRLIB MORPH 
************* USER ERROR **********
available signatures for Rep: 
    NONE
NEED Rep: () -> ?
****** comp fails at level 1 with expression: ******
((DEF (|Rep|) (NIL) (|Mapping| |target| |source|)))
****** level 1  ******
x:= (DEF (Rep) (NIL) (Mapping target source))
m:= $EmptyMode
f:=
((((~= #) (= #) (|coerce| #) (|latex| #) ...)))

   >> Apparent user error:
   unspecified error

If we define elt (commented out above), then f(1.1) gives the following:

  Error: Caught fatal error [memory may be damaged]
  Fast links are on: do (si::use-fast-links nil) for debugging
  Error signalled by RETURN.
  Broken at APPLY.  Type :H for Help.
  BOOT>>
  Error: The variable QUIT is unbound.
  Fast links are on: do (si::use-fast-links nil) for debugging
  Error signalled by EVALHOOK.
  Backtrace: system:universal-error-handler > evalhook > lambda > lambda-closure > block > apply > APPLY

But the same coercion works in the interpreter.

fricas

)show Morphism(Float,Integer)

   Morphism is an unknown constructor and so is unavailable. Did you 
      mean to use -> but type something different instead?

Next let's study how this sort of thing is accomplished in a similar domain.

spad

)abbrev domain AUTOM Automorph
++ Author: Manuel Bronstein
++ Date Created: 31 January 1994
++ Date Last Updated: 31 January 1994
++ References:
++ Description:
++       Automorph R is the domain of automorphisms on R.
Automorph(R:Ring): Join(SetCategory, Eltable(R, R)) with
      morphism: (R -> R) -> %
        ++ morphism(f) returns the non-invertible morphism given by f.
      morphism: (R -> R, R -> R) -> %
        ++ morphism(f, g) returns the invertible morphism given by f, where
        ++ g is the inverse of f..
   == add

      Rep := (R -> R)

      f = g                           == EQ(f, g)$Lisp

      elt(f, r) == (g := f pretend (R -> R); g(r))

      coerce(f:%):OutputForm          == message("R -> R")
      morphism(f:R -> R):%            == f

spad

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/8658851672039808957-25px003.spad
      using system compiler.
   AUTOM abbreviates domain Automorph 
------------------------------------------------------------------------
   initializing NRLIB AUTOM for Automorph 
   compiling into NRLIB AUTOM 
   compiling exported = : (%,%) -> Boolean
      AUTOM;=;2%B;1 is replaced by EQ 
Time: 0 SEC.

   compiling exported elt : (%,R) -> R
Time: 0 SEC.

   compiling exported coerce : % -> OutputForm
Time: 0 SEC.

   compiling exported morphism : R -> R -> %
      AUTOM;morphism;M%;4 is replaced by f 
Time: 0 SEC.

(time taken in buildFunctor:  0)
Time: 0 SEC.

   Warnings: 
      [1] elt: pretend(Mapping R R) -- should replace by @

   Cumulative Statistics for Constructor Automorph
      Time: 0 seconds

   finalizing NRLIB AUTOM 
   Processing Automorph for Browser database:
--------constructor---------
--------(morphism (% (Mapping R R)))---------
--------(morphism (% (Mapping R R) (Mapping R R)))---------
; compiling file "/var/aw/var/LatexWiki/AUTOM.NRLIB/AUTOM.lsp" (written 20 OCT 2026 05:12:35 PM):

; wrote /var/aw/var/LatexWiki/AUTOM.NRLIB/AUTOM.fasl
; compilation finished in 0:00:00.008
------------------------------------------------------------------------
   Automorph is now explicitly exposed in frame initial 
   Automorph will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/AUTOM.NRLIB/AUTOM

fricas

p:=morphism((x:Integer):Integer +-> x+1)$Automorphism(Integer)

$\label{eq1}\mbox{\rm R - > R}$

(1)

Type: Automorphism(Integer)

fricas

p(2)

$\label{eq2}3$

(2)

Type: PositiveInteger?