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In a domain and in categories referenced in a domain the notation % represents "this domain" (or self in some programming languages). So we commonly write for example:

  with
    f: (%,%) -> %

to indicate a function f which takes a pair of values in this domain and returns a value in this same domain - whatever domain we happen to be talking about in this context.

But what if we are interested in the domain as a functor? Suppose I was writing an "endofunctor" domain constructor like Set and I wanted to treat constructions like Set Set R, i.e. sets of sets as something special. E.g.

spad
)abbrev domain MYSET MySet
--#pile
--#include "axiom"
--MonadCat(T:SetCategory,M:SetCategory->SetCategory):Category == with
--  join: M M T -> M T
MySet(T:SetCategory): SetAggregate(T) with finiteAggregate join: MySet % -> % == add Rep := List T rep(x:%):Rep == x pretend Rep per(x:Rep):% == x pretend %
Rep2 == List List T rep2(x:MySet %):Rep2 == x pretend Rep2 per2(x:Rep2):MySet % == x pretend MySet %
coerce(x:%):OutputForm == brace(map(coerce$T, rep x)$ListFunctions2(T,OutputForm)) ((x:%) = (y:%)):Boolean == (rep(x) = rep(y))$Rep construct(x:List T):% == per(removeDuplicates(x)$Rep) parts(x:%):List T == rep x join(x:MySet %):% == construct(concat(rep2 x)$List(T))$%
copy(x:%):% == per(copy(rep x)$Rep) empty():% == per(empty()$Rep) map(f:T->T,x:%):% == per(map(f,rep x)$Rep) brace():% == empty()$% brace(x:List(T)):% == construct(x)$% set():% == empty()$% set(x:List(T)):% == construct(x)$%
-- dummy exports as required by Aldor _<(x:%, y:%):Boolean == true intersect(x:%, y:%):% == empty()$% difference(x:%, y:%):% == empty()$% subset?(x:%, y:%):Boolean == true union(x:%, y:%):% == empty()$% if T has ConvertibleTo(InputForm) then convert(x:%):InputForm == convert(rep(x))$Rep
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/8570179011506836244-25px001.spad
      using old system compiler.
   MYSET abbreviates domain MySet 
------------------------------------------------------------------------
   initializing NRLIB MYSET for MySet 
   compiling into NRLIB MYSET 
   compiling local rep : $ -> Rep
      MYSET;rep is replaced by x 
Time: 0.01 SEC.
compiling local per : Rep -> $ MYSET;per is replaced by x Time: 0 SEC.
************* USER ERROR ********** available signatures for Rep2: NONE NEED Rep2: () -> ? ****** comp fails at level 1 with expression: ****** ((DEF (|Rep2|) (NIL) (NIL) (|List| (|List| T$)))) ****** level 1 ****** $x:= (DEF (Rep2) (NIL) (NIL) (List (List T$))) $m:= $EmptyMode $f:= ((((|per| #)) ((|per| #) (|rep| #)) ((|rep| #) (|#| #) (< #) (<= #) ...)))
>> Apparent user error: unspecified error

axiom
)sh MySet
MySet is a domain constructor Abbreviation for MySet is MYSET This constructor is not exposed in this frame. ------------------------------- Operations --------------------------------
>> System error: The function BOOT::|MySet| is undefined.

axiom
MySet(Integer) has MonadCat(Integer,MySet)
MySet is an unknown constructor and so is unavailable. Did you mean to use -> but type something different instead?

axiom
m1:MySet(Integer) := construct([1,2,3])$MySet(Integer)
MySet is an unknown constructor and so is unavailable. Did you mean to use -> but type something different instead? m2:MySet(Integer) := construct([4,5,6])$MySet(Integer)
MySet is an unknown constructor and so is unavailable. Did you mean to use -> but type something different instead? ml:=[m1,m2]

\label{eq1}\left[ m 1, \: m 2 \right](1)
Type: List(OrderedVariableList?([m1,m2]))
axiom
mm:MySet MySet Integer := construct([m1,m2])$MySet(MySet(Integer))
MySet is an unknown constructor and so is unavailable. Did you mean to use -> but type something different instead? join mm
There are no exposed library operations named join but there are 2 unexposed operations with that name. Use HyperDoc Browse or issue )display op join to learn more about the available operations.
Cannot find a definition or applicable library operation named join with argument type(s) Variable(mm)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.




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