  
  [1X74 [33X[0;0YUnknowns[133X[101X
  
  [33X[0;0YSometimes  the  result  of  an operation does not allow further computations
  with  it.  In many cases, then an error is signalled, and the computation is
  stopped.[133X
  
  [33X[0;0YThis  is  not  appropriate  for  some  applications in character theory. For
  example,  if  one  wants  to  induce  a character of a group to a supergroup
  (see [2XInducedClassFunction[102X   ([14X72.9-3[114X))   but  the  class  fusion  is  only  a
  parametrized  map  (see  Chapter [14X73[114X),  there  may  be  values of the induced
  character  which  are determined by the fusion map, whereas other values are
  not known.[133X
  
  [33X[0;0YFor this and other situations, [5XGAP[105X provides the data type [13Xunknown[113X. An object
  of  this  type,  further  on called an [13Xunknown[113X, may stand for any cyclotomic
  (see Chapter [14X18[114X), in particular its family (see [14X13.1[114X) is [10XCyclotomicsFamily[110X.[133X
  
  [33X[0;0YUnknowns  are  parametrized  by  positive  integers.  When  a [5XGAP[105X session is
  started, no unknowns exist.[133X
  
  [33X[0;0YThe  only  ways to create unknowns are to call the function [2XUnknown[102X ([14X74.1-1[114X)
  or a function that calls it, or to do arithmetical operations with unknowns.[133X
  
  [33X[0;0Y[5XGAP[105X  objects  containing  unknowns will contain [13Xfixed[113X unknowns when they are
  printed  to  files, i.e., function calls [10XUnknown([110X[22Xn[122X[10X)[110X instead of [10XUnknown()[110X. So
  be  careful to read files printed in different [5XGAP[105X sessions, since there may
  be the same unknown at different places.[133X
  
  [33X[0;0YThe rest of this chapter contains information about the unknown constructor,
  the  category,  and  comparison of and arithmetical operations for unknowns.
  More is not known about unknowns in [5XGAP[105X.[133X
  
  
  [1X74.1 [33X[0;0YMore about Unknowns[133X[101X
  
  [1X74.1-1 Unknown[101X
  
  [29X[2XUnknown[102X( [[3Xn[103X] ) [32X operation
  
  [33X[0;0YCalled  without  argument,  [2XUnknown[102X  returns  a new unknown value, i.e., the
  first  one  that  is larger than all unknowns which exist in the current [5XGAP[105X
  session.[133X
  
  [33X[0;0YCalled  with a positive integer [3Xn[103X, [2XUnknown[102X returns the [3Xn[103X-th unknown; if this
  did not exist yet, it is created.[133X
  
  [1X74.1-2 LargestUnknown[101X
  
  [29X[2XLargestUnknown[102X[32X global variable
  
  [33X[0;0Y[2XLargestUnknown[102X  is  the  largest  [3Xn[103X  that is used in any [10XUnknown( [3Xn[103X[10X )[110X in the
  current  [5XGAP[105X session. This is used in [2XUnknown[102X ([14X74.1-1[114X) which increments this
  value when asked to make a new unknown.[133X
  
  [1X74.1-3 IsUnknown[101X
  
  [29X[2XIsUnknown[102X( [3Xobj[103X ) [32X Category
  
  [33X[0;0Yis the category of unknowns in [5XGAP[105X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XUnknown();  List( [ 1 .. 20 ], i -> Unknown() );;[127X[104X
    [4X[28XUnknown(1)[128X[104X
    [4X[25Xgap>[125X [27XUnknown();   # note that we have already created 21 unknowns.[127X[104X
    [4X[28XUnknown(22)[128X[104X
    [4X[25Xgap>[125X [27XUnknown(2000);  Unknown();[127X[104X
    [4X[28XUnknown(2000)[128X[104X
    [4X[28XUnknown(2001)[128X[104X
    [4X[25Xgap>[125X [27XLargestUnknown;[127X[104X
    [4X[28X2001[128X[104X
    [4X[25Xgap>[125X [27XIsUnknown( Unknown );  IsUnknown( Unknown() );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X74.1-4 [33X[0;0YComparison of Unknowns[133X[101X
  
  [33X[0;0YUnknowns  can  be [13Xcompared[113X via [10X=[110X and [10X<[110X with all cyclotomics and with certain
  other  [5XGAP[105X  objects  (see [14X4.12[114X). We have [10XUnknown( [3Xn[103X[10X ) >= Unknown( [3Xm[103X[10X )[110X if and
  only  if [10X[3Xn[103X[10X >= [3Xm[103X[10X[110X holds, unknowns are larger than all cyclotomics that are not
  unknowns.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XUnknown() >= Unknown();  Unknown(2) < Unknown(3);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XUnknown() > 3;  Unknown() > E(3);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XUnknown() > Z(8);  Unknown() > [];[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  
  [1X74.1-5 [33X[0;0YArithmetical Operations for Unknowns[133X[101X
  
  [33X[0;0YThe  usual  arithmetic  operations  [10X+[110X,  [10X-[110X, [10X*[110X and [10X/[110X are defined for addition,
  subtraction,  multiplication  and  division of unknowns and cyclotomics. The
  result will be a new unknown except in one of the following cases.[133X
  
  [33X[0;0YMultiplication  with  zero  yields  zero,  and  multiplication  with  one or
  addition  of  zero  yields the old unknown. [13XNote[113X that division by an unknown
  causes an error, since an unknown might stand for zero.[133X
  
  [33X[0;0YAs  unknowns  are cyclotomics, dense lists of unknowns and other cyclotomics
  are  row  vectors  and  they  can  be added and multiplied in the usual way.
  Consequently,  lists  of  such  row  vectors  of equal length are (ordinary)
  matrices (see [2XIsOrdinaryMatrix[102X ([14X24.2-2[114X)).[133X
  
