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%W  unknown.msk                  GAP documentation              Thomas Breuer
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%H  @(#)$Id: unknown.msk,v 1.8 2002/04/15 10:02:34 sal Exp $
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%Y  (C) 1999 School Math and Comp. Sci., University of St.  Andrews, Scotland
%Y  Copyright (C) 2002 The GAP Group
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\Chapter{Unknowns}

\index{data type!unknown}

Sometimes 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.

This is not appropriate for some applications in character theory.
For example, if a character of a group sill be induced to a supergroup
(see~"InducedClassFunction") but the class fusion is only a parametrized
map (see Chapter~"Maps Concerning Character Tables"),
there may be values of the induced character which are determined by the
fusion map, whereas other values are not known.

For this and other situations, {\GAP} provides the data type *unknown*.
An object of this type, further on called an *unknown*,
may stand for any cyclotomic (see Chapter~"Cyclotomic Numbers"),
in particular its family (see~"Families") is `CyclotomicsFamily'.

Unknowns are parametrized by positive integers.
When a {\GAP} session is started, no unknowns exist.

The only ways to create unknowns are to call the function `Unknown'
or a function that calls it,
or to do arithmetical operations with unknowns.

{\GAP} objects containing unknowns will contain *fixed* unknowns
when they are printed to files, i.e.,
function calls `Unknown( <n> )' instead of `Unknown()'.
So be careful to read files printed in different {\GAP} sessions,
since there may be the same unknown at different places.

The 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 {\GAP}.



\>Unknown( ) O
\>Unknown( <n> ) O

In the first form `Unknown' returns a new unknown value, i.e., the first
one that is larger than all unknowns which exist in the current {\GAP}
session.

In the second form `Unknown' returns the <n>-th unknown;
if it did not exist already, it is created.


\>`LargestUnknown' V

`LargestUnknown' is the largest <n> that is used in any `Unknown( <n> )'
in the current {\GAP} session.
This is used in `Unknown' which increments this value when asked to make
a new unknown.


\>IsUnknown( <obj> ) C

is the category of unknowns in {\GAP}.



\beginexample 
gap> Unknown();  List( [ 1 .. 20 ], i -> Unknown() );;
Unknown(1)
gap> Unknown();   # note that we have already created 21 unknowns.
Unknown(22)
gap> Unknown(2000);  Unknown();
Unknown(2000)
Unknown(2001)
gap> LargestUnknown;
2001
gap> IsUnknown( Unknown );  IsUnknown( Unknown() );
false
true
\endexample

Unknowns can be *compared* via `=' and `\<' with all cyclotomics
and with certain other {\GAP} objects (see~"Comparisons").
We have `Unknown( <n> ) >= Unknown( <m> )' if and only if `<n> >= <m>'
holds; unknowns are larger than all cyclotomics that are not unknowns.

\beginexample 
gap> Unknown() >= Unknown();  Unknown(2) < Unknown(3);
false
true
gap> Unknown() > 3;  Unknown() > E(3);
true
true
gap> Unknown() > Z(8);  Unknown() > [];
false
false
\endexample

The usual arithmetic operations `+', `-', `*' and `/' 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.

Multiplication with zero yields zero,
and multiplication with one or addition of zero yields the old unknown.
*Note* that division by an unknown causes an error, since an unknown
might stand for zero.

As 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~"IsOrdinaryMatrix").



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