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##############################################################################
##
#W unknown.gd GAP Library Martin Schoenert
##
#H @(#)$Id: unknown.gd,v 4.15 2002/08/23 14:39:20 gap Exp $
##
#Y Copyright (C) 1997, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
#Y (C) 1998 School Math and Comp. Sci., University of St. Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
## This file declares the operations for unknowns.
##
Revision.unknown_gd :=
"@(#)$Id: unknown.gd,v 4.15 2002/08/23 14:39:20 gap Exp $";
#############################################################################
#1
## 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 one wants to induce a character of a group 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}.
##
#############################################################################
##
#2
## 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.
#############################################################################
##
#3
## 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").
#############################################################################
##
#C IsUnknown( <obj> )
##
## is the category of unknowns in {\GAP}.
##
DeclareCategory( "IsUnknown", IsCyclotomic );
#############################################################################
##
#V LargestUnknown . . . . . . . . . . . . largest used index for an unknown
##
## `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.
##
LargestUnknown := 0;
#############################################################################
##
#O Unknown()
#O Unknown( <n> )
##
## 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 yet, it is created.
##
DeclareOperation( "Unknown", [] );
DeclareOperation( "Unknown", [ IsPosInt ] );
#############################################################################
##
#E
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