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#############################################################################
##
#W grptbl.gd GAP library Thomas Breuer
##
#H @(#)$Id: grptbl.gd,v 4.11.4.1 2005/05/06 08:46:30 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 contains the implementation of magmas, monoids, and groups from
## a multiplication table.
##
Revision.grptbl_gd :=
"@(#)$Id: grptbl.gd,v 4.11.4.1 2005/05/06 08:46:30 gap Exp $";
#############################################################################
##
#F MagmaByMultiplicationTableCreator( <A>, <domconst> )
##
## This is a utility for the uniform construction of a magma,
## a magma-with-one, or a magma-with-inverses from a multiplication table.
##
DeclareGlobalFunction( "MagmaByMultiplicationTableCreator" );
#############################################################################
##
#F MagmaByMultiplicationTable( <A> )
##
## For a square matrix <A> with $n$ rows such that all entries of <A> are
## in the range $[ 1 \.\. n ]$, `MagmaByMultiplicationTable' returns a magma
## $M$ with multiplication `\*' defined by $A$.
## That is, $M$ consists of the elements $m_1, m_2, \ldots, m_n$,
## and $m_i \* m_j = m_{A[i][j]}$.
##
## The ordering of elements is defined by $m_1 \< m_2 \< \cdots \< m_n$,
## so $m_i$ can be accessed as `MagmaElement( <M>, <i> )',
## see~"MagmaElement".
##
DeclareGlobalFunction( "MagmaByMultiplicationTable" );
#############################################################################
##
#F MagmaWithOneByMultiplicationTable( <A> )
##
## The only differences between `MagmaByMultiplicationTable' and
## `MagmaWithOneByMultiplicationTable' are that the latter returns a
## magma-with-one (see~"MagmaWithOne") if the magma described by the matrix
## <A> has an identity,
## and returns `fail' if not.
##
DeclareGlobalFunction( "MagmaWithOneByMultiplicationTable" );
#############################################################################
##
#F MagmaWithInversesByMultiplicationTable( <A> )
##
## `MagmaByMultiplicationTable' and `MagmaWithInversesByMultiplicationTable'
## differ only in that the latter returns
## magma-with-inverses (see~"MagmaWithInverses") if each element in the
## magma described by the matrix <A> has an inverse,
## and returns `fail' if not.
##
DeclareGlobalFunction( "MagmaWithInversesByMultiplicationTable" );
#############################################################################
##
#F MagmaElement( <M>, <i> ) . . . . . . . . . . <i>-th element of magma <M>
##
## For a magma <M> and a positive integer <i>, `MagmaElement' returns the
## <i>-th element of <M>, w.r.t.~the ordering `\<'.
## If <M> has less than <i> elements then `fail' is returned.
##
DeclareGlobalFunction( "MagmaElement" );
#############################################################################
##
#F SemigroupByMultiplicationTable( <A> )
##
## returns the semigroup whose multiplication is defined by the square
## matrix <A> (see~"MagmaByMultiplicationTable") if such a semigroup exists.
## Otherwise `fail' is returned.
##
DeclareGlobalFunction( "SemigroupByMultiplicationTable" );
#############################################################################
##
#F MonoidByMultiplicationTable( <A> )
##
## returns the monoid whose multiplication is defined by the square
## matrix <A> (see~"MagmaByMultiplicationTable") if such a monoid exists.
## Otherwise `fail' is returned.
##
DeclareGlobalFunction( "MonoidByMultiplicationTable" );
#############################################################################
##
#F GroupByMultiplicationTable( <A> )
##
## returns the group whose multiplication is defined by the square
## matrix <A> (see~"MagmaByMultiplicationTable") if such a group exists.
## Otherwise `fail' is returned.
##
DeclareGlobalFunction( "GroupByMultiplicationTable" );
#############################################################################
##
#A MultiplicationTable( <elms> )
#A MultiplicationTable( <M> )
##
## For a list <elms> of elements that form a magma $M$,
## `MultiplicationTable' returns a square matrix $A$ of positive integers
## such that $A[i][j] = k$ holds if and only if
## `<elms>[i] \* <elms>[j] = <elms>[k]'.
## This matrix can be used to construct a magma isomorphic to $M$,
## using `MagmaByMultiplicationTable'.
##
## For a magma <M>, `MultiplicationTable' returns the multiplication table
## w.r.t.~the sorted list of elements of <M>.
##
DeclareAttribute( "MultiplicationTable", IsHomogeneousList );
DeclareAttribute( "MultiplicationTable", IsMagma );
#############################################################################
##
#E
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