39.9 Elements of Finitely Presented Algebras

Zero and One of Finitely Presented Algebras

A finitely presented algebra A contains a zero element A.zero. If the number of generators of A is not zero, the multiplicative neutral element of A is A.one, which is the zero-th power of any nonzero element of A.

Comparisons of Elements of Finitely Presented Algebras

x = y
x < y

Elements of the same algebra can be compared in order to form sets. Note that probably it will be necessary to compute an isomorphic matrix representation in order to decide equality if x and y are not elements of a free algebra.

    gap> a:= FreeAlgebra( Rationals, 1 );;
    gap> a:= a / [ a.1^2 - a.one ];
    UnitalAlgebra( Rationals, [ a.1 ] )
    gap> [ a.1^3 = a.1, a.1^3 > a.1, a.1 > a.one, a.zero > a.one ];
    [ true, false, false, false ] 

Arithmetic Operations for Elements of Finitely Presented Algebras

x + y
x - y
x * y
x ^ n
x / c

The usual arithmetical operations for ring elements apply to elements of finitely presented algebras. Exponentiation ^ can be used to raise an element x to the n-th power. Division / is only defined for denominators in the base field of the algebra.

    gap> a:= FreeAlgebra( Rationals, 2 );;
    gap> x:= a.1 - a.2;
    a.1+-1*a.2
    gap> x^2;
    a.1^2+-1*a.1*a.2+-1*a.2*a.1+a.2^2
    gap> y:= 4 * x - a.1;
    3*a.1+-4*a.2
    gap> y^2;
    9*a.1^2+-12*a.1*a.2+-12*a.2*a.1+16*a.2^2 

IsFpAlgebraElement( obj )

returns true if obj is an element of a finitely presented algebra, and false otherwise.

    gap> IsFpAlgebraElement( a.zero );
    true
    gap> IsFpAlgebraElement( a.field.zero );
    false 

FpAlgebraElement( A, coeff, words )

Elements of finitely presented algebras normally arise from arithmetical operations. It is, however, possible to construct directly the element of the finitely presented algebra A that is the sum of the words in the list words, with coefficients given by the list coeff, by calling FpAlgebraElement( A, coeff, words ). Note that this function does not check whether some of the words are equal, or whether all coefficients are nonzero. So one should probably not use it.

    gap> a;
    UnitalAlgebra( Rationals, [ a.1, a.2 ] )
    gap> FpAlgebraElement( a, [ 1, 1 ], a.generators );
    a.1+a.2
    gap> FpAlgebraElement( a, [ 1, 1, 1 ], List( [ 1..3 ], i -> a.1^i ) );
    a.1+a.1^2+a.1^3 

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