38.18 Property Tests for Algebras

The following property tests (cf. Properties and Property Tests) are available for algebras.

IsAbelian( A ) :

returns true if the algebra A is abelian and false otherwise. An algebra A is abelian if and only if for every a, b in <A> the equation a* b = b* a holds.

IsCentral( A, U ) :

returns true if the algebra A centralizes the algebra U and false otherwise. An algebra A centralizes an algebra U if and only if for all a in <A> and for all u in <U> the equation a* u = u* a holds. Note that U need not to be a subalgebra of A but they must have a common parent algebra.

IsFinite( A ) :

returns true if the algebra A is finite, and false otherwise.

IsTrivial( A ) :

returns true if the algebra A consists only of the zero element, and false otherwise. If A is a unital algebra it is of course never trivial.

All tests expect a parent algebra or subalgebra and return true if the algebra has the property and false otherwise. Some functions may not terminate if the given algebra has an infinite set of elements. A warning may be printed in such cases.

    gap> IsAbelian( FreeAlgebra( GF(2), 2 ) );
    false
    gap> a:= UnitalAlgebra( Rationals, [ [ [ 1, 0 ], [ 0, 0 ] ] ] );
    UnitalAlgebra( Rationals, [ [ [ 1, 0 ], [ 0, 0 ] ] ] )
    gap> a.name:= "a";;
    gap> s1:= Subalgebra( a, [ One(a) ] );
    Subalgebra( a, [ [ [ 1, 0 ], [ 0, 1 ] ] ] )
    gap> IsCentral( a, s1 ); IsFinite( s1 );
    true
    false
    gap> s2:= Subalgebra( a, [] );
    Subalgebra( a, [  ] )
    gap> IsFinite( s2 ); IsTrivial( s2 );
    true
    true 

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