The following property tests (cf. Properties and Property Tests) are available for algebras.
IsAbelian( A ):
trueif the algebra A is abelian and
falseotherwise. 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 ):
trueif the algebra A centralizes the algebra U and
falseotherwise. 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 ):
trueif the algebra A is finite, and
IsTrivial( A ):
trueif the algebra A consists only of the zero element, and
falseotherwise. 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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