Let F be a field. A ring A is called an F-algebra if A is an F-vector space. All algebras in GAP are associative, that is, the multiplication is associative.
An algebra always contains a zero element that can be obtained by subtracting an arbitrary element from itself. A discussion of identity elements of algebras (and of the consequences for the implementation in GAP) can be found in Algebras and Unital Algebras.
Elements of the field F are not regarded as elements of A. The
practical reason (besides the obvious mathematical one) for this is that
even if the identity matrix is contained in the matrix algebra A it is
not possible to write
1 + a for adding the identity matrix to the
a, since independent of the algebra A the meaning in
GAP is already defined as to add
1 to all positions of the matrix
a. Thus one has to write
One( A ) + a or
a^0 + a instead.
The natural operation domains for algebras are modules (see Operation for Algebras, and chapter Modules).
Up Top Next