# 38.26 Algebra Elements

This section describes the operations and functions available for algebra elements.

Note that algebra elements may exist independently of an algebra, e.g., you can write down two matrices and compute their sum and product without ever defining an algebra that contains them.

Comparisons of Algebra Elements

`g = h`:

evaluates to `true` if the algebra elements g and h are equal and to `false` otherwise.

`g < h`:

evaluates to `true` if the algebra elements g and h are not equal and to `false` otherwise.

`g < h`
`g <= h`
`g = h`
`g h`

The operators `<`, `<=`, `=` and evaluate to `true` if the algebra element g is strictly less than, less than or equal to, greater than or equal to and strictly greater than the algebra element h. There is no general ordering on all algebra elements, so g and h should lie in the same parent algebra. Note that for elements of finitely presented algebra, comparison means comparison with respect to the underlying free algebra (see Elements of Finitely Presented Algebras).

Arithmetic Operations for Algebra Elements

`a * b`
`a + b`
`a - b`

The operators `*`, `+` and `-` evaluate to the product, sum and difference of the two algebra elements a and b. The operands must of course lie in a common parent algebra, otherwise an error is signalled. `a / c`

returns the quotient of the algebra element a by the nonzero element c of the base field of the algebra.

`a ^ i`

returns the i-th power of an algebra element a and a positive integer i. If i is zero or negative, perhaps the result is not defined, or not contained in the algebra generated by a.

`list + a`
`a + list`
`list * a`
`a * list`

In this form the operators `+` and `*` return a new list where each entry is the sum resp. product of a and the corresponding entry of list. Of course addition resp. multiplication must be defined between a and each entry of list.

GAP 3.4.4
April 1997