Editing Monoid (section)

=== Products and powers ===
For each nonnegative integer {{math|''n''}}, one can define the product <math>p_n = \textstyle \prod_{i=1}^n a_i</math> of any sequence {{math|(''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>)}} of {{math|''n''}} elements of a monoid recursively: let {{math|1=''p''<sub>0</sub> = ''e''}} and let {{math|1=''p''<sub>''m''</sub> = ''p''<sub>''m''−1</sub> • ''a''<sub>''m''</sub>}} for {{math|1 ≤ ''m'' ≤ ''n''}}.

As a special case, one can define nonnegative integer powers of an element {{math|''x''}} of a monoid: {{math|1=''x''<sup>0</sup> = 1}} and {{math|1=''x''<sup>''n''</sup> = ''x''<sup>''n''−1</sup> • ''x''}} for {{math|''n'' ≥ 1}}.  Then {{math|1=''x''<sup>''m''+''n''</sup> = ''x''<sup>''m''</sup> • ''x''<sup>''n''</sup>}} for all {{math|''m'', ''n'' ≥ 0}}.