Let be a -algebra, and let be a subset of its universe. The subset of generated by is defined to be the least subset of the universe which contains and is closed under the multiplication operation of the algebra. By associativity, this subset is the same thing as the image of the multiplication operation over the set . The following theorem shows that the elements generated by can be generated by using only the basic operations that were already used in the characterisation theorem from the parent page.
Theorem. Let be a -algebra with finite universe. For every subset of the universe, the subset generated by is the least subset that contains and is closed under binary product, shuffle, as well as the powers and .
The proof of this theorem is similar to the one in the parent page.