6. Relating the Submonoid and Subalgebra Membership Problems in Monoids and Monoid Rings

In the previous section we have seen how the subgroup problem is related to the membership problem for right respectively left ideals in group rings. Unfortunately, theorem 9 cannot be generalized for the submonoid problem as the following example shows:

for . But the submonoid itself in general is no longer the right congruence class . In our example we find that while . Hence the submonoid cannot be described adequately in the monoid ring using the right ideal congruence as in the subgroup case studied before. But there is another algebraic substructure of monoid rings which is appropriate to restate the submonoid problem in algebraic terms - the subalgebra.

- 1.
- ,
- 2.
- for all we have , and
- 3.
- for all we have .

- (1)
- .
- (2)
- .