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Hi! here's my thoughts/tentative solution on Exercise 3.3. it feels a little sus somehow, mainly because using the function sounds like cheating, so please let me know if there are any logical errors. thanks!
Problem: Construct a monoid that is nearly atomic but not atomic.
Solution: Consider the anomalous addition function defined by if and otherwise. Also, define the parity addition function such that we have and Let be the operation Then, we let , where
Using an argument similar to the one in 3.2, we find the atoms of Clearly, they must be in but since it follows that It thus easily follows that
The above implies that is not atomic: specifically, elements of the form with are not truly atomic, again using the same terms from 3.1.
However, is nearly atomic: letting and , we have that but implies that the above sum is , which is truly atomic as Otherwise, if , then is or ; since , we must have that is invertible. Thus is nearly atomic but not atomic, so we are done.
Problem: Construct a monoid that is nearly atomic but not atomic.
Solution: Consider the anomalous addition function defined by if and otherwise. Also, define the parity addition function such that we have and Let be the operation Then, we let , where
Using an argument similar to the one in 3.2, we find the atoms of Clearly, they must be in but since it follows that It thus easily follows that
The above implies that is not atomic: specifically, elements of the form with are not truly atomic, again using the same terms from 3.1.
However, is nearly atomic: letting and , we have that but implies that the above sum is , which is truly atomic as Otherwise, if , then is or ; since , we must have that is invertible. Thus is nearly atomic but not atomic, so we are done.
This post has been edited 1 time. Last edited by smileapple, May 14, 2023, 7:03 PM