& here's a solution to Exercise 3.2. as usual, pls let me know if there are any errors in my reasoning Problem: Construct a monoid that is almost atomic but not nearly atomic.
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----- Solution: Let contain the reciprocals of the powers of ; that is Now let In other words, if , we can write -----
Now we determine Clearly, if it must be in the generating set of ; that is, either or However, the former case is invalid, as It is easy to see that the latter case works. Thus
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Note that the some of the on the right hand side must terminate at some point; for instance, we cannot say that is in ; we can only conclude that there exist an infinitude of elements in that approach Under this logic, it follows that there must exist some such that for all We can thus write for some Thus, if , then we can write It is easy to see that all elements of this form must work as well.
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We claim that is almost atomic. Specifically, for let us define so that which is truly atomic, reusing the same notation from 3.1.
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However, we will also prove that is not nearly atomic. Suppose by contradiction that there exists some such that is truly atomic for all However, setting we have Since is odd and is even, it is clear that cannot be written in the form for some and thus is not truly atomic, a contradiction. Thus our works by construction.