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Here`s a solution to exercise 3.2. Please let me know if there are any errors.
Define . Now let The set K is an infinite subset of . Then it is countable, and we can construct a bijection f between and where is the set of prime numbers greater than 3. Define the monoid where . Firstly let's prove that . We can write Then any number k such that is not an atom. We can prove that the elements are atoms. Suppose that where a and b are elements of M. Then we can write these elements as sum of generators. Let and with all and different than k. Then we have but in this equality of the left side is -1 and for the right side 0. This contradiction proves the result. M is not atomic. Indeed 1/2 cannon be written as a sum of atoms. Suppose that . If does not divide then of the right side is -1 and for the left side 0, contradiction. Then for all i=1,2,...,n. Then we have that where . If 3 does not divide then of the right side is -1 and for the left side 0. Then but this implies that , a contradiction. Let`s prove that M is nearly atomic. Firstly we have that . We can write an arbitrary element of M as where and are atoms of M. Then we have that . Then M is nearly atomic.
Define . Now let The set K is an infinite subset of . Then it is countable, and we can construct a bijection f between and where is the set of prime numbers greater than 3. Define the monoid where . Firstly let's prove that . We can write Then any number k such that is not an atom. We can prove that the elements are atoms. Suppose that where a and b are elements of M. Then we can write these elements as sum of generators. Let and with all and different than k. Then we have but in this equality of the left side is -1 and for the right side 0. This contradiction proves the result. M is not atomic. Indeed 1/2 cannon be written as a sum of atoms. Suppose that . If does not divide then of the right side is -1 and for the left side 0, contradiction. Then for all i=1,2,...,n. Then we have that where . If 3 does not divide then of the right side is -1 and for the left side 0. Then but this implies that , a contradiction. Let`s prove that M is nearly atomic. Firstly we have that . We can write an arbitrary element of M as where and are atoms of M. Then we have that . Then M is nearly atomic.