Let P be the divisor poset of 15. This consists of all natural numbers > 0...

 

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Let P be the divisor poset of 15. This consists of all natural numbers > 0 that are divisible by 15, with the partial order relation given by a 3 b if a | b. Let A(P) be the incidence algebra of P. We say that f EA(P) is invertible if there is some g E.A (P) such that f *g = 6. (Note that this means that the convolution product f *g takes the same value as 8 on every single interval in the poset.) (a) Consider the function f e A(P) defined as f([x,y])=y - x. Is f invertible? If not, justify. If yes, find its inverse with justification. (b) Consider the function f e A(P) defined as f ([x, y]) = size of the set [x, y]. Is f invertible? If not, justify. If yes, find its inverse with justification. Let P be the divisor poset of 15. This consists of all natural numbers > 0 that are divisible by 15, with the partial order relation given by a 3 b if a | b. Let A(P) be the incidence algebra of P. We say that f EA(P) is invertible if there is some g E.A (P) such that f *g = 6. (Note that this means that the convolution product f *g takes the same value as 8 on every single interval in the poset.) (a) Consider the function f e A(P) defined as f([x,y])=y - x. Is f invertible? If not, justify. If yes, find its inverse with justification. (b) Consider the function f e A(P) defined as f ([x, y]) = size of the set [x, y]. Is f invertible? If not, justify. If yes, find its inverse with justification.

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Solution a Let A 0 1 2 3 4 and consider the following partition of A 0 3 4 1 2 the relation R ind... View the full answer