Could someone explain Exercise 4.1.1.7?

Find an operation on the set , i.e., a legitimate function , such that cannot be the multiplication formula for a monoid on . That is, either it is not associative or no element of can serve as a unit.

Sincerely,

Max

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Hi Max,

You can assign a value for each pair of inputs in such a way that either it won’t have identity element or fails associativity.

For example, if I take an elements (a,b) I can equate it to set {a,b}. Then the associativity (a*b)*c gets translated as (a*b) = (a,b) and *c will make it ((a,b),c). When we take a*(b*c), we get (a,(b,c)). Clearly associativity fails.

Oops! Above example is incorrect as domain is not M. One can define a function such that f(2,1)=1 and f(3,1)=4 as well as f(3,2)=2 and f(2,1)=2.

Then

3*(2*1)=3*f(2,1)=f(3,1)=4

(3*2)*1=f(3,2)*1=f(2,1)=2

which shows associativity fails!