Proving that every element of a monoid occurs exactly once
let (B,*) defines a monoid with a finite number of elements Let the elements of B be x1,x2,x3,x4 where every element of B occurs exactly once in this list.....let y be the invertible element of the monoid.. prove that every element of the monoid occurs exactly once in this list y*x1,y*x2...y*xn.
I have started by saying let x be an invertible element of B and let x^-1 be its inverse.This inverse element x^-1 is uniquely determined by x according to a theorem which states that every element of a monoid can have at most one inverse. To prove that every element of the monoid occurs once,I have to show that no two elements have the same inverse. let e be the identity element of B w * x1 = e w * x2 = e
I have to show that x1 and x2 are uniquely determined by x.
Knowing that an element of a monoid can have at most one inverse,
i would assume
w * x1 = x1 * w=e
w * x2= x2 * w=e
then x1= x1 * e = x1 * (w * x2)= (x1 * w )* x2= e * x2 = x2
thus x1=x2 which proves that an element of a monoid can have at most one inverse.I am not sure if this shows that every element in the list occurs exactly once