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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 yx1,yx2...yxn.

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 that
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

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    You should get acquainted with the [LaTeX guideline](http://meta.math.stackexchange.com/questions/107/faq-for-math-stackexchange/117#117). Also, use more line breaks and punctuation.2012-12-28

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Assume that $yx_i=yx_j$ for some distinct $i,j\in \{1,2,3,4\}$. Now multiply by $y^{-1}$ to get $x_i=x_j$ (contradiction)

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    I have edited my post.Would you consider it a good proof2012-12-28