The problem is to prove that the existence of an idempotent (an element whose "square" is itself, that is b * b = b) is a structural property.
I would just like someone to proofread my proof and give me some feedback....Thanx! (note: E="an element of")
Pf: Let <S,*> and <T,#> be isomorphic algebraic structures and suppose a*a=a, where a E S. Assume f: <S,*> --> <T,#> is an isomorphism. Then x=f(a) E T. Now,
x#x = f(a) # f(a) by substitution
= f(a*a) by operation preserving
= f(a) since a*a=a
= x as desired.
Hence, the existence of an idempotent is a srtuctural property.