Does the cartesian product have a neutral element? Let $A$ be any set. Is there a set $E$ such that $A \times E = E \times A = A$?
I thought of the empty set, but Wikipedia says otherwise. This operation changes dimension, so an isomorphism might be needed for such element to exist.

For "equality", no ... if $a \in A$ then it does not have the form $(a,e)$ of an element of $A \times E$, by the Axiom of Foundation.  
For "bijection", yes.  I leave that to you.
