Existence of universal gates (operation)

Take Boolean Algebra with two elements for instance, the underlying finite field is equivalent to or where NAND and NOR are considered as universal gates. This means that instead of defining two operations , I can work with either NAND or NOR and all things will fall in places. Is this property, which is true in 2 elements field, that (‘multiplication’) and (‘addition’) can be written in terms of a single universal binary relation (e.g. NAND or NOR) is true with every finite fields(rings)?
I have posted it on a online math community. The link is here, http://math.stackexchange.com/questions/24942/universal-binary-operation-and-finite-fields-ring