## Existence of universal gates (operation)

Take Boolean Algebra with two elements $(0, 1)$ for instance, the underlying finite field $0,1,\{.,+\}$ is equivalent to $0,1,\{NAND\}$ or $0,1,\{NOR\}$ 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