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

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