where "i" is the inclusion, and the dashed arrow denotes uniqueness. the idea is that once one chooses where to send the elements of "x", the laws for boolean algebra homomorphisms determine where to send everything else in the free algebra "fx". if "fx" contained elements inexpressible as combinations of elements of "x", then "f"′ wouldn't be unique, and if the elements of "x" weren't sufficiently independent, then "f"′ wouldn't be well defined! it is easily shown that "fx" is unique (up to isomorphism), so this definition makes sense. it is also easily shown that a free boolean algebra with generating set x, as defined originally, is isomorphic to "fx", so the two definitions agree.