above, we said that a free boolean algebra is a boolean algebra with a set of generators that behave a certain way; alternatively, one might start with a set and ask which algebra it generates. every set "x" generates a free boolean algebra "fx" defined as the algebra such that for every algebra "b" and function "f" : "x" → "b", there is a unique boolean algebra homomorphism "f"′ : "fx" → "b" that extends "f". diagrammatically,