The elements of every subset of every Boolean algebra satisfy various algebraic conditions (such, for example, as the distributive laws) just by virtue of belonging to the same Boolean algebra. If the elements of some particular set E satisfy no conditions except these necessary universal ones, it is natural to describe E by some such word as “free.” A crude but suggestive way to express the fact that the elements of E satisfy no special conditions is to say that the elements of E can be transferred to an arbitrary Boolean algebra in a completely arbitrary way with no danger of encountering a contradiction. In what follows we shall make these heuristic considerations precise. We shall restrict attention to sets that generate the entire algebra; from the practical point of view the loss of generality involved in doing so is negligible.
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