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Apr 12, 2014 at 18:14 comment added François G. Dorais The generalized Boolean algebras that the OP is talking about always have a bottom element. The difference is that they only have a relative complement operation $x - y$ rather than an absolute complement $-x$. Thus they always have a bottom $0 = x - x$ and they are Boolean algebras precisely when they have a top $1$, in which case the complement of $x$ is the relative complement $1-x$.
Apr 12, 2014 at 17:09 history answered Wouter Stekelenburg CC BY-SA 3.0