  We introduce the Boolean algebra of d-semialgebraic (more generally,
d-definable) sets and prove that its Stone space is naturally isomorphic to the
Ellis enveloping semigroup of the Stone space of the Boolean algebra of
semialgebraic (definable) sets. For definably connected o-minimal groups, we
prove that this family agrees with the one of externally definable sets in the
one-dimensional case. Nonetheless, we prove that in general these two families
differ, even in the semialgebraic case over the real algebraic numbers. On the
other hand, in the semialgebraic case we characterise real semialgrebraic
functions representing Boolean combinations of d-semialgebraic sets.

Logic;Algebraic Geometry