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 semialgebraic functions representing Boolean combinations of d-semialgebraic sets.

我们引入了d半代数（更一般地说，d可定义）集的布尔代数，并证明其 Stone 空间自然同构于半代数（可定义）集布尔代数的 Stone 空间的埃利斯包络半群。对于可定义连接的 o-最小群，我们证明该族与一维情况下的外部可定义集合之一一致。尽管如此，我们证明，总的来说，这两个族是不同的，即使在实代数数的半代数情况下也是如此。另一方面，在半代数情况下，我们描述代表d半代数集的布尔组合的实半代数函数。