# Is a proconstructible subsemigroup of $M_n(\mathbb{C})$ an intersection of constructible subsemigroups?

Let $S$ be a proconstructible subsemigroup of $M_n(\mathbb{C})$, that is a subsemigroup which is an intersection of constructible sets. Is $S$ an intersection of constructible subsemigroups?

The question arises from the model theory. The proconstructible sets are the type-definable sets and the constructible sets are the definable ones. It is well known that every type-definable subgroup of $M_n(\mathbb{C})$ is in fact definable (since it is $\omega$-stable) and in general in any stable theory a type-definable group is an intersection of definable subgroups.

Clarification: When I say "intersection of constructible sets", I mean countable intersection.

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