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I asked this question on the TCS stackexchange but have so far received no answer:

Is there a name for classes of languages closed under finite union, inverse generalised sequential machine mappings and intersection with regular languages? These have come up naturally in my research and I don't want to invent a new name for them if they already have one.

An example of a class of languages with these closure properties that is not a full trio (i.e. closed under homomorphism, inverse homomorphism and intersection with regular) is the poly-context-free languages (intersections of finitely many context-free languages).

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