Extensions of $ω$-Regular Languages

We consider extensions of monadic second order logic over $ω$-words, which are obtained by adding one language that is not $ω$-regular. We show that if the added language $L$ has a neutral letter, then the resulting logic is necessarily undecidable. A corollary is that the $ω$-regular languages are the only decidable Boolean-closed full trio over $ω$-words.

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