A note on algebras of languages

We study the Boolean algebras R,CS,D of regular languages, context-sensitive languages and decidable languages, respectively, over any alphabet. It is well known that R⊂CS⊂D, with proper inclusions. After observing that these Boolean algebras are all isomorphic, we study some immunity properties: for instance we prove that for every coinfinite decidable language L there exists a decidable language L′ such that L⊆L′, L′−L is infinite, and there is no context-sensitive language L″, with L″⊆L′ unless L″−L is finite; similarly, for every coinfinite regular language L there exists a context-sensitive language L′ such that L⊆L′, L′−L is infinite, and there is no regular language L″ such that L″⊆L′, unless L″−L is finite.