Some language families generated by one-letter languages

Abstract The following results are shown: Let A be an infinite set of positive integers such that for all x , y in A , x y implies 2 x ⩽ y ⩽ kx for some fixed k . Then the language family generated by { a n ¦ n in A}∪ {e} with trio and intersection operations contains { a n b n ¦ n in N }. If k is a multiple of 3 or 4, then the language family generated by { a n k ¦ n in N } with trio and intersection operations contains { a n b n ¦ n in N }. These language families are commutative.