A note on intersections of free submonoids of a free monoid

According to a theorem of Tilson [6] any intersection of free submonoids of a free monoid is free. Here we consider intersections of the form {x, y}* ∩ {u, v}*, where x, y, u and v are words in a finitely generated free monoid Σ*, and show that if both the monoids {x, y}* and {u, v}* are of the rank two, then the intersection is a free monoid generated either by (at most) two words or by a regular language of the form β0 + β((γ(1+ δ + ... δt))*ε for some words β0, β, γ, δ and ε, and some integer t≥0. An example is given showing that the latter possibility may occur for each t≥0 with nonempty values of the words.