A note on intersections of free submonoids of a free monoid

Abstract

According to a theorem of Tilson [7] 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 ⌆^*, 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+...+δ^t))^*ɛ for some words Β₀, Β, γ, δ and ɛ, 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. If {x,y} and {u,v} are prefixes then necessarily t=0 and Β₀=1.