On the pseudo-commutativity of words and languages ?

For a given function f : Σ∗ → Σ∗ over an alphabet Σ, a language X ⊆ Σ∗ is said to f -commute with a language Y ⊆ Σ∗ if the equality XY = f(Y )X holds. This paper investigates the maximal solution to such f -commutative language equations for both morphic and antimorphic involutions f , under the condition that the known quantity (either X or Y ) is a singleton. We prove that, in most cases, the maximal solutions to these equations can be completely described in terms of what we call the twin-roots of the known word. Moreover, for these cases, we propose linear time algorithms which find the maximal solutions. Some of the results provide closedform constructive solutions to conjugacy equations of the type L1L = LL2 for the particular case where L1 = {f(u)} and L2 = {u} and f is a morphic or antimorphic involution. Our results also provide affirmative answers for a particular singleton case of what we call the extended Conway’s problem: Given a regular language L, is the maximal language X that f -commutes with L, i.e., XL = f(L)X (or with which L f -commutes, i.e., LX = f(X)L) regular?