On morphisms preserving infinite Lyndon words

In a previous paper, we characterized free monoid morphisms preserving finite Lyndon words. In particular, we proved that such a morphism preserves the order on finite words. Here we study morphisms preserving in finite Lyndon words and morphisms preserving the order on in finite words. We characterize them and show relations with morphisms preserving Lyndon words or the order on finite words. We also briefly study morphisms preserving border-free words and those preserving the radix order.