beyond half synchronized systems

In irreducible subshifts, a word m is synchronizing if whenever vm and mw are admissible words,then vmw is admissible as well. a word m is (left) half (resp. weak) synchronizing, when there is a left transitive ray (resp. a left ray) x− such that if x−m and mw are admissible, then x−mw is also admissible. the respective subshifts are called half (resp. weak) synchronized. k. thomsen in [on the structure of a sofic shift space, american mathematical society, 356, number 9, 3557-3619] considers a synchronized component of a general subshift and investigates the approximation of entropy from inside of this component by some certain sfts. we, using a rather different approach, show how this result extends to weak synchronized systems.