An effective characterization of piecewise testable languages was given by Simon who proved that a language L is piecewise testable if and only if its syntactic monoid is J-trivial. Nowadays there exist several proofs of this result based on various methods from algebraic theory of regular languages. In the first part of the talk we show a new purely combinatorial proof.

The class of all piecewise testable languages is also known as level 1 in Straubing-Therien hierarchy. In the second part of the talk we give an overview of our results concerning certain subhierarchies of levels 1 and 2 in this hierarchy.