Piecewise testable languages via combinatorics on words

A regular language L over an alphabet A is called piecewise testable if it is a finite Boolean combination of languages of the form A ∗ a 1 A ∗ a 2 A ∗ … A ∗ a ℓ A ∗ , where a 1 , … , a ℓ ∈ A , ℓ ≥ 0 . An effective characterization of piecewise testable languages was given in 1972 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. Our contribution adds a new purely combinatorial proof.