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^∗a₁A^∗a₂A^∗…A^∗a_?A^∗, where a₁,…,a_?∈A, ?≥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.