Jean-Eric Pin ()

This main result of this paper states that a recognizable language is open in the Hall topology if and only if it belongs to the polynomial closure C of the group languages. A group language is a recognizable language accepted by a permutation automaton. The polynomial closure of a class of languages L is the set of languages that are finite unions of languages of the form L0a1L1 ... anLn, where the ai's are letters and the Li's are elements of L. The Hall topology is the topology in which the open sets are finite or infinite unions of group languages. We also give a combinatorial description of these languages and a syntactic characterization. Let L be a recognizable set of A*, let M be its syntactic monoid and let P be its syntactic image. Then L belongs to C if and only if, for every s, t in M and for every idempotent e of M, st in P implies set in P. Finally, we give a characterization on the minimal automaton of L that leads to a polynomial time algorithm to check, given a finite deterministic automaton, whether it recognizes of C.

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