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Equations defining the polynomial closure of a lattice of regular languages

Branco, Mário J. J.; Pin, J.-E.

Lecture Notes in Computer Science, 5556 (2009), 115-126

The polynomial closure Pol TeX of a class of languages TeX of A * is the set of languages that are finite unions of marked products of the form L 0 a 1 L 1 ... a n L n , where the a i are letters and the L i are elements of TeX .

The main result of this paper gives an equational description of Pol TeX , given an equational description of TeX , when TeX is a lattice of regular languages closed under quotients, or a quotienting algebra of languages, as we call it in the sequel. The term “equational description” refers to a recent paper [5], where it was shown that any lattice of regular languages can be defined by a set of profinite equations. More formally, our main result can be stated as follows: