Equations defining the polynomial closure of a lattice of regular languages

Abstract

The polynomial closure Pol (ℒ)of a class of languages ℒ 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 ℒ. The main result of this paper gives an equational description of Pol (ℒ), given an equational description of ℒ, when ℒ 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: © 2009 Springer Berlin Heidelberg.