Operations preserving regular languages

Given a strictly increasing sequence s of non-negative integers, filtering a word a0a1 · · · an by s consists in deleting the letters ai such that i is not in the set {s0, s1, . . .}. By a natural generalization, denote by L[s], where L is a language, the set of all words of L filtered by s. The filtering problem is to characterize the filters s such that, for every regular language L, L[s] is regular. In this paper, the filtering problem is solved, and a unified approach is provided to solve similar questions, including the removal problem considered by Seiferas and McNaughton. Our approach relies on a detailed study of various residual notions, notably residually ultimately periodic sequences and residually rational transductions.