Closures in Formal Languages and Kuratowski's Theorem

TL;DR

This paper explores the structure of formal languages generated by closure and complement operations, revealing a finite classification of possible algebras under these operations, inspired by Kuratowski's topological theorem.

Contribution

It extends Kuratowski's theorem to formal language settings, identifying exact counts of distinct algebraic structures generated by closure and complement.

Findings

Nine algebras with positive closure

Twelve algebras with Kleene closure

Finite classification of language algebras

Abstract

A famous theorem of Kuratowski states that in a topological space, at most 14 distinct sets can be produced by repeatedly applying the operations of closure and complement to a given set. We re-examine this theorem in the setting of formal languages, where closure is either Kleene closure or positive closure. We classify languages according to the structure of the algebra they generate under iterations of complement and closure. We show that there are precisely 9 such algebras in the case of positive closure, and 12 in the case of Kleene closure.