Recognizing pro-R closures of regular languages

TL;DR

This paper presents a method to construct a unary semigroup that recognizes the topological closure of a regular language within a specific algebraic framework, providing a new solution to the separation problem.

Contribution

It introduces an effective construction of a unary semigroup recognizing the pro-R closure of regular languages, advancing algebraic methods in formal language theory.

Findings

Constructed a unary semigroup recognizing the pro-R closure of regular languages.

Provided an effective solution to the separation problem for regular languages by R-languages.

Enhanced understanding of the algebraic structure of regular language closures.

Abstract

Given a regular language L, we effectively construct a unary semigroup that recognizes the topological closure of L in the free unary semigroup relative to the variety of unary semigroups generated by the pseudovariety R of all finite R-trivial semigroups. In particular, we obtain a new effective solution of the separation problem of regular languages by R-languages.