Well quasi-orders and the shuffle closure of finite sets

TL;DR

This paper investigates when the shuffle closure of a single word set forms a well quasi-order, providing a characterization for such cases and advancing understanding of the order structure in formal language theory.

Contribution

It offers a complete characterization of when the shuffle closure of a singleton set is a well quasi-order, addressing an open problem from prior research.

Findings

Characterization of when the shuffle closure of a single word set is a well quasi-order.

Extension of the theory of well quasi-orders in the context of shuffle operations.

Provides a solution to an open problem in formal language theory.

Abstract

Given a set I of word, the set of all words obtained by the shuffle of (copies of) words of I is naturally provided with a partial order. In [FS05], the authors have opened the problem of the characterization of the finite sets I such that the order is a well quasi-order . In this paper we give an answer in the case when I consists of a single word w.