$\alpha$-$\beta$-Factorization and the Binary Case of Simon's Congruence

TL;DR

This paper introduces the $ ext{ extalpha}$-$ ext{ extbeta}$-factorization, characterizes Simon's congruence for $k$-universal words, and provides a complete analysis for binary words, extending to some ternary cases.

Contribution

It presents a new $ ext{ extalpha}$-$ extbeta$-factorization, characterizes Simon's congruence in terms of $1$-universal words, and fully analyzes binary and some ternary cases.

Findings

Characterization of Simon's congruence for $k$-universal words.

Complete classification of binary word classes and congruence index.

Initial exploration of ternary case and $ ext{ extalpha} ext{ extbeta} ext{ extalpha}$-factors.

Abstract

In 1991 H\'ebrard introduced a factorization of words that turned out to be a powerful tool for the investigation of a word's scattered factors (also known as (scattered) subwords or subsequences). Based on this, first Karandikar and Schnoebelen introduced the notion of $k$-richness and later on Barker et al. the notion of $k$-universality. In 2022 Fleischmann et al. presented a generalization of the arch factorization by intersecting the arch factorization of a word and its reverse. While the authors merely used this factorization for the investigation of shortest absent scattered factors, in this work we investigate this new $\alpha$-$\beta$-factorization as such. We characterize the famous Simon congruence of $k$-universal words in terms of $1$-universal words. Moreover, we apply these results to binary words. In this special case, we obtain a full characterization of the classes and calculate the index of the congruence. Lastly, we start investigating the ternary case, present a full list of possibilities for $\alpha\beta\alpha$-factors, and characterize their congruence.