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.

该论文介绍了关于单词分解的几种概念，并对新的 α-β 分解进行了研究。该论文进一步阐明了 1 - 通用单词与 Simon 合同的关系，并对二进制单词进行了全面的分类和计算。最后，该论文还对三进制情况进行了研究，列出了所有可能的 αβα-factors 并描述了他们的合同关系。

α-β 分解及 Simon 同余式的二进制情况

$α$-$β$-Factorization and the Binary Case of Simon's Congruence

Determining the index of the Simon congruence is a long outstanding open
problem. Two words $u$ and $v$ are called Simon congruent if they have the same
set of scattered factors, which are parts of the word in the correct order but
not necessarily consecutive, e.g., $\mathtt{oath}$ is a scattered factor of
$\mathtt{logarithm}$. Following the idea of scattered factor $k$-universality,
we investigate $m$-nearly $k$-universality, i.e., words where $m$ scattered
factors of length $k$ are absent, w.r.t. Simon congruence. We present a full
characterisation as well as the index of the congruence for $m=1$. For $m\neq
1$, we show some results if in addition $w$ is $(k-1)$-universal as well as
some further insights for different $m$.

研究 Simon 同余中的 m - 近似 k - 普适性，提出了一种完整的特征描述及其 1 的同余指数，并对不同 m 值提供了一些进一步的见解。