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.

翻译：1991年Hébrard引入了一种单词分解方法，该方法后来被证明是研究单词散乱因子（亦称（散乱）子词或子序列）的有力工具。基于此，Karandikar与Schnoebelen首先提出了$k$-丰富性概念，随后Barker等人又提出了$k$-泛性概念。2022年Fleischmann等人通过将单词的拱形分解与其逆序拱形分解相交，提出了拱形分解的推广形式。尽管原作者仅将该分解用于最短缺失散乱因子的研究，但本文旨在对这一新型$α$-$β$分解本身进行探究。我们利用$1$-泛单词刻画了著名的$k$-泛单词的Simon同余关系。此外，我们将这些结果应用于二元单词，在此特例中获得了同余类的完全刻画并计算了同余指数。最后，我们初步探讨了三元情形，给出了$αβα$-因子的完整可能性列表，并刻画了其同余关系。