A prefix normal word is a binary word whose prefixes contain at least as many 1s as any of its factors of the same length. Introduced by Fici and Lipt\'ak in 2011 the notion of prefix normality is so far only defined for words over the binary alphabet. In this work we investigate possible generalisations for finite words over arbitrary finite alphabets, namely weighted and subset prefix normality. We prove that weighted prefix normality is more expressive than both binary and subset prefix normality and investigate the existence of a weighted prefix normal form. While subset prefix normality directly inherits most properties from the binary case, weighted prefix normality comes with several new peculiarities that did not already occur in the binary case. We characterise these issues and solve further questions regarding the weighted prefix normality and weighted prefix normal form.

中文翻译：

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加权前缀正常词：注意差距

前缀普通字是一个二进制字，其前缀至少包含与其相同长度的任何因子一样多的 1。Fici 和 Lipt\'ak 在 2011 年引入了前缀正态性的概念，目前仅针对二进制字母表上的单词定义。在这项工作中，我们研究了对任意有限字母表上有限单词的可能概括，即加权和子集前缀正态性。我们证明加权前缀正态性比二元和子集前缀正态性更具​​表现力，并研究了加权前缀范式的存在。虽然子集前缀正态性直接继承了二元情况的大多数属性，但加权前缀正态性带来了一些在二元情况下尚未出现的新特性。