For every fixed class of regular languages, there is a natural hierarchy of increasingly more general problems: Firstly, the membership problem asks whether a given language belongs to the fixed class of languages. Secondly, the separation problem asks for two given languages whether they can be separated by a language from the fixed class. And thirdly, the covering problem is a generalization of separation problem to more than two given languages. Most instances of such problems were solved by the connection of regular languages and finite monoids. Both the membership problem and the separation problem were also extended to ordered monoids. The computation of pointlikes can be interpreted as the algebraic counterpart of the covering problem. In this paper, we extend the computation of pointlikes to ordered monoids. This leads to the notion of conelikes for the corresponding algebraic framework. We apply this framework to the Trotter-Weil hierarchy and both the full and the half levels of the $\text{FO}^2$ quantifier alternation hierarchy. As a consequence, we solve the covering problem for the resulting subvarieties of $\mathbf{DA}$. An important combinatorial tool are uniform ranker characterizations for all subvarieties under consideration; these characterizations stem from order comparisons of ranker positions.

中文翻译：

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圆锥喜欢度和排名比较

对于固定语言的每个固定类别，都有一个自然的层次结构，它会出现越来越普遍的问题：首先，成员资格问题询问给定语言是否属于固定语言。其次，分离问题要求两种给定的语言是否可以用某种语言将其从固定类中分离出来。第三，覆盖问题是将分离问题推广到两种以上的给定语言。此类问题的大多数情况是通过将常规语言和有限的monoid相联系而解决的。隶属度问题和分离度问题也都扩展到有序半边形。点状的计算可以解释为覆盖问题的代数形式。在本文中，我们将点似然的计算扩展到有序半边形。这导致了相应代数框架的锥状概念。我们将此框架应用于Trotter-Weil层次结构以及$ \ text {FO} ^ 2 $量词替换层次结构的全部和一半级​​别。结果，我们解决了$ \ mathbf {DA} $的子变量的覆盖问题。一个重要的组合工具是对所考虑的所有子变种进行统一的分级表征。这些特征来自排名者位置的顺序比较。一个重要的组合工具是对所考虑的所有子变种进行统一的分级表征。这些特征来自排名者位置的顺序比较。一个重要的组合工具是对所考虑的所有子变种进行统一的分级表征。这些特征来自排名者位置的顺序比较。