Conelikes and Ranker Comparisons

Abstract

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 consider the extension 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.