Earlier papers \cite{VB2022,VB2023a} introduced the notions of a core and an index of a relation (an index being a special case of a core). A limited form of the axiom of choice was postulated -- specifically that all partial equivalence relations (pers) have an index -- and the consequences of adding the axiom to axiom systems for point-free reasoning were explored. In this paper, we define a partial ordering on relations, which we call the \textsf{thins} ordering. We show that our axiom of choice is equivalent to the property that core relations are the minimal elements of the \textsf{thins} ordering. We also postulate a novel axiom that guarantees that, when \textsf{thins} is restricted to non-empty pers, equivalence relations are maximal. This and other properties of \textsf{thins} provide further evidence that our axiom of choice is a desirable means of strengthening point-free reasoning on relations. Although our novel axiom is valid for concrete relations and is a sufficient condition for characterising maximality, we show that it is not a necessary condition in the abstract point-free algebra. This leaves open the problem of deriving a necessary and sufficient condition.

翻译：早期论文\cite{VB2022, VB2023a}引入了关系的核与索引概念（索引是核的特例）。我们假定了选择公理的有限形式——具体而言，所有偏等价关系均存在索引——并探讨了将该公理加入无点推理公理系统后的推论。本文定义了关系上的偏序关系，称之为\textsf{thins}序。我们证明：选择公理等价于"核关系是\textsf{thins}序的最小元素"这一性质。此外，我们提出一条新公理，该公理保证当\textsf{thins}序限制于非空偏等价关系时，等价关系是极大元素。\textsf{thins}序的这一性质及其他特征进一步表明：选择公理是强化关系无点推理的合意工具。尽管新公理对具体关系成立且是刻画极大性的充分条件，但我们在抽象无点代数中证明该条件并非必要条件。这留下了寻求充要条件的开放问题。