同余滤子对、伴随与Leibniz族
Congruence Filter Pairs, Adjoints and Leibniz Hierarchy
摘要：
我们回顾的概念（有限）过滤器对作为一种工具，用于创建和分析逻辑。过滤器对可以被看作是逻辑的表示，通过将其理论的格表示为格同态的图像来给出，具有某些属性确保所得到的逻辑是有限的和替换不变的。每一个有限的，替换不变的逻辑产生于一个过滤器对。特殊类别的逻辑可以被表征为产生于特殊类别的过滤器对。我们考虑所谓的同余过滤器对，即过滤器对的域的格同态是一个格的同余一些拟簇。我们表明，类的逻辑承认介绍这样的过滤器对正是类的逻辑具有代数语义。我们研究的性质，一定的伽罗瓦连接来这样的过滤器对。我们给出了一个同余过滤器对的标准，提出了一个逻辑在某些类的莱布尼茨层次通过这种伽罗瓦连接，其相互作用与莱布尼茨运营商。作为应用，我们证明了一个桥定理，说明合并属性意味着克雷格插值性质，为一定类别的逻辑，包括非原代数逻辑。
英文摘要：
We review the notion of (finitary) filter pair as a tool for creating and analyzing logics. A filter pair can be seen as a presentation of a logic, given by presenting its lattice of theories as the image of a lattice homomorphism, with certain properties ensuring that the resulting logic is finitary and substitution invariant. Every finitary, substitution invariant logic arises from a filter pair. Particular classes of logics can be characterized as arising from special classes of filter pairs. We consider so-called congruence filter pairs, i.e. filter pairs for which the domain of the lattice homomorphism is a lattice of congruences for some quasivariety. We show that the class of logics admitting a presentation by such a filter pair is exactly the class of logics having an algebraic semantics. We study the properties of a certain Galois connection coming with such filter pairs. We give criteria for a congruence filter pair to present a logic in some classes of the Leibniz hierarchy by means of this Galois connection, and its interplay with the Leibniz operator. As an application, we show a bridge theorem, stating that the amalgamation property implies the Craig interpolation property, for a certain class of logics including non-protoalgebraic logics.
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