The poset of all logics II: Leibniz classes and hierarchy

On the complexity of the Leibniz hierarchy

We prove that the problem of determining whether a finite logical matrix determines an algebraizable logic is complete for EXPTIME. The same result holds for the classes of order algebraizable, weakly algebraizable, equivalential and protoalgebraic logics. Finally, the same problem for the class of truth-equational logic is shown to be hard for EXPTIME.

Congruence Filter Pairs, Adjoints and Leibniz Hierarchy

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 lattic…

偏序集族

A Poset Hierarchy

本文扩展了一份文件的亚伯拉罕和邦纳概括了著名的豪斯多夫刻画类的分散线性订单。Abraham和Bonnet给出了一个偏序集层次，它描述了一类不具有无限反链的离散偏序集（缩写为FAC，表示有限反链条件）。这里的反链是在不可比性的意义上理解的。我们定义了一个比Abraham和Bonnet更大的偏序集层次，以包括更广泛的一类“散射”偏序集，我们称之为$κ$ -散射。这些偏序集不能嵌入任何顺序，使得对于每两个大小为$ < κ$的子集，一个严格小于另一个，其间有一个元素。如果一个线性序具有这个性质并且有$κ$大小，我们称这个集合为$\qkappa$。只有当$κ^{<κ}=κ$。具有如下性质的偏序：对于每个$a<b$，集合$\{x: a<x<b\}$的大小为$\geq κ$，称为弱$κ$ -稠密，不具有弱$κ$ -稠密子集的偏序称为强$κ$ -散布。我们证明了我们的层次包含所有的强$κ$ -分散的FAC偏序集，并且该层次包含在所有的FAC $κ$ -分散偏序集的类中。此外，我们证明了我们的层次结构实际上是所有$κ$ -良好成立的线性序下的反演，字典和和FAC削弱类的封闭。对于$κ=\aleph_0$，我们的层次结构与Abraham-Bonnet定理的层次结构一致。