A characterization theorem for injective model classes axiomatized by general rules
A characterization theorem for injective model classes axiomatized by general rules
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DOI：
10.1016/j.tcs.2006.02.025
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发表时间：
2006-08
期刊：
影响因子：
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通讯作者：
Zhaohui Zhu;Rong Zhang;Shan Lu
中科院分区：
文献类型：
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作者：
Zhaohui Zhu;Rong Zhang;Shan Lu
我们继续朱等人的工作。[正常条件的推理关系和内射模型，理论计算。Sci. 309（2003）287-311]。严格偏序结构（简称偏序集）类Ω称为可公理化的，如果来自Ω的所有内射偏好模型类可以用一般规则刻画。本文旨在得到可公理化类的一些特征。为此，提出了一种一元二阶框架语言。探讨了可公理化性与二阶可定义性之间的关系。然后引入了容许集的概念。基于这一概念，我们证明了任何不包含任何四结点子结构的偏好模型必定是某个内射模型的约简。此外，我们利用一般规则给出了一类内射偏好模型可公理化的充要条件。最后，我们证明了在某种意义下，不含四结点子结构的偏序集类是可公理化类中最大的一类。
We continue the work in Zhu et al. [Normal conditions for inference relations and injective models, Theoret. Comput. Sci. 309 (2003) 287–311]. A class Ω of strict partial order structures (posets, for short) is said to be axiomatizable if the class of all injective preferential models from Ω may be characterized in terms of general rules. This paper aims to obtain some characteristics of axiomatizable classes. To do this, a monadic second-order frame language is presented. The relationship between ℵ0-axiomatizability and second-order definability is explored. Then a notion of an admissible set is introduced. Based on this notion, we show that any preferential model, which does not contain any four-node substructure, must be a reduct of some injective model. Furthermore, we furnish a necessary and sufficient condition for the axiomatizability of classes of injective preferential models using general rules. Finally, we show that, in some sense, the class of all posets without any four-node substructure is the largest among axiomatizable classes.