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首页> 外文期刊>Order >Definability in Substructure Orderings, II: Finite Ordered Sets
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Definability in Substructure Orderings, II: Finite Ordered Sets

机译：子结构订购中的可定义性，II：有限订购集

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摘要

Let P be the ordered set of isomorphism types of finite ordered sets (posets), where the ordering is by embeddability. We study first-order definability in this ordered set. We prove among other things that for every finite poset P, the set {p, p~?} is definable, where p and P~? are the isomorphism types of P and its dual poset. We prove that the only non-identity automorphism of P is the duality map. Then we apply these results to investigate definability in the closely related lattice of universal classes of posets. We prove that this lattice has only one non-identity automorphism, the duality map; that the set of finitely generated and also the set of finitely axiomatizable universal classes are definable subsets of the lattice; and that for each member K of either of these two definable subsets, {K, K~?} is a definable subset of the lattice. Next, making fuller use of the techniques developed to establish these results, we go on to show that every isomorphism-invariant relation between finite posets that is definable in a certain strongly enriched second-order language L_2 is, after factoring by isomorphism, first-order definable up to duality in the ordered set P. The language L_2 has different types of quantifiable variables that range, respectively, over finite posets, their elements and order-relation, and over arbitrary subsets of posets, functions between two posets, subsets of products of finitely many posets (heteregenous relations), and can make reference to order relations between elements, the application of a function to an element, and the membership of a tuple of elements in a relation.
机译：令P为有限有序集（姿态）的同构类型的有序集，其中有序是通过可嵌入性实现的。我们在此有序集中研究一阶可定义性。我们证明，除其他外，对于每个有限的姿态P，集合{p，p〜？}是可定义的，其中p和P〜？是可定义的。是P及其对偶位姿的同构类型。我们证明P的唯一非同一性同构是对偶图。然后，我们将这些结果应用到普适型球类的紧密相关格中的可定义性。我们证明该格只有一个非同一性自同构，即对偶图；有限生成的集合以及有限可公理化的通用类的集合是晶格的可定义子集；对于这两个可定义子集的每个成员K，{K，K〜？}是晶格的可定义子集。接下来，我们将充分利用为建立这些结果而开发的技术，继续表明，在通过同构分解后，在某种高度丰富的二阶语言L_2中可以定义的有限姿态之间的所有同构不变性关系都是：语言L_2具有不同类型的可量化变量，其范围分别在有限的姿态，其元素和顺序关系以及姿态的任意子集，两个姿态之间的函数，子集的子集上有限多个位姿（异质关系）的乘积，并且可以引用元素之间的顺序关系，对元素的功能应用以及关系中元素元组的成员关系。

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  • 来源
    《Order》 |2010年第2期|P.115-145|共31页
  • 作者

    Jaroslav Jezek; Ralph McKenzie;

  • 作者单位

    Department of Mathematics, Charles University, Prague, Czech Republic;

    rnDepartment of Mathematics, Vanderbilt University, Nashville, TN 37235, USA;

  • 收录信息 美国《科学引文索引》(SCI);
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

    definability; ordered set; lattice; universal class; category;

    机译：可定义性有序集格子;通用类类别;

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