Elementary properties of ordered abelian groups
Elementary properties of ordered abelian groups
复制标题
有序交换群的基本性质
DOI：
10.1090/s0002-9947-1960-0114855-0
复制
发表时间：
1960
影响因子：
1.3
通讯作者：
E. Zakon
中科院分区：
文献类型：
--
作者：
A. Robinson;E. Zakon
导论. Szmielew [9]给出了交换群的基本性质（即可以在下谓词演算中形式化的性质）的完整分类。然而，迄今为止，还没有对有序群进行过这样的尝试。在本文件中的分类基本性质将进行所有阿基米德有序阿贝尔群，而且，为某一更一般的一类群体，我们将呼吁定期有序。同时，将建立两个这样的群在初等上等价的充分必要条件，即它们的所有初等性质是共同的（见定理4.7）。通过一个群类的完全分类，我们指的是它划分成不相交的子类，使得两个群属于一个子类，当且仅当它们是初等等价的。这一目标将通过建立一系列（有限或无限）完备公理系统来实现，每个系统定义有序交换群的某个子类。[8]的符号和术语将在全文中使用。特别地，将应用在[8]和[7]中引入的模型完备性的概念。本文的一个新奇之处是将基于模型完备性的方法与我们称之为“新关系的附加”相结合。为了说明这种方法的有用性，我们还将用它来证明Langford和Tarski关于某些涉及有序集的公理系统的完备性的一些定理。这将构成一个额外的结果的文件（见？2）。1.词汇表、术语和符号。有序集的概念可以通过以下公理系统在下谓词演算中形式化，该公理系统基于“等价”的二元关系E（x，y）（读作：“x等价于y”）和“序”的二元关系Q（x，y）（读作：“x小于或等价于y”）：
Introduction. A complete classification of abelian groups by their elementary properties (i.e. properties that can be formalized in the lower predicate calculus) was given by Szmielew [9]. No such attempt, however, has so far been made with respect to ordered groups. In the present paper the classification by elementary properties will be carried out for all archimedean ordered abelian groups and, moreover, for a certain more general class of groups which we shall call regularly ordered. Simultaneously, necessary and sufficient conditions will be established for two such groups to be elementarily equivalent, i.e. to have all their elementary properties in common (see Theorem 4.7). By a complete classification of a class of groups we mean its partition into disjoint subclasses in such a way that two groups belong to one subclass if, and only if, they are elementarily equivalent. This goal will be attained by setting up a series of (finite or infinite) complete systems of axioms, each system defining a certain subclass of ordered abelian groups. The notation and terminology of [8] will be used throughout. In particular, the concept of model-completeness introduced in [8] and [7] will be applied. A novelty feature of the present paper is that the method based on model-completeness will be combined with what we shall call "adjunction of new relations." To illustrate the usefulness of the method, we shall also apply it to give new simplified proofs of some theorems by Langford and Tarski on the completeness of certain systems of axioms referring to ordered sets. This will constitute an additional result of the paper (see ?2). 1. Preliminaries, terminology and notation. The concept of an ordered set can be formalized in the lower predicate calculus by means of the following system of axioms based on a binary relation of "equivalence," E(x, y), (read: "x is equivalent to y") and a binary relation of "order," Q(x, y) (read: "x is less than, or equivalent to, y"):