无限结构上的存在关系

IF 0.6 4区 数学 Q3 MATHEMATICS
Boris A. Romov
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引用次数: 0

摘要

我们为无限域上的结构M建立了一个标准,使其具有Galois闭包\({{\,\textrm{InvAut}\,}}(M)\)(M域上对M的所有自同构不变的所有关系的集合),我们通过来自M的所有存在关系的有限部分自同构,给出了M中的量词消去的准则,以及M的(弱)齐性的准则。然后,我们用可数签名描述了M的性质,对于该性质,由M上的量词费公式表示的所有关系的集合是弱归纳的,即,这个集合在相同arity关系的任意无限交集下都是封闭的。证明了最后一个条件是等价的:对于每个\(n\ge1\),由n个元素生成的M的子结构只有有限多个同构类型。在具有可数签名的代数的情况下,这种类型可以由有限方程组的所有解的集合和由这些代数上的n元项产生的不等式来定义。接下来,我们证明了对于具有有限签名的有限M,从\({{\,\textrm{InvAut}\,}}(M)\)通过表示它的M上的一阶公式描述任何关系的问题在算法上是可解的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Existential relations on infinite structures

We establish a criterion for a structure M on an infinite domain to have the Galois closure \({{\,\textrm{InvAut}\,}}(M)\) (the set all relations on the domain of M that are invariant to all automorphisms of M) defined via infinite Boolean combinations of infinite (constructed by infinite conjunction) existential relations from M. Based on this approach, we present criteria for quantifier elimination in M via finite partial automorphisms of all existential relations from M, as well as criteria for (weak) homogeneity of M. Then we describe properties of M with a countable signature, for which the set of all relations, expressed by quantifier-fee formulas over M, is weakly inductive, that is, this set is closed under any infinitary intersection of the same arity relations. It is shown that the last condition is equivalent: for every \(n \ge 1\) there are only finitely many isomorphism types for substructures of M generated by n elements. In case of algebras with a countable signature such type can be defined by the set of all solutions of a finite system of equations and inequalities produced by n-ary terms over those algebras. Next, we prove that for a finite M with a finite signature the problem of the description of any relation from \({{\,\textrm{InvAut}\,}}(M)\) via the first order formula over M, which expresses it, is algorithmically solvable.

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来源期刊
Algebra Universalis
Algebra Universalis 数学-数学
CiteScore
1.00
自引率
16.70%
发文量
34
审稿时长
3 months
期刊介绍: Algebra Universalis publishes papers in universal algebra, lattice theory, and related fields. In a pragmatic way, one could define the areas of interest of the journal as the union of the areas of interest of the members of the Editorial Board. In addition to research papers, we are also interested in publishing high quality survey articles.
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