标题： 稳定过一个谓词的融合和Gaifman性质 显示英文标题

标题： Stable amalgamation over a predicate and the Gaifman property

摘要： 我们考虑一阶理论T的一个性质，该理论有一个指定的单元谓词P：理论P的每个模型都作为T的某个模型的P部分出现。我们称这个性质为Gaifman性质。 Gaifman猜想，如果T在P上是相对范畴的，则它具有Gaifman性质。 我们提出这个猜想的一个推广版本：如果T不满足Gaifman性质，则它在P上表现出非结构，即在许多基数上具有许多不同构的模型。 我们针对可数理论来讨论这个猜想。 受分类理论思想的启发，我们将这个猜想分为两部分：1）在P上的稳定性（理论的结构性质）蕴含Gaifman性质，2）在P上的不稳定蕴含非结构。 在本文中，我们证明了这个猜想的第一部分。 事实上，我们证明了一个更强的陈述：适当版本的稳定性蕴含更高阶的稳定粘合性质。 显示英文摘要

摘要： We consider the following property of a first order theory T with a distinguished unary predicate P: every model of the theory of P occurs as the P-part of some model of T. We call this property the Gaifman property. Gaifman conjectured that if T is relatively categorical over P, then it has the Gaifman property. We propose a generalized version of this conjecture: if T fails the Gaifman property, then it exhibits non-structure over P, i.e., has many non-isomorphic models over P in many cardinalities. We address this conjecture for countable theories. Motivated by ideas from Classification Theory, we separate this conjecture into two parts: 1) stability over P (a structure property of theories) implies the Gaifman property, and 2) instability over P implies non-structure. In this paper prove the first part of this conjecture. In fact, we prove a stronger statement: an appropriate version of stability implies higher stable amalgamation properties.