带整数谓词与不带整数谓词的实数加法理论
Theories of real addition with and without a predicate for integers
AI总结：
本文研究了实数加法结构中整数谓词的可定义性，通过拓扑刻画证明了可判定性，并指出两结构间无中间结构，且给出了一个非有效但简单的判定条件。
摘要：
我们证明了在结构⟨ℝ,+,<,ℤ⟩中可定义的关系是否也能在结构⟨ℝ,+,<,1⟩中定义，这一问题是可判定的。该结果通过以下方式获得：首先在⟨ℝ,+,<,ℤ⟩可定义关系族中给出⟨ℝ,+,<,1⟩可定义关系的拓扑刻画，然后遵循Muchnik的方法，将关系X的刻画在⟨ℝ,+,<,1,X⟩的逻辑中表达出来。上述刻画使我们能够证明在⟨ℝ,+,<,ℤ⟩与⟨ℝ,+,<,1⟩之间不存在中间结构。我们还证明了：一个⟨ℝ,+,<,ℤ⟩可定义关系是⟨ℝ,+,<,1⟩可定义的，当且仅当它与每条⟨ℝ,+,<,1⟩可定义直线的交集是⟨ℝ,+,<,1⟩可定义的。这给出了⟨ℝ,+,<,1⟩可定义关系的一个非有效但简单的刻画。
英文摘要：
We show that it is decidable whether or not a relation on the reals definable in the structure $\langle \mathbb{R}, +,<, \mathbb{Z} \rangle$ can be defined in the structure $\langle \mathbb{R}, +,<, 1 \rangle$. This result is achieved by obtaining a topological characterization of $\langle \mathbb{R}, +,<, 1 \rangle$-definable relations in the family of $\langle \mathbb{R}, +,<, \mathbb{Z} \rangle$-definable relations and then by following Muchnik's approach of showing that the characterization of the relation $X$ can be expressed in the logic of $\langle \mathbb{R}, +,<,1, X \rangle$. The above characterization allows us to prove that there is no intermediate structure between $\langle \mathbb{R}, +,<, \mathbb{Z} \rangle$ and $\langle \mathbb{R}, +,<, 1 \rangle$. We also show that a $\langle \mathbb{R}, +,<, \mathbb{Z} \rangle$-definable relation is $\langle \mathbb{R}, +,<, 1 \rangle$-definable if and only if its intersection with every $\langle \mathbb{R}, +,<, 1 \rangle$-definable line is $\langle \mathbb{R}, +,<, 1 \rangle$-definable. This gives a noneffective but simple characterization of $\langle \mathbb{R}, +,<, 1 \rangle$-definable relations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
