弱寡纯结构的Ryll-Nardzewski型定理
Towards a Ryll-Nardzewski-type Theorem for weakly oligomorphic structures
摘要：
一个结构称为弱寡纯的，如果它对每个n只实现2/3个n元正存在类型。本文的目的是表明，同态齐性和弱寡形的概念不仅完全类似于超齐性和寡形的经典概念，但实际上密切相关。第一个结果是同态-齐次关系结构的Fraïssé型定理。进一步我们证明了每个弱寡纯同态齐次结构包含（直到同构）唯一的齐次、同态齐次核，与之同态等价。由此，我们得到了每一个可数弱寡纯结构与一个有限或ω-范畴结构同态等价。另一个结果是弱寡纯结构的正存在理论被刻画为ω-范畴理论的正存在部分。最后，我们证明了可数弱寡纯结构的可数模型是相互同态等价的（我们称具有此性质的一阶理论为弱ω范畴的）.这些结果与Engeler-Ryll-Nardzewski-Svenonius-定理的一部分类似。
英文摘要：
A structure is called weakly oligomorphic if it realizes only finitely many n-ary positive existential types for every n. The goal of this paper is to show that the notions of homomorphism-homogeneity, and weak oligomorphy are not only completely analogous to the classical notions of ultrahomogeneity and oligomorphy, but are actually closely related. A first result is a Fraïssé-type theorem for homomorphism-homogeneous relational structures. Further we show that every weakly oligomorphic homomorphism-homogeneous structure contains (up to isomorphism) a unique homogeneous, homomorphism-homogeneous core, to which it is homomorphism-equivalent. As a consequence, we obtain that every countable weakly oligomorphic structure is homomorphism-equivalent with a finite or ω-categorical structure. Another result is the characterization of positive existential theories of weakly oligomorphic structures as the positive existential parts of ω-categorical theories. Finally, we show, that the countable models of countable weakly oligomorphic structures are mutually homomorphism-equivalent (we call first order theories with this property weakly ω-categorical). These results are in analogy with part of the Engeler-Ryll-Nardzewski-Svenonius-theorem.
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