Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0406481

Interpreting groups and fields in some nonelementary classes

Tapani Hyttinen·Olivier Lessmann·Saharon Shelah
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper is concerned with extensions of geometric stability theory to some nonelementary classes. We prove the following theorem: Theorem: Let C be a large homogeneous model of a stable diagram D. Let p, q in S_D(A), where p is quasiminimal and q unbounded. Let P=p(C) and Q=q(C). Suppose that there exists an integer n<omega such that dim(a_1...a_n/A cup C)=n, for any independent a_1,..., a_n in P and finite subset C subseteq Q, but dim(a_1...a_n a_{n+1}/A cup C) <= n, for some independent a_1,...,a_n,a_{n+1} in P and some finite subset C subseteq Q. Then C interprets a group G which acts on the geometry P' obtained from P. Furthermore, either C interprets a non-classical group, or n=1,2,3 and * If n=1 then G is abelian and acts regularly on P'. * If n=2 the action of G on P' is isomorphic to the affine action of K times K^* on the algebraically closed field K. * If n = 3 the action of G on P' is isomorphic to the action of PGL_2(K) on the projective line {P}^1(K) of the algebraically closed field K .

Explore related subjects

math.LO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Tapani Hyttinen, Olivier Lessmann, Saharon Shelah. 2004-06-23. Interpreting groups and fields in some nonelementary classes. https://arxiv.org/abs/math/0406481

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Loops, Inverse Limits and Non-Determinism

We introduce an operator on problems in Weihrauch complexity, which we call the infinite loop or inverse limit, and which corresponds to an infinite compositional product. This operation arises naturally whenever one implements algorithms that produce a sequence of results in an infinite loop, using some fixed subroutine. We prove that the corresponding operator is monotone with respect to (strong) Weihrauch reducibility but that it is not a closure operator. One of our findings is that weak Kőnig's lemma is closed under infinite loops, which implies that the class of non-deterministically computable problems is also closed under this operation. Consequently, this class allows for a high degree of flexibility in programming. As our main technical tools, we present an injective version of the recursion theorem and an infinitary version of the so-called independent choice theorem. We also show that, in general, the infinite loop operator is more powerful than the composition of the diamond operator followed by the parallelization operator. However, in many practical scenarios, these compositions yield a result, which coincides with the application of the infinite loop operator. Finally, we discuss the special situation of loops for single-valued problems and for problems on Turing degrees.

math.LO↗
arXiv2026

Scott topologies on meet-continuous domains

We study Scott products and sobriety of countable meet-continuous domains, meaning meet-continuous dcpos without any additional continuity or least-element assumption. Using the complete-lattice test-family theorem of Xu and Ji, we prove finite-product equality for those domains that are $L$-dcpos, and for the weaker class whose principal ideals have suprema of all nonempty subsets. For an arbitrary family of nonempty countable meet-continuous $L$-dcpos, we prove that the Scott topology on the order product equals the product of the factor Scott topologies if and only if only finitely many factors lack a least element. We also establish sobriety under bounded completeness and under additional common-upper-bound conditions. Assuming square-product equality, sobriety is characterized by Scott closedness of common-upper-bound sections associated with irreducible Scott-closed sets, with an equivalent sequential formulation in the countable case. Two extraction lemmas extend to meet-semilattice dcpos. The finite-product and sobriety questions for general countable meet-continuous domains remain unresolved here.

math.LO↗
arXiv2026

Randomized Borel $(2d+1)$-coloring of digraphs

Let $G$ be a Borel digraph with maximum out-degree $d \in \mathbb{N}$. We show that $G$ admits a random Borel $(2d+1)$-coloring for which every edge is almost surely not monochromatic. This gives a simpler proof of a recent result of Pelayo-Gómez: such a graph $G$ admits a measurable proper $(2d+1)$-coloring with respect to any Borel probability measure on $V(G)$. Our proof is an adaptation of Pelayo-Gómez's proof to the randomized Borel setting.

math.LO↗