Boundedness and absoluteness of some dynamical invariants in model theory

Krupiński, Krzysztof; Newelski, Ludomir; Simon, Pierre

doi:10.1142/s0219061319500120

“…In [Hru20], Hrushovski proved the existence of locally compact and Lie models in a generalized sense involving quasi-homomorphisms, and used them to give complete classifications of approximate lattices in SL n pZq and SL n pQ p q. This work is very advanced; among various tools, it uses a new locally compact group attached to a theory invented by Hrushovski as a counterpart of the Ellis group (or rather its canonical Hausdorff quotient) of a first order theory which was defined and studied in [KPR18], [KNS19], and [KR20].…”

Section: Introductionmentioning

confidence: 99%

Locally compact models for approximate rings

Krupiński¹

2022

Preprint

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By an approximate subring of a ring we mean an additively symmetric subset X such that X ¨X Y pX `Xq is covered by finitely many additive translates of X. We prove that each approximate subring X of a ring has a locally compact model, i.e. a ring homomorphism f : xXy Ñ S for some locally compact ring S such that f rXs is relatively compact in S and there is a neighborhood U of 0 in S with f ´1rU s Ď 4X `X ¨4X (where 4X :" X `X `X `X). This S is obtained as the quotient of the ring xXy interpreted in a sufficiently saturated model by its type-definable ring connected component. The main point is to prove that this component always exists. In order to do that, we extend the basic theory of model-theoretic connected components of definable rings (developed in [GJK20] and [KR20]) to the case of rings generated by definable approximate subgrings and we answer a question from [KR20] in the more general context of approximate subrings. Namely, let X be a definable (in a structure M ) approximate subring of a ring and R :" xXy. Let X be the interpretation of X in a sufficiently saturated elementary extension and R :" x Xy. It follows from [MW15] that there exists the smallest M -type-definable subgroup of p R, `q of bounded index, which is denoted by p R, `q00M . We prove that p R, `q00 M `R ¨p R, `q00 M is the smallest M -type-definable two-sided ideal of R of bounded index, which we denote by R00M . Then S in the first sentence of the abstract is just R{ R00M and f : R Ñ R{ R00 M is the quotient map. In fact, f is the universal "definable" (in a suitable sense) locally compact model.

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“…The Lascar group only depends on the theory and it is a quasi-compact topological group with respect to a quotient topology of a certain Stone type space over a model ( [2] or [13]). More recently, the notions of the relativized Lascar groups were introduced in [3] (and studied also in [11] in the context of topological dynamics). Namely, given a type-definable set X in a large saturated model of the theory T, we consider the group of automorphisms restricted to the set X quotiented by the group of restricted automorphisms fixing the Lascar types of the sequences from X of length .…”

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confidence: 99%

The Relativized Lascar Groups, Type-Amalgamation, and Algebraicity

Dobrowolski¹,

Kim²,

Kolesnikov³

et al. 2021

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In this paper we study the relativized Lascar Galois group of a strong type. The group is a quasi-compact connected topological group, and if in addition the underlying theory T is G-compact, then the group is compact. We apply compact group theory to obtain model theoretic results in this note. For example, we use the divisibility of the Lascar group of a strong type to show that, in a simple theory, such types have a certain model theoretic property that we call divisible amalgamation. The main result of this paper is that if c is a finite tuple algebraic over a tuple a, the Lascar group of stp(ac) is abelian, and the underlying theory is G-compact, then the Lascar groups of stp(ac) and of stp(a) are isomorphic. To show this, we prove a purely compact group-theoretic result that any compact connected abelian group is isomorphic to its quotient by every finite subgroup. Several (counter)examples arising in connection with the theoretical development of this note are presented as well. For example, we show that, in the main result above, neither the assumption that the Lascar group of stp(ac) is abelian, nor the assumption of c being finite can be removed.

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Locally compact models for approximate rings

Krupiński

¹

2023

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By an approximate subring of a ring we mean an additively symmetric subset X such that $$X\cdot X \cup (X +X)$$ X · X ∪ ( X + X ) is covered by finitely many additive translates of X. We prove that each approximate subring X of a ring has a locally compact model, i.e. a ring homomorphism $$f :\langle X \rangle \rightarrow S$$ f : ⟨ X ⟩ → S for some locally compact ring S such that f[X] is relatively compact in S and there is a neighborhood U of 0 in S with $$f^{-1}[U] \subseteq 4X + X \cdot 4X$$ f - 1 [ U ] ⊆ 4 X + X · 4 X (where $$4X:=X+X+X+X$$ 4 X : = X + X + X + X ). This S is obtained as the quotient of the ring $$\langle X \rangle $$ ⟨ X ⟩ interpreted in a sufficiently saturated model by its type-definable ring connected component. The main point is to prove that this component always exists. In order to do that, we extend the basic theory of model-theoretic connected components of definable rings [developed in Gismatullin et al. (J Symb Log First View: 1–35, 2022, https://doi.org/10.1017/jsl.2022.10) and Krupiński et al. (Ann Pure Appl Logic 173.7(July):103119, 2022) to the case of rings generated by definable approximate subrings and we answer a question from Krupiński et al. (2022) in the more general context of approximate subrings. Namely, let X be a definable (in a structure M) approximate subring of a ring and $$R:=\langle X \rangle $$ R : = ⟨ X ⟩ . Let $${\bar{X}}$$ X ¯ be the interpretation of X in a sufficiently saturated elementary extension and $${\bar{R}}:= \langle {\bar{X}} \rangle $$ R ¯ : = ⟨ X ¯ ⟩ . It follows from Massicot and Wagner (J Éc Polytech Math 2:55–63, 2015) that there exists the smallest M-type-definable subgroup of $$({\bar{R}},+)$$ ( R ¯ , + ) of bounded index, which is denoted by $$({\bar{R}},+)^{00}_M$$ ( R ¯ , + ) M 00 . We prove that $$({\bar{R}},+)^{00}_M + {\bar{R}} \cdot ({\bar{R}},+)^{00}_M$$ ( R ¯ , + ) M 00 + R ¯ · ( R ¯ , + ) M 00 is the smallest M-type-definable two-sided ideal of $${\bar{R}}$$ R ¯ of bounded index, which we denote by $${\bar{R}}^{00}_M$$ R ¯ M 00 . Then S in the first sentence of the abstract is just $${\bar{R}}/{\bar{R}}^{00}_M$$ R ¯ / R ¯ M 00 and $$f: R \rightarrow {\bar{R}}/{\bar{R}}^{00}_M$$ f : R → R ¯ / R ¯ M 00 is the quotient map. In fact, f is the universal “definable” (in a suitable sense) locally compact model. The existence of locally compact models can be seen as a general structural result about approximate subrings: every approximate subring X can be recovered up to additive commensurability as the preimage by a locally compact model $$f :\langle X \rangle \rightarrow S$$ f : ⟨ X ⟩ → S of any relatively compact neighborhood of 0 in S. It should also have various applications to get more precise structural or even classification results. For example, in this paper, we deduce that every [definable] approximate subring X of a ring of positive characteristic is additively commensurable with a [definable] subring contained in $$4X + X \cdot 4X$$ 4 X + X · 4 X . This easily implies that for any given $$K,L \in \mathbb {N}$$ K , L ∈ N there exists a constant C(K, L) such that every K-approximate subring X (i.e. K additive translates of X cover $$X \cdot X \cup (X+X)$$ X · X ∪ ( X + X ) ) of a ring of positive characteristic $$\le L$$ ≤ L is additively C(K, L)-commensurable with a subring contained in $$4X + X \cdot 4X$$ 4 X + X · 4 X . Another application of the existence of locally compact models is a classification of finite approximate subrings of rings without zero divisors: for every $$K \in \mathbb {N}$$ K ∈ N there exists $$N(K) \in \mathbb {N}$$ N ( K ) ∈ N such that for every finite K-approximate subring X of a ring without zero divisors either $$|X| <N(K)$$ | X | < N ( K ) or $$4X + X \cdot 4X$$ 4 X + X · 4 X is a subring which is additively $$K^{11}$$ K 11 -commensurable with X.

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Boundedness and absoluteness of some dynamical invariants in model theory

Cited by 5 publications

References 17 publications

Locally compact models for approximate rings

Locally compact models for approximate rings

The Relativized Lascar Groups, Type-Amalgamation, and Algebraicity

Locally compact models for approximate rings

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