Searcharxiv⌕ Search

arXiv subjects

Junguk Lee

Publications and source records attributed to Junguk Lee.

17 recordsLinked to original sources
arXiv2025

The Ax-Kochen-Ershov principles via the higher valued hyperfield

In this paper, we concern the model theory of finitely ramified henselian valued fields via higher valued hyperfields. Most of all, we provide a number of Ax-Kochen-Ershov Theorems for finitely ramified henselian valued fields relative to higher valued hyperfields. As corollaries, we deduce a transfer of decidability for full theories and existential theories of a finitely ramified henselian valued fields relative to higher valued hyperfields.

math.LO↗
arXiv2024

Relativized Galois groups of first order theories over a hyperimaginary

We study relativized Lascar groups, which are formed by relativizing Lascar groups to the solution set of a partial type $Σ$. We introduce the notion of a Lascar tuple for $Σ$ and by considering the space of types over a Lascar tuple for $Σ$, the topology for a relativized Lascar group is (re-)defined and some fundamental facts about the Galois groups of first-order theories are generalized to the relativized context. In particular, we prove that any closed subgroup of a relativized Lascar group corresponds to a stabilizer of a bounded hyperimaginary having at least one representative in the solution set of the given partial type $Σ$. Using this, we find the correspondence between subgroups of the relativized Lascar group and the relativized strong types.

math.LO↗
arXiv2024

Preservation of NATP

We prove several preservation theorems for NATP and furnish several examples of NATP. First, we prove preservation of NATP for the parametrization and sum of the theories of Fraïssé limits of Fraïssé classes satisfying strong amalgamation property. Second, we prove preservation of NATP for two kinds of dense/co-dense expansions, that is, the theories of lovely pairs and of H-structures for geometric theories and dense/co-dense expansion on vector spaces. Third, we prove preservation of NATP for the generic predicate expansion and the pair of an algebraically closed field and its distinguished subfield; for the latter, not only NATP, but also preservations of NTP$_1$ and NTP$_2$ are considered. Fourth, we present some proper examples of NATP using the results proved in this paper. Most of all, we show that the model companion of the theory of algebraically closed fields with circular orders (ACFO) is NATP.

math.LO↗
arXiv2023

The embedding property for sorted profinite groups

We study the embedding property in the category of sorted profinite groups. We introduce a notion of the sorted embedding property (SEP), analogous to the embedding property for profinite groups. We show that any sorted profinite group has a universal SEP-cover. Our proof gives an alternative proof for the existence of a universal embedding cover of a profinite group. Also our proof works for any full subcategory of the sorted profinite groups, which is closed under taking finite quotients, fibre product, and inverse limit. We also introduce a weaker notion of finitely sorted embedding property (FSEP), and it turns out to be equivalent to SEP. The advantage of FSEP is to be able to be axiomatized in the first order language of sorted complete systems. Using this, we show that any sorted profinite group having SEP has the sorted complete system whose theory is $ω$-stable under the assumption that the set of sorts is countable. In this case, as a byproduct, we get the uniqueness of a universal SEP-cover of a sorted profinite group, which generalizes the uniqueness of an embedding cover of a profinite group.

math.LO↗
arXiv2022

On the Antichain Tree Property

In this note, we investigate a new model theoretical tree property, called the antichain tree property (ATP). We develop combinatorial techniques for ATP. First, we show that ATP is always witnessed by a formula in a single free variable, and for formulas, not having ATP is closed under disjunction. Second, we show the equivalence of ATP and $k$-ATP, and provide a criterion for theories to have not ATP (being NATP). Using these combinatorial observations, we find algebraic examples of ATP and NATP, including pure group, pure fields, and valued fields. More precisely, we prove Mekler's construction for groups, Chatzidakis' style criterion for PAC fields, and the AKE-style principle for valued fields preserving NATP. And we give a construction of an antichain tree in the Skolem arithmetic and atomless Boolean algebras.

math.LO↗
arXiv2021

Ramsey theory and topological dynamics for first order theories

We investigate interactions between Ramsey theory, topological dynamics, and model theory. We introduce various Ramsey-like properties for first order theories and characterize them in terms of the appropriate dynamical properties of the theories in question (such as [extreme] amenability of a theory or some properties of the associated Ellis semigroups). Then we relate them to profiniteness and triviality of the Ellis groups of first order theories. In particular, we find various criteria for [pro]finiteness and for triviality of the Ellis group of a given theory from which we obtain wide classes of examples of theories with [pro]finite or trivial Ellis groups. As an initial motivation, we note that profiniteness of the Ellis group of a theory implies that the Kim-Pillay Galois group of this theory is also profinite, which in turn is equivalent to the equality of the Shelah and Kim-Pillay strong types. We also find several concrete examples illustrating the lack of implications between some fundamental properties. In the appendix, we give a full computation of the Ellis group of the theory of the random hypergraph with one binary and one 4-ary relation. This example shows that the assumption of NIP in the version of Newelski's conjecture for amenable theories (proved in [15]) cannot be dropped.

math.LO↗
arXiv2021

Co-theory of sorted profinite groups for PAC structures

We achieve several results. First, we develop a variant of the theory of absolute Galois groups in the context of many sorted structures. Second, we provide a method for coding absolute Galois groups of structures, so they can be interpreted in some monster model with an additional predicate. Third, we prove the "Weak Independence Theorem" for PAC substructures of an ambient structure with nfcp and the property B(3). Fourth, we describe Kim-dividing in these PAC substructures and show several results related to the SOPn hierarchy. Fifth, we characterize the algebraic closure in PAC structures

math.LO↗
arXiv2020

On the structure of certain valued fields

In this article, we study the structure of finitely ramified mixed characteristic valued fields. For any two complete discrete valued fields $K_1$ and $K_2$ of mixed characteristic with perfect residue fields, we show that if the $n$-th residue rings are isomorphic for each $n\ge 1$, then $K_1$ and $K_2$ are isometric and isomorphic. More generally, for $n_1\ge 1$, there is $n_2$ depending only on the ramification indices of $K_1$ and $K_2$ such that any homomorphism from the $n_1$-th residue ring of $K_1$ to the $n_2$-th residue ring of $K_2$ can be lifted to a homomorphism between the valuation rings. Moreover, we get a functor from the category of certain principal Artinian local rings of length $n$ to the category of certain complete discrete valuation rings of mixed characteristic with perfect residue fields, which naturally generalizes the functorial property of unramified complete discrete valuation rings. Our lifting result improves Basarab's relative completeness theorem for finitely ramified henselian valued fields, which solves a question posed by Basarab, in the case of perfect residue fields.

math.LO↗
arXiv2019

Elementary Equivalence Theorem for PAC structures

We generalize a well-known theorem binding the elementary equivalence relation on the level of PAC fields and the isomorphism class of their absolute Galois groups. Our results concern two cases: saturated PAC structures and non-saturated PAC structures.

math.LO↗
arXiv2018

Hyperfields, truncated DVRs and valued fields

For any two complete discrete valued fields $K_1$ and $K_2$ of mixed characteristic with perfect residue fields, we show that if the $n$-th valued hyperfields of $K_1$ and $K_2$ are isomorphic over $p$ for each $n\ge1$, then $K_1$ and $K_2$ are isomorphic. More generally, for $n_1,n_2\ge 1$, if $n_2$ is large enough, then any homomorphism, which is over $p$, from the $n_1$-th valued hyperfield of $K_1$ to the $n_2$-th valued hyperfield of $K_2$ can be lifted to a homomorphism from $K_1$ to $K_2$. We compute such $n_2$ effectively, which depends only on the ramification indices of $K_1$ and $K_2$. Moreover, if $K_1$ is tamely ramified, then any homomorphism over $p$ between the first valued hyperfields is induced from a unique homomorphism of valued fields. Using this lifting result, we deduce a relative completeness theorem of AKE-style in terms of valued hyperfields. We also study some relationships between valued hyperfields, truncated discrete valuation rings, and complete discrete valued fields of mixed characteristic. For a prime number $p$ and a positive integer $e$ and for large enough $n$, we show that a certain category of valued hyperfields is equivalent to the category of truncated discrete valuation rings of length $n$ and the ramification indices $e$ having perfect residue fields of characteristic $p$. Furthermore, in the tamely ramified case, we show that a subcategory of this category of valued hyperfields is equivalent to the category of complete discrete valued rings of mixed characteristic $(0,p)$ having perfect residue fields.

math.AC↗
arXiv2018

Geometric stability theory for $μ$-structures

We introduce a notion of $μ$-structures which are certain locally compact group actions and prove some counterparts of results on Polish structures(introduced by Krupinski in \cite{Kru5}). Using the Haar measure of locally compact groups, we introduce an independence, called $μ$-independence, in $μ$-structures having good properties. With this independence notion, we develop geometric stability theory for $μ$-structures. Then we see some structural theorems for compact groups which are $μ$-structure. We also give examples of profinite structures where $μ$-independence is different from $nm$-independence introduced by Krupinski for Polish structures.

math.LO↗
arXiv2017

The Lascar groups and the 1st homology groups in model theory

Let $p$ be a strong type of an algebraically closed tuple over $B=\acl^{\eq}(B)$ in any theory $T$. Depending on a ternary relation $\indo^*$ satisfying some basic axioms (there is at least one such, namely the trivial independence in $T$), the first homology group $H^*_1(p)$ can be introduced, similarly to \cite{GKK1}. We show that there is a canonical surjective homomorphism from the Lascar group over $B$ to $H^*_1(p)$. We also notice that the map factors naturally via a surjection from the `relativised' Lascar group of the type (which we define in analogy with the Lascar group of the theory) onto the homology group, and we give an explicit description of its kernel. Due to this characterization, it follows that the first homology group of $p$ is independent from the choice of $\indo^*$, and can be written simply as $H_1(p)$. As consequences, in any $T$, we show that $|H_1(p)|\geq 2^{\aleph_0}$ unless $H_1(p)$ is trivial, and we give a criterion for the equality of stp and Lstp of algebraically closed tuples using the notions of the first homology group and a relativised Lascar group. We also argue how any abelian connected compact group can appear as the first homology group of the type of a model.

math.LO↗
arXiv2016

Some arithmetic properties on nonstandard rationals

For a given number field $K$, we show that the ranks of nonsingular elliptic curves over $K$ are uniformly finitely bounded if and only if weak Mordell-Weil property holds in all(some) ultrpowers $^*K$ of $K$. Also we introduce Nonstandard Mordell-Weil property for $^*K$ considering each Mordell-Weil group as $^*Z$-module, where $^*Z$ is an ultrapower of $Z$, and we show that Nonstandard Mordell-Weil property is equivalent to weak Mordell-Weil property in $^*K$. In Appendix, we showed that it is possible to consider definable abelian groups as $^*Z$-modules in a saturated nonstandard rational number field $^*Q$ so that nonstandard Mordell-Weil property is well-defined, and thus we showed that nonstandard Mordell-Weil property and weak Mordell-Weil property are equivalent. Next we focus on priems and prime ideals of nonstandard raional number fields. We give an infinite factorization theorem on $^*Q$ using valuations induced from primes of $^*Z$, and we classify maximal and prime ideal of $^*Z$ in terms of maximal filter on the set of primes of $^*Z$ and ordered semigroups of the valuation semigroup induced from maximal ideals of $^*Z$.

math.LO↗
arXiv2015

Lascar groups and the first homology groups of strong types in rosy theories

For a rosy theory, we give a canonical surjective homomorphism from a Lascar group over $A=\acl^{eq}(A)$ to a first homology group of a strong type over $A$, and we describe its kernel by an invariant equivalence relation. As a consequence, we show that the first homology groups of strong types in rosy theories have the cardinalities of one or at least $2^{\aleph_0}$. We give two examples of rosy theories having non trivial first homology groups of strong types over $\acl^{eq}(\emptyset)$. In these examples, these two homology groups are exactly isomorphic to their Lascar group over $\acl^{eq}(\emptyset)$.

math.LO↗
arXiv2015

Non-commutative groupoids obtained from the failure of $3$-uniqueness in stable theories

We construct a possibly non-commutative groupoid from the failure of $3$-uniqueness of a strong type. The commutative groupoid constructed by John Goodrick and Alexei Kolesnikov in \cite{GK} lives in the center of the groupoid. A certain automorphism group approximated by the vertex groups of the non-commutative groupoids is suggested as a "fundamental group" of the strong type.

math.LO↗
arXiv2015

A classification of 2-chains having 1-shell boundaries in rosy theories

We classify, in a non-trivial amenable collection of functors, all 2-chains up to the relation of having the same 1-shell boundary. In particular, we prove that in a rosy theory, every 1-shell of a Lascar strong type is the boundary of some 2-chain, hence making the 1st homology group trivial. We also show that, unlike in simple theories, in rosy theories there is no upper bound on the minimal lengths of $2$-chains whose boundary is a $1$-shell.

math.LO↗