First, an example of a 2‐dependent group without a minimal subgroup of bounded index is given. Second, all infinite n ‐dependent fields are shown to be Artin‐Schreier closed. Furthermore, the …
First, an example of a 2‐dependent group without a minimal subgroup of bounded index is given. Second, all infinite n ‐dependent fields are shown to be Artin‐Schreier closed. Furthermore, the theory of any non separably closed PAC field has the IP n property for all natural numbers n and certain properties of dependent (NIP) valued fields extend to the n ‐dependent context.
We continue the study of $n$-dependent groups, fields and related structures, largely motivated by the conjecture that every $n$-dependent field is dependent. We provide evidence towards this conjecture by showing …
We continue the study of $n$-dependent groups, fields and related structures, largely motivated by the conjecture that every $n$-dependent field is dependent. We provide evidence towards this conjecture by showing that every infinite $n$-dependent valued field of positive characteristic is henselian, obtaining a variant of Shelah's Henselianity Conjecture in this case and generalizing a recent result of Johnson for dependent fields. Additionally, we prove a result on intersections of type-definable connected components over generic sets of parameters in $n$-dependent groups, generalizing Shelah's absoluteness of $G^{00}$ in dependent theories and relative absoluteness of $G^{00}$ in $2$-dependent theories. In an effort to clarify the scope of this conjecture, we provide new examples of strictly $2$-dependent fields with additional structure, showing that Granger's examples of non-degenerate bilinear forms over dependent fields are $2$-dependent. Along the way, we obtain some purely model-theoretic results of independent interest: we show that $n$-dependence is witnessed by formulas with all but one variable singletons; provide a type-counting criterion for $2$-dependence and use it to deduce $2$-dependence for compositions of dependent relations with arbitrary binary functions (the Composition Lemma); and show that an expansion of a geometric theory $T$ by a generic predicate is dependent if and only if it is $n$-dependent for some $n$, if and only if the algebraic closure in $T$ is disintegrated. An appendix by Martin Bays provides an explicit isomorphism in the Kaplan-Scanlon-Wagner theorem.
The notion of almost centralizer and almost commutator are introduced and basic properties are established. They are used to study $\widetilde{\mathfrak M}\_c$-groups, i. e.groups for which every descending chain of …
The notion of almost centralizer and almost commutator are introduced and basic properties are established. They are used to study $\widetilde{\mathfrak M}\_c$-groups, i. e.groups for which every descending chain of centralizers each having infinite index in its predecessor stabilizes after finitely many steps. The Fitting subgroup of such groups is shown to be nilpotent and a theorem of Hall for nilpotent groups is generalized to ind-definable almost nilpotent subgroups of $\widetilde{\mathfrak M}\_c$-groups.
Any superrosy division ring is shown to be centrally finite. Furthermore, division rings satisfying a generalized chain condition on definable subgroups are studied. In particular, a division ring of burden …
Any superrosy division ring is shown to be centrally finite. Furthermore, division rings satisfying a generalized chain condition on definable subgroups are studied. In particular, a division ring of burden <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has dimension at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over its center, and any definable group of definable automorphisms of a field of burden <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has size at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Additionally, an alternative proof that division rings interpretable in o-minimal structures are algebraically closed, real closed or the quaternions over a real closed field is given.
The notion of bounded FC-nilpotent group is introduced and it is shown that any such group is nilpotent-by-finite, generalizing a result of Neumann on bounded FC-groups.
The notion of bounded FC-nilpotent group is introduced and it is shown that any such group is nilpotent-by-finite, generalizing a result of Neumann on bounded FC-groups.
This thesis is dedicated to the study of groups and fields whose definable sets do not admit certain combinatorial patterns. Given a group G, one particular problem we are interested …
This thesis is dedicated to the study of groups and fields whose definable sets do not admit certain combinatorial patterns. Given a group G, one particular problem we are interested in is to find definable envelopes for arbitrary abelian, nilpotent or solvable subgroups of G which admit the same algebraic properties. Such evelopes exists if G is stable and even if G is merely dependent but sufficiently saturated, with the additional hypothesis of normality in the solvable case. In groups with a simple theory, one obtains definable envelopes up to finite index.We introduce the notion of an almost centralizer and establish some of its basic properties. This enables us to extend the aforementioned results to Mc~ groups, i. e. groups in which any definable section satisfies a chain condition on centralizers up to finite index. These include any definable group in a rosy and in particular in a simple theory. Furthermore, inspired from the proof in dependent theories as well as using techniques developed for almost centralizers in this thesis, we are able to find definable envelopes up to finite index for abelian, nilpotent and normal solvable subgroups of any enough saturated NTP2 group. Moreover, using envelopes for nilpotent subgroups of Mc~ groups and the chain condition on centralizer up to finite index, we show additionally that the Fitting subgroup of any Mc~ group is nilpotent and that its almost Fitting subgroup is virtually solvable.The second part of this thesis focuses on the study of n-dependent fields. We prove that any n-dependent field is Artin-Schreier closed and that non separably closed PAC fields are not n-dependent for any natural number n
We prove the existence of abelian, solvable and nilpotent definable envelopes for groups definable in models of an NTP2 theory.
We prove the existence of abelian, solvable and nilpotent definable envelopes for groups definable in models of an NTP2 theory.
Any superrosy division ring (i.e. a division ring equipped with an abstract notion of rank) is shown to be centrally finite. Furthermore, division rings satisfying a generalized chain condition on …
Any superrosy division ring (i.e. a division ring equipped with an abstract notion of rank) is shown to be centrally finite. Furthermore, division rings satisfying a generalized chain condition on definable subgroups are studied. In particular, a division ring of burden n has dimension at most n over its center, and any definable group of definable automorphisms of a field of burden n has size at most n. Additionally, interpretable division rings in o-minimal structures are shown to be algebraically closed, real closed or the quaternions over a real closed field.
The role of finite centralizers of involutions in pseudo-finite groups is analyzed. Using basic techniques from infinite group theory, it is shown that a pseudo-finite group admitting a definable involutory …
The role of finite centralizers of involutions in pseudo-finite groups is analyzed. Using basic techniques from infinite group theory, it is shown that a pseudo-finite group admitting a definable involutory automorphism fixing only finitely many elements is finite-by-abelian-by-finite. As a consequence, an alternative proof of the corresponding result for periodic groups due to Hartley and Meixner is given, as well as a gently improvement regarding definable properties. Furthermore, it is shown that any pseudo-finite group has an infinite abelian subgroup and consequently that there are only finitely many finite groups with a maximal abelian subgroup of a given size.
Mekler's construction gives an interpretation of any structure in a finite relational language in a group (nilpotent of class $2$ and exponent $p>2$, but not finitely generated in general). Even …
Mekler's construction gives an interpretation of any structure in a finite relational language in a group (nilpotent of class $2$ and exponent $p>2$, but not finitely generated in general). Even though this construction is not a bi-interpretation, it is known to preserve some model-theoretic tameness properties of the original structure including stability and simplicity. We demonstrate that $k$-dependence of the theory is preserved, for all $k \in \mathbb{N}$, and that NTP$_2$ is preserved. We apply this result to obtain first examples of strictly $k$-dependent groups (with no additional structure).
The notion of bounded FC-nilpotent group is introduced and it is shown that any such group is nilpotent-by-finite, generalizing a result of Neumann on bounded FC-groups.
The notion of bounded FC-nilpotent group is introduced and it is shown that any such group is nilpotent-by-finite, generalizing a result of Neumann on bounded FC-groups.
The notion of bounded FC-nilpotent group is introduced and it is shown that any such group is nilpotent-by-finite, generalizing a result of Neumann on bounded FC-groups.
The notion of bounded FC-nilpotent group is introduced and it is shown that any such group is nilpotent-by-finite, generalizing a result of Neumann on bounded FC-groups.
Any superrosy division ring (i.e. a division ring equipped with an abstract notion of rank) is shown to be centrally finite. Furthermore, division rings satisfying a generalized chain condition on …
Any superrosy division ring (i.e. a division ring equipped with an abstract notion of rank) is shown to be centrally finite. Furthermore, division rings satisfying a generalized chain condition on definable subgroups are studied. In particular, a division ring of burden n has dimension at most n over its center, and any definable group of definable automorphisms of a field of burden n has size at most n. Additionally, interpretable division rings in o-minimal structures are shown to be algebraically closed, real closed or the quaternions over a real closed field.
First, an example of a 2-dependent group without a minimal subgroup of bounded index is given. Second, all infinite n-dependent fields are shown to be Artin-Schreier closed. Furthermore, the theory …
First, an example of a 2-dependent group without a minimal subgroup of bounded index is given. Second, all infinite n-dependent fields are shown to be Artin-Schreier closed. Furthermore, the theory of any non separably closed PAC field has the IPn property for all natural numbers n and certain properties of dependent (NIP) valued fields extend to the n-dependent context.
Mekler's construction gives an interpretation of any structure in a finite relational language in a group (nilpotent of class $2$ and exponent $p>2$, but not finitely generated in general). Even …
Mekler's construction gives an interpretation of any structure in a finite relational language in a group (nilpotent of class $2$ and exponent $p>2$, but not finitely generated in general). Even though this construction is not a bi-interpretation, it is known to preserve some model-theoretic tameness properties of the original structure including stability and simplicity. We demonstrate that $k$-dependence of the theory is preserved, for all $k \in \mathbb{N}$, and that NTP$_2$ is preserved. We apply this result to obtain first examples of strictly $k$-dependent groups (with no additional structure).
The role of finite centralizers of involutions in pseudo-finite groups is analyzed. It is shown that a pseudo-finite group admitting a definable involutory automorphism fixing only finitely many elements is …
The role of finite centralizers of involutions in pseudo-finite groups is analyzed. It is shown that a pseudo-finite group admitting a definable involutory automorphism fixing only finitely many elements is finite-by-abelian-by-finite. As a consequence, we give a model-theoretic proof of a result for periodic groups due to Hartley and Meixner. Furthermore, it is shown that any pseudo-finite group has an infinite abelian subgroup.
We develop some model theory of multi-linear forms, generalizing Granger in the bi-linear case. In particular, after proving a quantifier elimination result, we show that for an NIP field K, …
We develop some model theory of multi-linear forms, generalizing Granger in the bi-linear case. In particular, after proving a quantifier elimination result, we show that for an NIP field K, the theory of infinite dimensional non-degenerate alternating n-linear spaces over K is strictly n-dependent; and it is NSOP1 if K is. This relies on a new Composition Lemma for functions of arbitrary arity and NIP relations (which in turn relies on certain higher arity generalizations of Sauer-Shelah lemma). We also study the invariant connected components $G^{\infty}$ in n-dependent groups, demonstrating their relative absoluteness in the abelian case.
We develop some model theory of multi-linear forms, generalizing Granger in the bi-linear case. In particular, after proving a quantifier elimination result, we show that for an NIP field K, …
We develop some model theory of multi-linear forms, generalizing Granger in the bi-linear case. In particular, after proving a quantifier elimination result, we show that for an NIP field K, the theory of infinite dimensional non-degenerate alternating n-linear spaces over K is strictly n-dependent; and it is NSOP1 if K is. This relies on a new Composition Lemma for functions of arbitrary arity and NIP relations (which in turn relies on certain higher arity generalizations of Sauer-Shelah lemma). We also study the invariant connected components $G^{\infty}$ in n-dependent groups, demonstrating their relative absoluteness in the abelian case.
We continue the study of $n$-dependent groups, fields and related structures, largely motivated by the conjecture that every $n$-dependent field is dependent. We provide evidence towards this conjecture by showing …
We continue the study of $n$-dependent groups, fields and related structures, largely motivated by the conjecture that every $n$-dependent field is dependent. We provide evidence towards this conjecture by showing that every infinite $n$-dependent valued field of positive characteristic is henselian, obtaining a variant of Shelah's Henselianity Conjecture in this case and generalizing a recent result of Johnson for dependent fields. Additionally, we prove a result on intersections of type-definable connected components over generic sets of parameters in $n$-dependent groups, generalizing Shelah's absoluteness of $G^{00}$ in dependent theories and relative absoluteness of $G^{00}$ in $2$-dependent theories. In an effort to clarify the scope of this conjecture, we provide new examples of strictly $2$-dependent fields with additional structure, showing that Granger's examples of non-degenerate bilinear forms over dependent fields are $2$-dependent. Along the way, we obtain some purely model-theoretic results of independent interest: we show that $n$-dependence is witnessed by formulas with all but one variable singletons; provide a type-counting criterion for $2$-dependence and use it to deduce $2$-dependence for compositions of dependent relations with arbitrary binary functions (the Composition Lemma); and show that an expansion of a geometric theory $T$ by a generic predicate is dependent if and only if it is $n$-dependent for some $n$, if and only if the algebraic closure in $T$ is disintegrated. An appendix by Martin Bays provides an explicit isomorphism in the Kaplan-Scanlon-Wagner theorem.
The notion of bounded FC-nilpotent group is introduced and it is shown that any such group is nilpotent-by-finite, generalizing a result of Neumann on bounded FC-groups.
The notion of bounded FC-nilpotent group is introduced and it is shown that any such group is nilpotent-by-finite, generalizing a result of Neumann on bounded FC-groups.
Any superrosy division ring is shown to be centrally finite. Furthermore, division rings satisfying a generalized chain condition on definable subgroups are studied. In particular, a division ring of burden …
Any superrosy division ring is shown to be centrally finite. Furthermore, division rings satisfying a generalized chain condition on definable subgroups are studied. In particular, a division ring of burden <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has dimension at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over its center, and any definable group of definable automorphisms of a field of burden <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has size at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Additionally, an alternative proof that division rings interpretable in o-minimal structures are algebraically closed, real closed or the quaternions over a real closed field is given.
Mekler's construction gives an interpretation of any structure in a finite relational language in a group (nilpotent of class $2$ and exponent $p>2$, but not finitely generated in general). Even …
Mekler's construction gives an interpretation of any structure in a finite relational language in a group (nilpotent of class $2$ and exponent $p>2$, but not finitely generated in general). Even though this construction is not a bi-interpretation, it is known to preserve some model-theoretic tameness properties of the original structure including stability and simplicity. We demonstrate that $k$-dependence of the theory is preserved, for all $k \in \mathbb{N}$, and that NTP$_2$ is preserved. We apply this result to obtain first examples of strictly $k$-dependent groups (with no additional structure).
The role of finite centralizers of involutions in pseudo-finite groups is analyzed. Using basic techniques from infinite group theory, it is shown that a pseudo-finite group admitting a definable involutory …
The role of finite centralizers of involutions in pseudo-finite groups is analyzed. Using basic techniques from infinite group theory, it is shown that a pseudo-finite group admitting a definable involutory automorphism fixing only finitely many elements is finite-by-abelian-by-finite. As a consequence, an alternative proof of the corresponding result for periodic groups due to Hartley and Meixner is given, as well as a gently improvement regarding definable properties. Furthermore, it is shown that any pseudo-finite group has an infinite abelian subgroup and consequently that there are only finitely many finite groups with a maximal abelian subgroup of a given size.
Mekler's construction gives an interpretation of any structure in a finite relational language in a group (nilpotent of class $2$ and exponent $p>2$, but not finitely generated in general). Even …
Mekler's construction gives an interpretation of any structure in a finite relational language in a group (nilpotent of class $2$ and exponent $p>2$, but not finitely generated in general). Even though this construction is not a bi-interpretation, it is known to preserve some model-theoretic tameness properties of the original structure including stability and simplicity. We demonstrate that $k$-dependence of the theory is preserved, for all $k \in \mathbb{N}$, and that NTP$_2$ is preserved. We apply this result to obtain first examples of strictly $k$-dependent groups (with no additional structure).
The role of finite centralizers of involutions in pseudo-finite groups is analyzed. It is shown that a pseudo-finite group admitting a definable involutory automorphism fixing only finitely many elements is …
The role of finite centralizers of involutions in pseudo-finite groups is analyzed. It is shown that a pseudo-finite group admitting a definable involutory automorphism fixing only finitely many elements is finite-by-abelian-by-finite. As a consequence, we give a model-theoretic proof of a result for periodic groups due to Hartley and Meixner. Furthermore, it is shown that any pseudo-finite group has an infinite abelian subgroup.
The notion of bounded FC-nilpotent group is introduced and it is shown that any such group is nilpotent-by-finite, generalizing a result of Neumann on bounded FC-groups.
The notion of bounded FC-nilpotent group is introduced and it is shown that any such group is nilpotent-by-finite, generalizing a result of Neumann on bounded FC-groups.
This thesis is dedicated to the study of groups and fields whose definable sets do not admit certain combinatorial patterns. Given a group G, one particular problem we are interested …
This thesis is dedicated to the study of groups and fields whose definable sets do not admit certain combinatorial patterns. Given a group G, one particular problem we are interested in is to find definable envelopes for arbitrary abelian, nilpotent or solvable subgroups of G which admit the same algebraic properties. Such evelopes exists if G is stable and even if G is merely dependent but sufficiently saturated, with the additional hypothesis of normality in the solvable case. In groups with a simple theory, one obtains definable envelopes up to finite index.We introduce the notion of an almost centralizer and establish some of its basic properties. This enables us to extend the aforementioned results to Mc~ groups, i. e. groups in which any definable section satisfies a chain condition on centralizers up to finite index. These include any definable group in a rosy and in particular in a simple theory. Furthermore, inspired from the proof in dependent theories as well as using techniques developed for almost centralizers in this thesis, we are able to find definable envelopes up to finite index for abelian, nilpotent and normal solvable subgroups of any enough saturated NTP2 group. Moreover, using envelopes for nilpotent subgroups of Mc~ groups and the chain condition on centralizer up to finite index, we show additionally that the Fitting subgroup of any Mc~ group is nilpotent and that its almost Fitting subgroup is virtually solvable.The second part of this thesis focuses on the study of n-dependent fields. We prove that any n-dependent field is Artin-Schreier closed and that non separably closed PAC fields are not n-dependent for any natural number n
Any superrosy division ring (i.e. a division ring equipped with an abstract notion of rank) is shown to be centrally finite. Furthermore, division rings satisfying a generalized chain condition on …
Any superrosy division ring (i.e. a division ring equipped with an abstract notion of rank) is shown to be centrally finite. Furthermore, division rings satisfying a generalized chain condition on definable subgroups are studied. In particular, a division ring of burden n has dimension at most n over its center, and any definable group of definable automorphisms of a field of burden n has size at most n. Additionally, interpretable division rings in o-minimal structures are shown to be algebraically closed, real closed or the quaternions over a real closed field.
First, an example of a 2‐dependent group without a minimal subgroup of bounded index is given. Second, all infinite n ‐dependent fields are shown to be Artin‐Schreier closed. Furthermore, the …
First, an example of a 2‐dependent group without a minimal subgroup of bounded index is given. Second, all infinite n ‐dependent fields are shown to be Artin‐Schreier closed. Furthermore, the theory of any non separably closed PAC field has the IP n property for all natural numbers n and certain properties of dependent (NIP) valued fields extend to the n ‐dependent context.
The notion of bounded FC-nilpotent group is introduced and it is shown that any such group is nilpotent-by-finite, generalizing a result of Neumann on bounded FC-groups.
The notion of bounded FC-nilpotent group is introduced and it is shown that any such group is nilpotent-by-finite, generalizing a result of Neumann on bounded FC-groups.
Any superrosy division ring (i.e. a division ring equipped with an abstract notion of rank) is shown to be centrally finite. Furthermore, division rings satisfying a generalized chain condition on …
Any superrosy division ring (i.e. a division ring equipped with an abstract notion of rank) is shown to be centrally finite. Furthermore, division rings satisfying a generalized chain condition on definable subgroups are studied. In particular, a division ring of burden n has dimension at most n over its center, and any definable group of definable automorphisms of a field of burden n has size at most n. Additionally, interpretable division rings in o-minimal structures are shown to be algebraically closed, real closed or the quaternions over a real closed field.
The notion of almost centralizer and almost commutator are introduced and basic properties are established. They are used to study $\widetilde{\mathfrak M}\_c$-groups, i. e.groups for which every descending chain of …
The notion of almost centralizer and almost commutator are introduced and basic properties are established. They are used to study $\widetilde{\mathfrak M}\_c$-groups, i. e.groups for which every descending chain of centralizers each having infinite index in its predecessor stabilizes after finitely many steps. The Fitting subgroup of such groups is shown to be nilpotent and a theorem of Hall for nilpotent groups is generalized to ind-definable almost nilpotent subgroups of $\widetilde{\mathfrak M}\_c$-groups.
We prove the existence of abelian, solvable and nilpotent definable envelopes for groups definable in models of an NTP2 theory.
We prove the existence of abelian, solvable and nilpotent definable envelopes for groups definable in models of an NTP2 theory.
First, an example of a 2-dependent group without a minimal subgroup of bounded index is given. Second, all infinite n-dependent fields are shown to be Artin-Schreier closed. Furthermore, the theory …
First, an example of a 2-dependent group without a minimal subgroup of bounded index is given. Second, all infinite n-dependent fields are shown to be Artin-Schreier closed. Furthermore, the theory of any non separably closed PAC field has the IPn property for all natural numbers n and certain properties of dependent (NIP) valued fields extend to the n-dependent context.
Baer [2] and Neumann [5] have discussed groups in which there is a limitation on the number of conjugates which an element may have. For a given group G, let …
Baer [2] and Neumann [5] have discussed groups in which there is a limitation on the number of conjugates which an element may have. For a given group G, let H 1 be the set of all elements of G which have only a finite number of conjugates in G, let H 2 be the set of those elements of G, the conjugates of each of which lie in only a finite number of cosets of H 1 in G; and in this fashion define H 3 , H 4 , …. We shall show that the H i are strictly characteristic subgroups of G.
First, an example of a 2‐dependent group without a minimal subgroup of bounded index is given. Second, all infinite n ‐dependent fields are shown to be Artin‐Schreier closed. Furthermore, the …
First, an example of a 2‐dependent group without a minimal subgroup of bounded index is given. Second, all infinite n ‐dependent fields are shown to be Artin‐Schreier closed. Furthermore, the theory of any non separably closed PAC field has the IP n property for all natural numbers n and certain properties of dependent (NIP) valued fields extend to the n ‐dependent context.
We study model theoretic tree properties ([Formula: see text]) and their associated cardinal invariants ([Formula: see text], respectively). In particular, we obtain a quantitative refinement of Shelah’s theorem ([Formula: see …
We study model theoretic tree properties ([Formula: see text]) and their associated cardinal invariants ([Formula: see text], respectively). In particular, we obtain a quantitative refinement of Shelah’s theorem ([Formula: see text]) for countable theories, show that [Formula: see text] is always witnessed by a formula in a single variable (partially answering a question of Shelah) and that weak [Formula: see text] is equivalent to [Formula: see text] (answering a question of Kim and Kim). Besides, we give a characterization of [Formula: see text] via a version of independent amalgamation of types and apply this criterion to verify that some examples in the literature are indeed [Formula: see text].
For an infinite cardinal ${\it\kappa}$ , let $\text{ded}\,{\it\kappa}$ denote the supremum of the number of Dedekind cuts in linear orders of size ${\it\kappa}$ . It is known that ${\it\kappa}<\text{ded}\,{\it\kappa}\leqslant 2^{{\it\kappa}}$ …
For an infinite cardinal ${\it\kappa}$ , let $\text{ded}\,{\it\kappa}$ denote the supremum of the number of Dedekind cuts in linear orders of size ${\it\kappa}$ . It is known that ${\it\kappa}<\text{ded}\,{\it\kappa}\leqslant 2^{{\it\kappa}}$ for all ${\it\kappa}$ and that $\text{ded}\,{\it\kappa}<2^{{\it\kappa}}$ is consistent for any ${\it\kappa}$ of uncountable cofinality. We prove however that $2^{{\it\kappa}}\leqslant \text{ded}(\text{ded}(\text{ded}(\text{ded}\,{\it\kappa})))$ always holds. Using this result we calculate the Hanf numbers for the existence of two-cardinal models with arbitrarily large gaps and for the existence of arbitrarily large models omitting a type in the class of countable dependent first-order theories. Specifically, we show that these bounds are as large as in the class of all countable theories.
$\operatorname {NTP}_{2}$ is a large class of first-order theories defined by Shelah generalizing simple and NIP theories. Algebraic examples of $\operatorname {NTP}_{2}$ structures are given by ultra-products of $p$-adics and …
$\operatorname {NTP}_{2}$ is a large class of first-order theories defined by Shelah generalizing simple and NIP theories. Algebraic examples of $\operatorname {NTP}_{2}$ structures are given by ultra-products of $p$-adics and certain valued difference fields (such as a non-standard Frobenius automorphism living on an algebraically closed valued field of characteristic 0). In this note we present some results on groups and fields definable in $\operatorname {NTP}_{2}$ structures. Most importantly, we isolate a chain condition for definable normal subgroups and use it to show that any $\operatorname {NTP}_{2}$ field has only finitely many Artin-Schreier extensions. We also discuss a stronger chain condition coming from imposing bounds on burden of the theory (an appropriate analogue of weight) and show that every strongly dependent valued field is Kaplansky.
In this article, we develop and clarify some of the basic combinatorial properties of the new notion of n-dependence (for 1≤n<ω) recently introduced by Shelah. In the same way as …
In this article, we develop and clarify some of the basic combinatorial properties of the new notion of n-dependence (for 1≤n<ω) recently introduced by Shelah. In the same way as dependence of a theory means its inability to encode a bipartite random graph with a definable edge relation, n-dependence corresponds to the inability to encode a random (n+1)-partite (n+1)-hypergraph with a definable edge relation. We characterize n-dependence by counting φ-types over finite sets (generalizing the Sauer–Shelah lemma, answering a question of Shelah), and in terms of the collapse of random ordered (n+1)-hypergraph indiscernibles down to order-indiscernibles (which implies that the failure of n-dependence is always witnessed by a formula in a single free variable).
Let $T$ be a (first order complete) dependent theory, ${\mathfrak{C}}$ a $\barκ$-saturated model of $T$ and $G$ a definable subgroup which is abelian. Among subgroups of bounded index which are …
Let $T$ be a (first order complete) dependent theory, ${\mathfrak{C}}$ a $\barκ$-saturated model of $T$ and $G$ a definable subgroup which is abelian. Among subgroups of bounded index which are the union of $
Abstract. The famous Tits alternative states that a linear group either contains a non-abelian free group or is soluble-by-(locally finite). In this paper we study similar alternatives in pseudofinite groups. …
Abstract. The famous Tits alternative states that a linear group either contains a non-abelian free group or is soluble-by-(locally finite). In this paper we study similar alternatives in pseudofinite groups. We show, for instance, that an ℵ
Non-forking is one of the most important notions in modern model theory capturing the idea of a generic extension of a type (which is a far-reaching generalization of the concept …
Non-forking is one of the most important notions in modern model theory capturing the idea of a generic extension of a type (which is a far-reaching generalization of the concept of a generic point of a variety). To a countable first-order theory we associate its non-forking spectrum – a function of two cardinals \kappa and \lambda giving the supremum of the possible number of types over a model of size \lambda that do not fork over a sub-model of size \kappa . This is a natural generalization of the stability function of a theory. We make progress towards classifying the non-forking spectra. On the one hand, we show that the possible values a non-forking spectrum may take are quite limited. On the other hand, we develop a general technique for constructing theories with a prescribed non-forking spectrum, thus giving a number of examples. In particular, we answer negatively a question of Adler whether NIP is equivalent to bounded non-forking. In addition, we answer a question of Keisler regarding the number of cuts a linear order may have. Namely, we show that it is possible that ded \kappa < (ded _\kappa) ^{\omega} .
It is proved that any supersimple field has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types …
It is proved that any supersimple field has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types in groups and fields whose theory is simple.
This is a survey, intended both for group theorists and model theorists, concerning the structure of pseudofinite groups, that is, infinite models of the first-order theory of finite groups. The …
This is a survey, intended both for group theorists and model theorists, concerning the structure of pseudofinite groups, that is, infinite models of the first-order theory of finite groups. The focus is on concepts from stability theory and generalisations in the context of pseudofinite groups, and on the information this might provide for finite group theory.
Abstract This paper investigates the logical stability of various groups. Theorem 1: If a group G is stable and locally nilpotent then it is solvable. Theorem 2: Every non-Abelian variety …
Abstract This paper investigates the logical stability of various groups. Theorem 1: If a group G is stable and locally nilpotent then it is solvable. Theorem 2: Every non-Abelian variety of groups is unstable.
A group is small if it has countably many complete $n$-types over the empty set for each natural number n. More generally, a group $G$ is weakly small if it …
A group is small if it has countably many complete $n$-types over the empty set for each natural number n. More generally, a group $G$ is weakly small if it has countably many complete 1-types over every finite subset of G. We show here that in a weakly small group, subgroups which are definable with parameters lying in a finitely generated algebraic closure satisfy the descending chain conditions for their traces in any finitely generated algebraic closure. An infinite weakly small group has an infinite abelian subgroup, which may not be definable. A small nilpotent group is the central product of a definable divisible group with a definable one of bounded exponent. In a group with simple theory, any set of pairwise commuting elements is contained in a definable finite-by-abelian subgroup. First corollary : a weakly small group with simple theory has an infinite definable finite-by-abelian subgoup. Secondly, in a group with simple theory, a normal solvable group A of derived length n is contained in an A-definable almost solvable group of class n.
We prove that every ultraproduct of $p$-adics is inp-minimal (i.e., of burden $1$). More generally, we prove an Ax-Kochen type result on preservation of inp-minimality for Henselian valued fields of …
We prove that every ultraproduct of $p$-adics is inp-minimal (i.e., of burden $1$). More generally, we prove an Ax-Kochen type result on preservation of inp-minimality for Henselian valued fields of equicharacteristic $0$ in the RV language.
The Fitting subgroup of a type-definable group in a simple theory is relatively definable and nilpotent. Moreover, the Fitting subgroup of a supersimple hyperdefinable group has a normal hyperdefinable nilpotent …
The Fitting subgroup of a type-definable group in a simple theory is relatively definable and nilpotent. Moreover, the Fitting subgroup of a supersimple hyperdefinable group has a normal hyperdefinable nilpotent subgroup of bounded index, and is itself of bounded index in a hyperdefinable subgroup.
We consider groups G interpretable in a supersimple finite rank theory T such that Teq eliminates ∃∞. It is shown that G has a definable soluble radical. If G has …
We consider groups G interpretable in a supersimple finite rank theory T such that Teq eliminates ∃∞. It is shown that G has a definable soluble radical. If G has rank 2, then if G is pseudofinite, it is soluble-by-finite, and partial results are obtained under weaker hypotheses, such as ‘functional unimodularity’ of the theory. A classification is obtained when T is pseudofinite and G has a definable and definably primitive action on a rank 1 set.
Let an element of a group be called an FC element if it has only a finite number of conjugates in the group. Baer(1) and Neumann (8) have discussed groups …
Let an element of a group be called an FC element if it has only a finite number of conjugates in the group. Baer(1) and Neumann (8) have discussed groups in which every element is FC, and called them FC- groups . Both Abelian and finite groups are trivially FC-groups; Neumann has studied the properties common to FC-groups and Abelian groups, and Baer the properties common to FC-groups and finite groups. Baer has also shown that, for an arbitrary group G , the set H 1 of all FC elements is a characteristic subgroup. Haimo (3) has defined the FC- chain of a group G by H i / H i −1 is the subgroup of all FC elements in G / H i −1 .
Abstract Let p be an odd prime. A method is described which given a structure M of finite similarity type produces a nilpotent group of class 2 and exponent p …
Abstract Let p be an odd prime. A method is described which given a structure M of finite similarity type produces a nilpotent group of class 2 and exponent p which is in the same stability class as M . Theorem. There are nilpotent groups of class 2 and exponent p in all stability classes . Theorem. The problem of characterizing a stability class is equivalent to characterizing the (nilpotent, class 2, exponent p) groups in that class .
It is proved that if a Lie ring L admits an automorphism of prime order p with a finite number m of fixed points and with pL = L, then …
It is proved that if a Lie ring L admits an automorphism of prime order p with a finite number m of fixed points and with pL = L, then L has a nilpotent subring of index bounded in terms of p and m and whose nilpotency class is bounded in terms of p. It is also shown that if a nilpotent periodic group admits an automorphism of prime order p which has a finite number m of fixed points, then it has a nilpotent subgroup of finite index bounded in terms of m and p and whose class is bounded in terms of p (this gives a positive answer to Hartley's Question 8.81b in the Kourovka Notebook). From this and results of Fong, Hartley, and Meixner, modulo the classification of finite simple groups the following corollary is obtained: a locally finite group in which there is a finite centralizer of an element of prime order is almost nilpotent (with the same bounds on the index and nilpotency class of the subgroup). The proof makes use of the Higman-Kreknin-Kostrikin theorem on the boundedness of the nilpotency class of a Lie ring which admits an automorphism of prime order with a single (trivial) fixed point.
Let <italic>G</italic> be a profinite group in which every centralizer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript upper G Baseline left-parenthesis x right-parenthesis left-parenthesis x element-of upper G right-parenthesis"> <mml:semantics> …
Let <italic>G</italic> be a profinite group in which every centralizer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript upper G Baseline left-parenthesis x right-parenthesis left-parenthesis x element-of upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>G</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="thickmathspace" /> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{C_G}(x)\;(x \in G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is either finite or of finite index. It is shown that <italic>G</italic> is finite-by-abelian-by-finite. Moreover, if, in addition, <italic>G</italic> is a just-infinite pro-<italic>p</italic> group, then it has the structure of a <italic>p</italic>-adic space group whose point group is cyclic or generalized quaternion.
Abstract It is shown that a stable division ring with positive characteristic has finite dimension over its centre. This is then extended to simple division rings.
Abstract It is shown that a stable division ring with positive characteristic has finite dimension over its centre. This is then extended to simple division rings.
We prove that in an arbitrary o-minimal structure, every interpretable group is definably isomorphic to a definable one. We also prove that every definable group lives in a cartesian product …
We prove that in an arbitrary o-minimal structure, every interpretable group is definably isomorphic to a definable one. We also prove that every definable group lives in a cartesian product of one-dimensional definable group-intervals (or one-dimensional definable groups). We discuss the general open question of elimination of imaginaries in an o-minimal structure.
A collection ${\mathcal C}$ of finite $\mathcal {L}$-structures is a 1-dimensional asymptotic class if for every $m \in {\mathbb N}$ and every formula $\varphi (x,\bar {y})$, where $\bar {y}=(y_1,\ldots ,y_m)$: …
A collection ${\mathcal C}$ of finite $\mathcal {L}$-structures is a 1-dimensional asymptotic class if for every $m \in {\mathbb N}$ and every formula $\varphi (x,\bar {y})$, where $\bar {y}=(y_1,\ldots ,y_m)$: [(i)] There is a positive constant $C$ and a finite set $E\subset {\mathbb R}^{>0}$ such that for every $M\in {\mathcal C}$ and $\bar {a}\in M^m$, either $|\varphi (M,\bar {a})|\leq C$, or for some $\mu \in E$, \[ \big ||\varphi (M,\bar {a})|-\mu |M|\big | \leq C|M|^{\frac {1}{2}}.\] [(ii)] For every $\mu \in E$, there is an $\mathcal {L}$-formula $\varphi _{\mu }(\bar {y})$, such that $\varphi _{\mu }(M^m)$ is precisely the set of $\bar {a}\in M^m$ with \[ \big ||\varphi (M,\bar {a})|-\mu |M|\big | \leq C|M|^{\frac {1}{2}}.\] One-dimensional asymptotic classes are introduced and studied here. These classes come equipped with a notion of dimension that is intended to provide for the study of classes of finite structures a concept that is central in the development of model theory for infinite structures. Connections with the model theory of infinite structures are also drawn.
Let () G=G0>G>G>... >G=I be chain of subgroups of the group G. Following Klou]nine [1], we define the stability group of the chain (1) to be the group A of …
Let () G=G0>G>G>... >G=I be chain of subgroups of the group G. Following Klou]nine [1], we define the stability group of the chain (1) to be the group A of 11 uutomorphisms of G such that (2) x holds for ll x e G_ nd for each i 1, 2, m.If the subgroups G re ll normal in G, then it is easy to show that A is nilpotent nd of class t most m 1.But without some such ussumption of normality, the nture of the group A is not so clear.In [1], however, Klouinine proved that A is lwys ut least u soluble group, nd the length d of its derived series cnnot exceed m-1.He remarks of this result that it is "whrscheinlich nicht endgiiltig."In fct, we shll find that A is still nilpotent even in the general cse.This is stated in THEOREM 1.The stability group A of any subgroup-chain (1) of length m is nilpoent and of class at mos 1/2m(m 1).It ws shown in [3] that nilpotent group A of derived length d must be of class t least 2-.Thus Theorem 1 yields the bound (3) d -< [logs m(m-1)] for the derived length of the stability group A. This bound never exceeds m 1 and is smaller than m 1 for m > 5. Indeed, it is of a smaller order of magnitude as m --.Hence Kaloujnine's theorem follows from (3).For the class of A we have the bounds m 1 and 1/2m(m 1) which apply in the normal case and the general case, respectively.These bounds first differ when m 3.That the difference is significant we show by constructing a group with a subgroup-chain of length 3 for which the stability group is of class 3. It will also be proved that the subgroup of G generated by all the commutators x-ix with x e G and e A is always locally nilpotent.This commutator subgroup is known to be always nilpotent in the normal case:cf. [1], Satz 4. We show by an example that this need not be so in the general case.