Searcharxiv⌕ Search

arXiv subjects

Michael C. Laskowski

Publications and source records attributed to Michael C. Laskowski.

At least 19 recordsLinked to original sources
arXiv2026

Borel completeness of $R$-modules when $R$ fails the DCC on pp-definable subgroups

We prove that for any countable ring $R$ (not necessarily commutative), if the associated left $R$-module ${}_R R$ has a strictly descending sequence of pp-definable subgroups, then the theory $Th(R^{(\omega)})$ of the infinite dimensional direct sum is Borel complete. From this, we conclude that if $R$ is countable and not left perfect, then the theory of $R$-modules is Borel complete, and we give a full characterization of which countable simple rings have Borel complete theories. One special case is that the complete theory $Th({\mathbb Z}^{(\omega)})$ is Borel complete, which strengthens the existing proofs of the Borel completeness of TFAB, the theory of torsion free abelian groups. The proof also introduces, relative to the chosen pp-chain, a proper two-sided ideal $L^R$, and a notion of f.g. hulls which, for countable rings and countable parameter sets in theories satisfying $T=T^{\aleph_0}$ exist and are unique up to isomorphism. These constructions may be of independent interest in the model theory of modules.

math.LO↗
arXiv2025

Existential characterizations of monadic NIP

We show that if a universal theory is not monadically NIP, then this is witnessed by a canonical configuration defined by an existential formula. As a consequence, we show that a hereditary class of relational structures is NIP (resp. stable) if and only if it is monadically NIP (resp. monadically stable). As another consequence, we show that if such a class is not monadically NIP, then it has superexponential growth rate.

math.LO↗
arXiv2025

Indiscernibles in monadically NIP theories

We prove various results around indiscernibles in monadically NIP theories. First, we provide several characterizations of monadic NIP in terms of indiscernibles, mirroring previous characterizations in terms of the behavior of finite satisfiability. Second, we study (monadic) distality in hereditary classes and complete theories. Here, via finite combinatorics, we prove a result implying that every planar graph admits a distal expansion. Finally, we prove a result implying that no monadically NIP theory interprets an infinite group, and note an example of a (monadically) stable theory with no distal expansion that does not interpret an infinite group.

math.LO↗
arXiv2025

Equivalents of NOTOP

Working within the context of countable, superstable theories, we give many equivalents of a theory having NOTOP. In particular, NOTOP is equivalent to V-DI, the assertion that any type $V$-dominated by an independent triple is isolated over the triple. If $T$ has NOTOP, then every model $N$ is atomic over an independent tree of countable, elementary substructures, and hence is determined up to back-and-forth equivalence over such a tree. We also verify Shelah's assertion from Chapter XII of \cite{Shc} that NOTOP implies PMOP (without using NDOP).

math.LO↗
arXiv2025

An analogue of U-rank for atomic classes

For a countable, complete, first-order theory $T$, we study $At$, the class of atomic models of $T$. We develop an analogue of $U$-rank and prove two results. On one hand, if some tp(d/a) is not ranked, then there are $2^{\aleph_1}$ non-isomorphic models in $At$ of size $\aleph_1$. On the other hand, if all types have finite rank, then the rank is fully additive and every finite tuple is dominated by an independent set of realizations of pseudo-minimal types.

math.LO↗
arXiv2024

Characterizations of monadic NIP

We give several characterizations of when a complete first-order theory $T$ is monadically NIP, i.e. when expansions of $T$ by arbitrary unary predicates do not have the independence property. The central characterization is a condition on finite satisfiability of types. Other characterizations include decompositions of models, the behavior of indiscernibles, and a forbidden configuration. As an application, we prove non-structure results for hereditary classes of finite substructures of non-monadically NIP models that eliminate quantifiers.

math.LO↗
arXiv2024

Borel complexity of families of finite equivalence relations via large cardinals

We consider a large family of theories of equivalence relations, each with finitely many classes, and assuming the existence of an $ω$-Erdos cardinal, we determine which of these theories are Borel complete. We develop machinery, including {\em forbidding nested sequences} which implies a tight upper bound on Borel complexity, and {\em admitting cross-cutting absolutely indiscernible sets} which in our context implies Borel completeness. In the Appendix we classify the reducts of theories of refining equivalence relations, possibly with infinite splitting.

math.LO↗
arXiv2024

Non-locally modular regular types in classifiable theories

We introduce the notion of strong $p$-semi-regularity and show that if $p$ is a regular type which is not locally modular then any $p$-semi-regular type is strongly $p$-semi-regular. Moreover, for any such $p$-semi-regular type, "domination implies isolation" which allows us to prove the following: Suppose that $T$ is countable, classifiable and $M$ is any model. If $p\in S(M)$ is regular but not locally modular and $b$ is any realization of $p$ then every model $N$ containing $M$ that is dominated by $b$ over $M$ is both constructible and minimal over $Mb$.

math.LO↗
arXiv2023

Characterizing the existence of a Borel complete expansion

We develop general machinery to cast the class of potential canonical Scott sentences of an infinitary sentence $Φ$ as a class of structures in a related language. From this, we show that $Φ$ has a Borel complete expansion if and only if $S_\infty$ divides $Aut(M)$ for some countable model $M\models Φ$. Using this, we prove that for theories $T_h$ asserting that $\{E_n\}$ is a countable family of cross cutting equivalence relations with $h(n)$ classes, if $h(n)$ is uniformly bounded then $T_h$ is not Borel complete, providing a converse to Theorem~2.1 of \cite{LU}.

math.LO↗
arXiv2022

Borel complexity of modules

We prove that for a countable, commutative ring $R$, the class of countable $R$-modules either has only countably many isomorphism types, or else it is Borel complete. The machinery gives a succinct proof of the Borel completeness of TFAB, the class of torsion-free abelian groups. We also prove that for any countable ring $R$, both the class of left $R$-modules endowed with an endomorphism and the class of left $R$-modules with four named submodules are Borel complete.

math.LO↗
arXiv2022

Worst case expansions of complete theories

Given a complete theory $T$ and a subset $Y \subseteq X^k$, we precisely determine the {\em worst case complexity}, with respect to further monadic expansions, of an expansion $(M,Y)$ by $Y$ of a model $M$ of $T$ with universe $X$. In particular, although by definition monadically stable/NIP theories are robust under arbitrary monadic expansions, we show that monadically NFCP (equivalently, mutually algebraic) theories are the largest class that is robust under anything beyond monadic expansions. We also exhibit a paradigmatic structure for the failure of each of monadic NFCP/stable/NIP and prove each of these paradigms definably embeds into a monadic expansion of a sufficiently saturated model of any theory without the corresponding property.

math.LO↗
arXiv2022

Jumps in speeds of hereditary properties in finite relational languages

Given a finite relational language $\mathcal{L}$, a hereditary $\mathcal{L}$-property is a class of finite $\mathcal{L}$-structures closed under isomorphism and substructure. The speed of $\mathcal{H}$ is the function which sends an integer $n\geq 1$ to the number of distinct elements in $\mathcal{H}$ with underlying set $\{1, . . . , n\}$. In this paper we give a description of many new jumps in the possible speeds of a hereditary $\mathcal{L}$-property, where $\mathcal{L}$ is any finite relational language. In particular, we characterize the jumps in the polynomial and factorial ranges, and show they are essentially the same as in the case of graphs. The results in the factorial range are new for all examples requiring a language of arity greater than two, including the setting of hereditary properties of $k$-uniform hypergraphs for $k>2$. Further, adapting an example of Balogh, Bollobás, and Weinreich, we show that for all $k\geq 2$, there are hereditary properties of $k$-uniform hypergraphs whose speeds oscillate between functions near the upper and lower bounds of the penultimate range, ruling out many natural functions as jumps in that range. Our theorems about the factorial range use model theoretic tools related to the notion of mutual algebricity.

math.CO↗
arXiv2021

Mutual algebraicity and cellularity

We prove two results intended to streamline proofs about cellularity that pass through mutual algebraicity. First, we show that a countable structure $M$ is cellular if and only if $M$ is $ω$-categorical and mutually algebraic. Second, if a countable structure $M$ in a finite relational language is mutually algebraic non-cellular, we show it admits an elementary extension adding infinitely many infinite MA-connected components. Towards these results, we introduce MA-presentations of a mutually algebraic structure, in which every atomic formula is mutually algebraic. This allows for an improved quantifier elimination and a decomposition of the structure into independent pieces. We also show this decomposition is largely independent of the MA-presentation chosen.

math.LO↗
arXiv2021

Most(?) theories have Borel complete reducts

We prove that many seemingly simple theories have Borel complete reducts. Specifically, if a countable theory has uncountably many complete 1-types, then it has a Borel complete reduct. Similarly, if $Th(M)$ is not small, then $M^{eq}$ has a Borel complete reduct, and if a theory $T$ is not $ω$-stable, then the elementary diagram of some countable model of $T$ has a Borel complete reduct.

math.LO↗
arXiv2021

Counting siblings in universal theories

We show that if a countable structure $M$ in a finite relational language is not cellular, then there is an age-preserving $N \supseteq M$ such that $2^{\aleph_0}$ many structures are bi-embeddable with $N$. The proof proceeds by a case division based on mutual algebraicity.

math.LO↗
arXiv2019

Uniformly bounded arrays and mutually algebraic structures

We define an easily verifiable notion of an atomic formula having uniformly bounded arrays in a structure $M$. We prove that if $T$ is a complete $L$-theory, then $T$ is mutually algebraic if and only if there is some model $M$ of $T$ for which every atomic formula has uniformly bounded arrays. Moreover, an incomplete theory $T$ is mutually algebraic if and only if every atomic formula has uniformly bounded arrays in every model $M$ of $T$.

math.LO↗
arXiv2019

Weakly minimal groups with a new predicate

Fix a weakly minimal (i.e., superstable $U$-rank $1$) structure $\mathcal{M}$. Let $\mathcal{M}^*$ be an expansion by constants for an elementary substructure, and let $A$ be an arbitrary subset of the universe $M$. We show that all formulas in the expansion $(\mathcal{M}^*,A)$ are equivalent to bounded formulas, and so $(\mathcal{M},A)$ is stable (or NIP) if and only if the $\mathcal{M}$-induced structure $A_{\mathcal{M}}$ on $A$ is stable (or NIP). We then restrict to the case that $\mathcal{M}$ is a pure abelian group with a weakly minimal theory, and $A_{\mathcal{M}}$ is mutually algebraic (equivalently, weakly minimal with trivial forking). This setting encompasses most of the recent research on stable expansions of $(\mathbb{Z},+)$. Using various characterizations of mutual algebraicity, we give new examples of stable structures of the form $(\mathcal{M},A)$. Most notably, we show that if $(G,+)$ is a weakly minimal additive subgroup of the algebraic numbers, $A\subseteq G$ is enumerated by a homogeneous linear recurrence relation with algebraic coefficients, and no repeated root of the characteristic polynomial of $A$ is a root of unity, then $(G,+,B)$ is superstable for any $B\subseteq A$.

math.LO↗