arXiv subjects

Rachael Alvir

Publications and source records attributed to Rachael Alvir.

7 recordsLinked to original sources

arXiv2025

On the computability of optimal Scott sentences

Given a countable mathematical structure, its Scott sentence is a sentence of the infinitary logic $\mathcal{L}_{ω_1 ω}$ that characterizes it among all countable structures. We can measure the complexity of a structure by the least complexity of a Scott sentence for that structure. It is known that there can be a difference between the least complexity of a Scott sentence and the least complexity of a computable Scott sentence; for example, Alvir, Knight, and McCoy showed that there is a computable structure with a $Π_2$ Scott sentence but no computable $Π_2$ Scott sentence. It is well known that a structure with a $Π_2$ Scott sentence must have a computable $Π_4$ Scott sentence. We show that this is best possible: there is a computable structure with a $Π_2$ Scott sentence but no computable $Σ_4$ Scott sentence. We also show that there is no reasonable characterization of the computable structures with a computable $Π_n$ Scott sentence by showing that the index set of such structures is $Π^1_1$-$m$-complete.

math.LO↗

arXiv2024

Scott Complexity of Reduced Abelian $p$-Groups

Given a reduced abelian $p$-group, we give an upper bound on the Scott complexity of the group in terms of its Ulm invariants. For limit ordinals, we show that this upper bound is tight. This gives an explicit sequence of such groups with arbitrarily high Scott complexity below $ω_1$. Along the way, we give a largely algebraic characterization of the back-and-forth relations on reduced abelian $p$-groups, making progress on an open problem from the book by Ash and Knight.

math.LO↗

arXiv2023

Effectiveness of Walker's Cancellation Theorem

Walker's Cancellation theorem for abelian groups tells us that if $A$ is finitely generated and $G$ and $H$ are such that $A \oplus G \cong A \oplus H$, then $G \cong H$. Michael Deveau showed that the theorem can be effectivized, but not uniformly. In this paper, we expand on Deveau's initial analysis to show that the complexity of uniformly outputting an index of an isomorphism between $G$ and $H$, given indices for $A$, $G$, $H$, the isomorphism between $A \oplus G$ and $A \oplus H$, and the rank of $A$, is $\mathbf{0'}$.

math.LO↗

arXiv2022

Interpreting a field in its Heisenberg group

We improve on and generalize a 1960 result of Maltsev. For a field $F$, we denote by $H(F)$ the Heisenberg group with entries in $F$. Maltsev showed that there is a copy of $F$ defined in $H(F)$, using existential formulas with an arbitrary non-commuting pair $(u,v)$ as parameters. We show that $F$ is interpreted in $H(F)$ using computable $Σ_1$ formulas with no parameters. We give two proofs. The first is an existence proof, relying on a result of Harrison-Trainor, Melnikov, R. Miller, and Montalbán. This proof allows the possibility that the elements of $F$ are represented by tuples in $H(F)$ of no fixed arity. The second proof is direct, giving explicit finitary existential formulas that define the interpretation, with elements of $F$ represented by triples in $H(F)$. Looking at what was used to arrive at this parameter-free interpretation of $F$ in $H(F)$, we give general conditions sufficient to eliminate parameters from interpretations.

math.LO↗

arXiv2019

The complexity of Scott sentences of scattered linear orders

Given a countable scattered linear order $L$ of Hausdorff rank $α< ω_1$ we show that it has a $d\text{-}Σ_{2α+1}$ Scott sentence. Ash calculated the back and forth relations for all countable well-orders. From this result we obtain that this upper bound is tight, i.e., for every $α< ω_1$ there is a linear order whose optimal Scott sentence has this complexity. We further show that for all countable $α$ the class of Hausdorff rank $α$ linear orders is $\pmb Σ_{2α+2}$ complete.

math.LO↗

arXiv2018

Complexity of Scott Sentences

We give effective versions of some results on Scott sentences. We show that if $\mathcal{A}$ has a computable $Π_α$ Scott sentence, then the orbits of all tuples are defined by formulas that are computable $Σ_β$ for some $β<α$. (This is an effective version of a result of Montalbán.) We show that if a countable structure $\mathcal{A}$ has a computable $Σ_α$ Scott sentence and one that is computable $Π_α$, then it has one that is computable $d$-$Σ_β$ for some $β< α$. (This is an effective version of a result of A. Miller.) We also give an effective version of a result of D. Miller. Using the non-effective results of Montalbán and A. Miller, we show that a finitely generated group has a $d$-$Σ_2$ Scott sentence iff the orbit of some (or every) generating tuple is defined by a $Π_1$ formula. Using our effective results, we show that for a computable finitely generated group, there is a computable $d$-$Σ_2$ Scott sentence iff the orbit of some (every) generating tuple is defined by a computable $Π_1$ formula.

math.LO↗

arXiv2015

Zero-Divisor Graphs of Quotient Rings

The compressed zero-divisor graph $Γ_C(R)$ associated with a commutative ring $R$ has vertex set equal to the set of equivalence classes $\{ [r] \mid r \in Z(R), r \neq 0 \}$ where $r \sim s$ whenever $ann(r) = ann(s)$. Distinct classes $[r],[s]$ are adjacent in $Γ_C(R)$ if and only if $xy = 0$ for all $x \in [r], y \in [s]$. In this paper, we explore the compressed zero-divisor graph associated with quotient rings of unique factorization domains. Specifically, we prove several theorems which exhibit a method of constructing $Γ(R)$ for when one quotients out by a principal ideal, and prove sufficient conditions for when two such compressed graphs are graph-isomorphic. We show these conditions are not necessary unless one alters the definition of the compressed graph to admit looped vertices, and conjecture necessary and sufficient conditions for two compressed graphs with loops to be isomorphic when considering any quotient ring of a unique factorization domain.

math.AC↗