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Peter Mayr

Publications and source records attributed to Peter Mayr.

At least 19 recordsLinked to original sources
arXiv2025

Ample generics in automorphism groups of Boolean powers of simple Mal'cev algebras

Let $\mathbf{A}$ be a finite simple Mal'cev algebra, such as for example a finite simple group, module, ring, associative or Lie algebra, loop or quasigroup. We show that the automorphism group of a filtered Boolean power of continuous functions from the Cantor space $2^\omega$ to $\mathbf{A}$ has ample generics. The proof splits into the abelian and non-abelian cases. In the abelian case, we use a representation by modules and the theory of $n$-systems developed by Kechris and Rosendal. In the non-abelian case, the proof relies on the decomposition of the automorphism group as a semidirect product of a certain closure of a filtered Boolean power of continuous functions from $2^\omega$ to the automorphism group of $\mathbf{A}$ and the stabiliser of finitely many points in the homeomorphism group $\mathrm{Homeo}\, 2^\omega$. As an intermediate step, we show that pointwise stabilisers in $\mathrm{Homeo}\, 2^\omega$ have ample generics, which extends Kwiatkowska's result that $\mathrm{Homeo}\, 2^\omega$ has ample generics.

math.RA↗
arXiv2025

Solvable, nilpotent and supernilpotent semigroups with completely simple ideal and monoids

Around 1980 commutator theory was generalized from groups to arbitrary algebras using the socalled term condition commutator. The semigroups that are abelian with respect to this commutator were classified by Warne (1994). We study what solvability, nilpotence, and supernilpotence in the sense of commutator theory mean for semigroups and how these notions relate to classical concepts in semigroup theory. We show that a semigroup with a completely simple ideal is solvable (left nilpotent or right nilpotent or supernilpotent) in the sense of commutator theory iff it is a nilpotent extension in the classical sense of semigroup theory of a completely simple semigroup with solvable (nilpotent) subgroups. These characterizations hold in particular for finite semigroups and for eventually regular semigroups, i.e., semigroups in which every element has some regular power. We also show that a monoid is (left and right) nilpotent in the sense of commutator theory iff it embeds into a nilpotent group.

math.GR↗
arXiv2024

Every finite nilpotent loop has a supernilpotent loop as reduct

A basic fact taught in undergraduate algebra courses is that every finite nilpotent group is a direct product of $p$-groups. Already Bruck observed that this does not generalize to loops. In particular, there exist nilpotent loops of size $6$ which are not direct products of loops of size $2$ and $3$. Still we show that every finite nilpotent loop $(A,\cdot)$ has a binary term operation $*$ such that $(A,*)$ is a direct product of nilpotent loops of prime power order, i.e., $(A,*)$ is supernilpotent. As an application we obtain that every nilpotent loop of order $pq$ for primes $p,q$ has a finite basis for its equational theory.

math.GR↗
arXiv2024

Filtered Boolean powers of finite simple non-abelian Mal'cev algebras

Let $\mathbf{A}$ be a finite simple non-abelian Mal'cev algebra (e.g. a group, loop, ring). We investigate the Boolean power $\mathbf{D}$ of $\mathbf{A}$ by the countable atomless Boolean algebra $\mathbf{B}$ filtered at some idempotents $e_1,\dots,e_n$ of $\mathbf{A}$. When $e_1,\dots,e_n$ are all idempotents of $\mathbf{A}$ we establish two concrete representations of $\mathbf{D}$: as the Fra\"iss\'e limit of the class of finite direct powers of $\mathbf{A}$, and as congruence classes of the countable free algebra in the variety generated by $\mathbf{A}$. Further, for arbitrary $e_1,\dots,e_n$, we show that $\mathbf{D}$ is $\omega$-categorical and that its automorphism group has the small index property, strong uncountable cofinality and the Bergman property. As necessary background we establish some general properties of congruences and automorphisms of filtered Boolean powers of $\mathbf{A}$ by any Boolean algebra $\mathbf{B}$, including a semidirect decomposition for their automorphism groups.

math.LO↗
arXiv2023

Clonoids between modules

Clonoids are sets of finitary functions from an algebra $\mathbb{A}$ to an algebra $\mathbb{B}$ that are closed under composition with term functions of $\mathbb{A}$ on the domain side and with term functions of $\mathbb{B}$ on the codomain side. For $\mathbb{A},\mathbb{B}$ (polynomially equivalent to) finite modules we show: If $\mathbb{A},\mathbb{B}$ have coprime order and the congruence lattice of $\mathbb{A}$ is distributive, then there are only finitely many clonoids from $\mathbb{A}$ to $\mathbb{B}$. This is proved by establishing for every natural number $k$ a particular linear equation that all $k$-ary functions from $\mathbb{A}$ to $\mathbb{B}$ satisfy. Else if $\mathbb{A},\mathbb{B}$ do not have coprime order, then there exist infinite ascending chains of clonoids from $\mathbb{A}$ to $\mathbb{B}$ ordered by inclusion. Consequently any extension of $\mathbb{A}$ by $\mathbb{B}$ has countably infinitely many $2$-nilpotent expansions up to term equivalence.

math.RA↗
arXiv2022

Vaughan--Lee's nilpotent loop of size $12$ is finitely based

In 1983 Vaughan--Lee showed that if a finite nilpotent loop splits into a direct product of factors of prime power order, then its equational theory has a finite basis. Whether the condition on the direct decomposition is necessary has remained open since. In the same paper, Vaughan--Lee gives an explicit example of a nilpotent loop of order $12$ that does not factor into loops of prime power order and asks whether it is finitely based. We give a finite basis for his example by explicitly characterizing its term functions. This also allows us to show that the subpower membership problem for this loop can be solved in polynomial time.

math.RA↗
arXiv2021

Small Promise CSPs that reduce to large CSPs

For relational structures A, B of the same signature, the Promise Constraint Satisfaction Problem PCSP(A,B) asks whether a given input structure maps homomorphically to A or does not even map to B. We are promised that the input satisfies exactly one of these two cases. If there exists a structure C with homomorphisms $A\to C\to B$, then PCSP(A,B) reduces naturally to CSP(C). To the best of our knowledge all known tractable PCSPs reduce to tractable CSPs in this way. However Barto showed that some PCSPs over finite structures A, B require solving CSPs over infinite C. We show that even when such a reduction to finite C is possible, this structure may become arbitrarily large. For every integer $n>1$ and every prime p we give A, B of size n with a single relation of arity $n^p$ such that PCSP(A, B) reduces via a chain of homomorphisms $ A\to C\to B$ to a tractable CSP over some C of size p but not over any smaller structure. In a second family of examples, for every prime $p\geq 7$ we construct A, B of size $p-1$ with a single ternary relation such that PCSP(A, B) reduces via $A\to C\to B$ to a tractable CSP over some C of size p but not over any smaller structure. In contrast we show that if A, B are graphs and PCSP(A,B) reduces to tractable CSP(C) for some finite digraph C, then already A or B has a tractable CSP. This extends results and answers a question of Deng et al.

cs.CC↗
arXiv2021

BPS Indices, Modularity and Perturbations in Quantum K-theory

We study a perturbation family of N=2 3d gauge theories and its relation to quantum K-theory. A 3d version of the Intriligator-Vafa formula is given for the quantum K-theory ring of Grassmannians. The 3d BPS half-index of the gauge theory is connected to the theory of bilateral hypergeometric q-series, and to modular q-characters of a class of conformal field theories in a certain massless limit. Turning on 3d Wilson lines at torsion points leads to mock modular behavior. Perturbed correlators in the IR regime are computed by determining the UV-IR map in the presence of deformations.

hep-th↗
arXiv2020

Sandwiches for Promise Constraint Satisfaction

Promise Constraint Satisfaction Problems (PCSP) were proposed recently by Brakensiek and Guruswami arXiv:1704.01937 as a framework to study approximations for Constraint Satisfaction Problems (CSP). Informally a PCSP asks to distinguish between whether a given instance of a CSP has a solution or not even a specified relaxation can be satisfied. All currently known tractable PCSPs can be reduced in a natural way to tractable CSPs. Barto arXiv:1909.04878 presented an example of a PCSP over Boolean structures for which this reduction requires solving a CSP over an infinite structure. We give a first example of a PCSP over Boolean structures which reduces to a tractable CSP over a structure of size $3$ but not smaller. Further we investigate properties of PCSPs that reduce to systems of linear equations or to CSPs over structures with semilattice or majority polymorphism.

cs.CC↗
arXiv2019

Wilson loop algebras and quantum K-theory for Grassmannians

We study the algebra of Wilson line operators in three-dimensional N=2 supersymmetric U(M) gauge theories with a Higgs phase related to a complex Grassmannian Gr(M,N), and its connection to K-theoretic Gromov-Witten invariants for Gr(M,N). For different Chern-Simons levels, the Wilson loop algebra realizes either the quantum cohomology of Gr(M,N), isomorphic to the Verlinde algebra for U(M), or the quantum K-theoretic ring of Schubert structure sheaves studied by mathematicians, or closely related algebras.

hep-th↗
arXiv2019

Algebras from Congruences

We present a functorial construction which, starting from a congruence $\alpha$ of finite index in an algebra A, yields a new algebra C with the following properties: the congruence lattice of C is isomorphic to the interval of congruences between 0 and $\alpha$ on A, this isomorphism preserves higher commutators and TCT types, and C inherits all idempotent Maltsev conditions from A. As applications of this construction, we first show that supernilpotence is decidable for congruences of finite algebras in varieties that omit type 1. Secondly, we prove that the subpower membership problem for finite algebras with a cube term can be effectively reduced to membership questions in subdirect products of subdirectly irreducible algebras with central monoliths. As a consequence, we obtain a polynomial time algorithm for the subpower membership problem for finite algebras with a cube term in which the monolith of every subdirectly irreducible section has a supernilpotent centralizer.

math.LO↗
arXiv2019

Bounded homomorphisms and finitely generated fiber products of lattices

We investigate when fiber products of lattices are finitely generated and obtain a new characterization of bounded lattice homomorphisms onto lattices satisfying a property we call Dean's condition (D) which arises from Dean's solution to the word problem for finitely presented lattices. In particular, all finitely presented lattices and those satisfying Whitman's condition satisfy (D). For lattice epimorphisms $g\colon A\to D$, $h\colon B\to D$, where $A$, $B$ are finitely generated and $D$ satisfies (D), we show the following: If $g$ and $h$ are bounded, then their fiber product (pullback) $C=\{(a,b)\in A\times B\ |\ g(a)=h(b)\}$ is finitely generated. While the converse is not true in general, it does hold when $A$ and $B$ are free. As a consequence we obtain an (exponential time) algorithm to decide boundedness for finitely presented lattices and their finitely generated sublattices satisfying (D). This generalizes an unpublished result of Freese and Nation.

math.LO↗
arXiv2019

Quantum K-Theory of Calabi-Yau Manifolds

The disk partition function of certain 3d N=2 supersymmetric gauge theories computes a quantum K-theoretic ring for Kahler manifolds X. We study the 3d gauge theory/quantum K-theory correspondence for global and local Calabi-Yau manifolds with several Kahler moduli. We propose a multi-cover formula that relates the 3d BPS world-volume degeneracies computed by quantum K-theory to Gopakumar-Vafa invariants.

hep-th↗
arXiv2019

Generating subdirect products

We study conditions under which subdirect products of various types of algebraic structures are finitely generated or finitely presented. In the case of two factors, we prove general results for arbitrary congruence permutable varieties, which generalise previously known results for groups, and which apply to modules, rings, $K$-algebras and loops. For instance, if $C$ is a fiber product of $A$ and $B$ over a common quotient $D$, and if $A$, $B$ and $D$ are finitely presented, then $C$ is finitely generated. For subdirect products of more than two factors we establish a general connection with projections on pairs of factors and higher commutators. More detailed results are provided for groups, loops, rings and $K$-algebras. In particular, let $C$ be a subdirect product of $K$-algebras $A_1,\dots,A_n$ for a Noetherian ring $K$ such that the projection of $C$ onto any $A_i\times A_j$ has finite co-rank in $A_i\times A_j$. Then $C$ is finitely generated (resp. finitely presented) if and only if all $A_i$ are finitely generated (resp. finitely presented). Finally, examples of semigroups and lattices are provided which indicate further complications as one ventures beyond congruence permutable varieties.

math.RA↗
arXiv2019

Presentations for subrings and subalgebras of finite co-rank

Let $K$ be a commutative Noetherian ring with identity, let $A$ be a $K$-algebra, and let $B$ be a subalgebra of $A$ such that $A/B$ is finitely generated as a $K$-module. The main result of the paper is that $A$ is finitely presented (resp. finitely generated) if and only if $B$ is finitely presented (resp. finitely generated). As corollaries we obtain: a subring of finite index in a finitely presented ring is finitely presented; a subalgebra of finite co-dimension in a finitely presented algebra over a field is finitely presented (already shown by Voden in 2009). We also discuss the role of the Noetherian assumption on $K$, and show that for finite generation it can be replaced by a weaker condition that the module $A/B$ be finitely presented. Finally, we demonstrate that the results do not readily extend to non-associative algebras, by exhibiting an ideal of co-dimension $1$ of the free Lie algebra of rank 2 which is not finitely generated as a Lie algebra.

math.RA↗
arXiv2018

A 3d Gauge Theory/Quantum K-Theory Correspondence

The 2d gauged linear sigma model (GLSM) gives a UV model for quantum cohomology on a Kahler manifold X, which is reproduced in the IR limit. We propose and explore a 3d lift of this correspondence, where the UV model is the N=2 supersymmetric 3d gauge theory and the IR limit is given by Givental's permutation equivariant quantum K-theory on X. This gives a one-parameter deformation of the 2d GLSM/quantum cohomology correspondence and recovers it in a small radius limit. We study some novelties of the 3d case regarding integral BPS invariants, chiral rings, deformation spaces and mirror symmetry.

hep-th↗
arXiv2018

The Subpower Membership Problem for Finite Algebras with Cube Terms

The subalgebra membership problem is the problem of deciding if a given element belongs to an algebra given by a set of generators. This is one of the best established computational problems in algebra. We consider a variant of this problem, which is motivated by recent progress in the Constraint Satisfaction Problem, and is often referred to as the Subpower Membership Problem (SMP). In the SMP we are given a set of tuples in a direct product of algebras from a fixed finite set $\mathcal{K}$ of finite algebras, and are asked whether or not a given tuple belongs to the subalgebra of the direct product generated by a given set. Our main result is that the subpower membership problem SMP($\mathcal{K}$) is in P if $\mathcal{K}$ is a finite set of finite algebras with a cube term, provided $\mathcal{K}$ is contained in a residually small variety. We also prove that for any finite set of finite algebras $\mathcal{K}$ in a variety with a cube term, each one of the problems SMP($\mathcal{K}$), SMP($\mathbb{HS} \mathcal{K}$), and finding compact representations for subpowers in $\mathcal{K}$, is polynomial time reducible to any of the others, and the first two lie in NP.

cs.LO↗