Identities in polymorphism algebras of infinite structures
Identities in polymorphism algebras of infinite structures
Disciplines
Computer Sciences (10%); Mathematics (90%)
Keywords
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Clone,
Polymorphism,
Oligomorphic Permutation Group,
Algebraic Identity,
Constraint Satisfaction Problem,
Variety Of Algebras
An algebra consists of a set together with functions on this set. Examples are the set of integers together with the functions of addition and multiplication; the set of real numbers together with addition, multiplication, and the power function; or the set which has only two elements, 0 and 1, together with the maximum function. It is clear from their general definition that algebras are ubiquitous in mathematics, and naturally model real-world situations where we have objects (modeled by the elements of the set of the algebra) and some interaction between these objects (modeled by the functions on these elements). The structure of each algebra which appears in mathematics could, in theory, be studied separately, and this is also done for important algebras. The field of universal algebra, on the other hand, aims at understanding the structure of an algebra abstractly from the equations which hold in it, without considering a concrete algebra. For example, the set of integers together with addition satisfies the equations x-x+y=y=y-x+x, which has structural consequences in general: any algebra in which a similar equation holds for some its functions shares certain structural properties with this particular algebra. The theory of equations in finite algebras, such as the above-mentioned algebra on the set with elements 0,1 and the maximum function, has been studied since the beginnings of universal algebra, but has evolved rapidly recently since the discovery of important applications in theoretical computer science: finite algebras model the complexity of certain computational problems, and in fact the equations alone which hold in an algebra determine how complex the problem is. After almost twenty years of research on this connection, last year it was finally established precisely which equations imply that a computational problem can be computed efficiently. This project aims at lifting the strong recent methods for finite algebras to certain infinite algebras. While some sporadic surprising results for infinite algebras have been obtained recently, a general method for proving such results, as has been developed for finite algebras, is still missing. Our research is motivated by the above-mentioned application in theoretical computer science, but also has its own interest, since infinite algebras appear naturally; see the examples above. We also hope that by studying the finite methods in a wider context, we can gain a better understanding of them even in the finite.
An algebra consists of a set together with functions on this set. Examples are the set of integers together with the functions of addition and multiplication; the set of real numbers together with addition, multiplication, and the power function; or the set which has only two elements, 0 and 1, together with the maximum function. It is clear from their general definition that algebras are ubiquitous in mathematics, and naturally model real-world situations where we have objects (modeled by the elements of the set of the algebra) and some interaction between these objects (modeled by the functions on these elements). The structure of each algebra which appears in mathematics could, in theory, be studied separately, and this is also done for important algebras. The field of universal algebra, on the other hand, aims at understanding the structure of an algebra abstractly from the equations which hold in it, without considering a concrete algebra. For example, the set of integers together with addition satisfies the equations x-x+y=y=y-x+x, which has structural consequences in general: any algebra in which a similar equation holds for some its functions shares certain structural properties with this particular algebra. The theory of equations in finite algebras, such as the above-mentioned algebra on the set with elements 0,1 and the maximum function, has been studied since the beginnings of universal algebra, but has evolved rapidly recently since the discovery of important applications in theoretical computer science: finite algebras model the complexity of certain computational problems, and in fact the equations alone which hold in an algebra determine how complex the problem is. After almost twenty years of research on this connection, in 2017 it was finally established precisely which equations imply that a computational problem can be computed efficiently. This project lifted some of the strong recent methods for finite algebras to certain infinite algebras.
- Technische Universität Wien - 100%
- Andrei Bulatov, Simon Fraser University - Canada
- Libor Barto, Charles University Prague - Czechia
- Manuel Bodirsky, Technische Universität Dresden - Germany
- Marcin Kozik, Jagellonian University - Poland
- Keith Kearnes, University of Colorado Boulder - USA
Research Output
- 21 Citations
- 28 Publications
- 5 Scientific Awards
- 2 Fundings
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2020
Title ? \omega -categorical structures avoiding height 1 identities DOI 10.1090/tran/8179 Type Journal Article Author Bodirsky M Journal Transactions of the American Mathematical Society Pages 327-350 Link Publication -
2022
Title The VC-dimension of axis-parallel boxes on the Torus DOI 10.1016/j.jco.2021.101600 Type Journal Article Author Gillibert P Journal Journal of Complexity Pages 101600 Link Publication -
2022
Title When Symmetries Are Not Enough: A Hierarchy of Hard Constraint Satisfaction Problems DOI 10.1137/20m1383471 Type Journal Article Author Gillibert P Journal SIAM Journal on Computing Pages 175-213 Link Publication -
2022
Title Current Challenges in Infinite-Domain Constraint Satisfaction: Dilemmas of the Infinite Sheep DOI 10.1109/ismvl52857.2022.00019 Type Conference Proceeding Abstract Author Pinsker M Pages 80-87 Link Publication -
2022
Title Smooth approximations and CSPs over finitely bounded homogeneous structures DOI 10.1145/3531130.3533353 Type Conference Proceeding Abstract Author Mottet A Pages 1-13 Link Publication -
2023
Title Symmetries of Graphs and Structures that Fail to Interpret a Finite Thing DOI 10.1109/lics56636.2023.10175732 Type Conference Proceeding Abstract Author Barto L Pages 1-13 -
2023
Title Polish topologies on endomorphism monoids of relational structures DOI 10.1016/j.aim.2023.109214 Type Journal Article Author Elliott L Journal Advances in Mathematics Pages 109214 Link Publication -
2021
Title Galois covers of ℙ1 and number fields with large class groups DOI 10.1142/s1793042122500646 Type Journal Article Author Gillibert J Journal International Journal of Number Theory -
2021
Title CORES OVER RAMSEY STRUCTURES DOI 10.1017/jsl.2021.6 Type Journal Article Author Mottet A Journal The Journal of Symbolic Logic -
2023
Title Corrigendum to "-categorical structures avoiding height 1 identities" DOI 10.1090/tran/8501 Type Journal Article Author Bodirsky M Journal Transactions of the American Mathematical Society -
2021
Title When symmetries are enough: collapsing the bounded width hierarchy for infinite-domain CSPs DOI 10.48550/arxiv.2102.07531 Type Preprint Author Mottet A -
2022
Title Polish topologies on endomorphism monoids of relational structures DOI 10.48550/arxiv.2203.11577 Type Preprint Author Elliott L -
2022
Title Current Challenges in Infinite-Domain Constraint Satisfaction: Dilemmas of the Infinite Sheep DOI 10.48550/arxiv.2203.17182 Type Preprint Author Pinsker M -
2023
Title An order out of nowhere: a new algorithm for infinite-domain CSPs DOI 10.48550/arxiv.2301.12977 Type Preprint Author Mottet A -
2023
Title Submaximal clones over a three-element set up to minor-equivalence DOI 10.48550/arxiv.2304.12807 Type Preprint Author Vucaj A -
2023
Title The semigroup of increasing functions on the rational numbers has a unique Polish topology DOI 10.48550/arxiv.2305.04921 Type Preprint Author Pinsker M -
2023
Title On the Zariski topology on endomorphism monoids of omega-categorical structures DOI 10.48550/arxiv.2308.09466 Type Preprint Author Pinsker M -
2020
Title When symmetries are not enough: a hierarchy of hard Constraint Satisfaction Problems DOI 10.48550/arxiv.2002.07054 Type Preprint Author Gillibert P -
2020
Title Cores over Ramsey structures DOI 10.48550/arxiv.2004.05936 Type Other Author Mottet A -
2020
Title The VC-Dimension of Axis-Parallel Boxes on the Torus DOI 10.48550/arxiv.2004.13861 Type Journal Article Author Gillibert Pierre Journal arXiv e-prints -
2020
Title Smooth approximations and CSPs over finitely bounded homogeneous structures DOI 10.48550/arxiv.2011.03978 Type Preprint Author Mottet A -
2023
Title AN ORDER OUT OF NOWHERE: A NEW ALGORITHM FOR INFINITE-DOMAIN CSPS Type Other Author Mottet -
2020
Title Cores over Ramsey structures Type Other Author Mottet -
2021
Title Smooth approximations and relational width collapses Type Other Author Mottet -
2020
Title ω-Categorical structures avoiding height 1 identities Type Other Author Bodirsky -
2020
Title Smooth approximations and CSPS over finitely bounded homogeneous structures Type Other Author Mottet -
2023
Title ON THE ZARISKI TOPOLOGY ON ENDOMORPHISM MONOIDS OF OMEGA-CATEGORICAL STRUCTURES DOI 10.1017/jsl.2023.81 Type Journal Article Author Pinsker M Journal The Journal of Symbolic Logic -
0
DOI 10.1145/3531130 Type Other
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2022
Title Plenary talk at the IEEE International Symposium on Multiple-Valued Logic 2022 Type Personally asked as a key note speaker to a conference Level of Recognition Continental/International -
2021
Title Plenary talk at the 100th Arbeitstagung Allgemeine Algebra Type Personally asked as a key note speaker to a conference Level of Recognition Continental/International -
2023
Title Plenary talk at the Algebra Week 2023 Type Personally asked as a key note speaker to a conference Level of Recognition Continental/International -
2019
Title Two invited plenary talks at the 57th Summer School on General Algebra and Ordered Sets in Karolinka, Czech Republic. Type Personally asked as a key note speaker to a conference Level of Recognition Continental/International -
2023
Title Distinguished paper award Type Poster/abstract prize Level of Recognition Continental/International
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2023
Title ERC Synergy Grant Type Research grant (including intramural programme) Start of Funding 2023 Funder European Research Council (ERC) -
2022
Title WEAVE Type Research grant (including intramural programme) Start of Funding 2022 Funder Austrian Science Fund (FWF)