{"title":"Functors on Relational Structures Which Admit Both Left and Right Adjoints","authors":"Víctor Dalmau, Andrei Krokhin, Jakub Opršal","doi":"10.1137/23m1555223","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2041-2068, September 2024. <br/> Abstract. This paper describes several cases of adjunction in the homomorphism preorder of relational structures. We say that two functors [math] and [math] between thin categories of relational structures are adjoint if for all structures [math] and [math], we have that [math] maps homomorphically to [math] if and only if [math] maps homomorphically to [math]. If this is the case, [math] is called the left adjoint to [math] and [math] the right adjoint to [math]. Foniok and Tardif [Discrete Math., 338 (2015), pp. 527–535] described some functors on the category of digraphs that allow both left and right adjoints. The main contribution of Foniok and Tardif is a construction of right adjoints to some of the functors identified as right adjoints by Pultr [Reports of the Midwest Category Seminar IV, Lecture Notes in Math. 137, Springer, 1970, pp. 100–113]. We generalize results of Foniok and Tardif to arbitrary relational structures, and coincidently, we also provide more right adjoints on digraphs, and since these constructions are connected to finite duality, we also provide a new construction of duals to trees. Our results are inspired by an application in promise constraint satisfaction—it has been shown that such functors can be used as efficient reductions between these problems.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"6 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/23m1555223","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2041-2068, September 2024. Abstract. This paper describes several cases of adjunction in the homomorphism preorder of relational structures. We say that two functors [math] and [math] between thin categories of relational structures are adjoint if for all structures [math] and [math], we have that [math] maps homomorphically to [math] if and only if [math] maps homomorphically to [math]. If this is the case, [math] is called the left adjoint to [math] and [math] the right adjoint to [math]. Foniok and Tardif [Discrete Math., 338 (2015), pp. 527–535] described some functors on the category of digraphs that allow both left and right adjoints. The main contribution of Foniok and Tardif is a construction of right adjoints to some of the functors identified as right adjoints by Pultr [Reports of the Midwest Category Seminar IV, Lecture Notes in Math. 137, Springer, 1970, pp. 100–113]. We generalize results of Foniok and Tardif to arbitrary relational structures, and coincidently, we also provide more right adjoints on digraphs, and since these constructions are connected to finite duality, we also provide a new construction of duals to trees. Our results are inspired by an application in promise constraint satisfaction—it has been shown that such functors can be used as efficient reductions between these problems.
期刊介绍:
SIAM Journal on Discrete Mathematics (SIDMA) publishes research papers of exceptional quality in pure and applied discrete mathematics, broadly interpreted. The journal''s focus is primarily theoretical rather than empirical, but the editors welcome papers that evolve from or have potential application to real-world problems. Submissions must be clearly written and make a significant contribution.
Topics include but are not limited to:
properties of and extremal problems for discrete structures
combinatorial optimization, including approximation algorithms
algebraic and enumerative combinatorics
coding and information theory
additive, analytic combinatorics and number theory
combinatorial matrix theory and spectral graph theory
design and analysis of algorithms for discrete structures
discrete problems in computational complexity
discrete and computational geometry
discrete methods in computational biology, and bioinformatics
probabilistic methods and randomized algorithms.