We first consider the class K of graphs on a zero-dimensional metrizable compact space with continuous chromatic number at least three. We provide a concrete basis of size continuum for K made up of countable graphs, comparing them with the quasi-order associated with injective continuous homomorphisms. We prove that the size of such a basis is sharp, using odometers. However, using odometers again, we prove that there is no antichain basis in K, and provide infinite descending chains in K. Our method implies that the equivalence relation of flip conjugacy of minimal homeomorphisms of the Cantor space is Borel reducible to the equivalence relation associated with our quasi-order. We also prove that there is no antichain basis in the class of graphs on a zero-dimensional Polish space with continuous chromatic number at least three. We study the graphs induced by a continuous function, and show that any basis for the class of graphs induced by a homeomorphism of a zero-dimensional metrizable compact space with continuous chromatic number at least three must have size continuum, using odometers or subshifts.
{"title":"Continuous 2-colorings and topological dynamics","authors":"D. Lecomte","doi":"10.4064/dm870-7-2023","DOIUrl":"https://doi.org/10.4064/dm870-7-2023","url":null,"abstract":"We first consider the class K of graphs on a zero-dimensional metrizable compact space with continuous chromatic number at least three. We provide a concrete basis of size continuum for K made up of countable graphs, comparing them with the quasi-order associated with injective continuous homomorphisms. We prove that the size of such a basis is sharp, using odometers. However, using odometers again, we prove that there is no antichain basis in K, and provide infinite descending chains in K. Our method implies that the equivalence relation of flip conjugacy of minimal homeomorphisms of the Cantor space is Borel reducible to the equivalence relation associated with our quasi-order. We also prove that there is no antichain basis in the class of graphs on a zero-dimensional Polish space with continuous chromatic number at least three. We study the graphs induced by a continuous function, and show that any basis for the class of graphs induced by a homeomorphism of a zero-dimensional metrizable compact space with continuous chromatic number at least three must have size continuum, using odometers or subshifts.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47623353","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Scott Carson, Igor Dolinka, James East, Victoria Gould, Rida-e Zenab
This paper concerns a class of semigroups that arise as products $US$, associated to what we call ‘action pairs’. Here $U$ and $S$ are subsemigroups of a common monoid and, roughly speaking, $S$ has an action on the monoid completion $U^1$ that is sui
{"title":"Product decompositions of semigroups induced by action pairs","authors":"Scott Carson, Igor Dolinka, James East, Victoria Gould, Rida-e Zenab","doi":"10.4064/dm871-8-2023","DOIUrl":"https://doi.org/10.4064/dm871-8-2023","url":null,"abstract":"This paper concerns a class of semigroups that arise as products $US$, associated to what we call ‘action pairs’. Here $U$ and $S$ are subsemigroups of a common monoid and, roughly speaking, $S$ has an action on the monoid completion $U^1$ that is sui","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135009733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We discuss the curvilinear web $boldsymbol{mathcal W}_{0,n+3}$ on the moduli space $mathcal M_{0,n+3}$ of projective configurations of $n+3$ points on $mathbf P^1$ defined by the $n+3$ forgetful maps $mathcal M_{0,n+3}rightarrow mathcal M_{0,n+2}$. We recall classical results which show that this web is linearizable when $n$ is odd, or is equivalent to a web by conics when $n$ is even. We then turn to the abelian relations (ARs) of these webs. After recalling the well-known case when $n=2$ (related to the 5-terms functional identity of the dilogarithm), we focus on the case of the 6-web $boldsymbol{mathcal W}_{{0,6}}$. We show that this web is isomorphic to the web formed by the lines contained in Segre's cubic primal $boldsymbol{S}subset mathbf P^4$ and that a kind of `Abel's theorem' allows to describe the ARs of $boldsymbol{mathcal W}_{{0,6}}$ by means of the abelian 2-forms on the Fano surface $F_1(boldsymbol{S})subset G_1(mathbf P^4)$ of lines contained in $boldsymbol{S}$. We deduce from this that $boldsymbol{mathcal W}_{{0,6}}$ has maximal rank with all its ARs rational, and that these span a space which is an irreducible $mathfrak S_6$-module. Then we take up an approach due to Damiano that we correct in the case when $n$ is odd: it leads to an abstract description of the space of ARs of $boldsymbol{mathcal W}_{0,n+3}$ as a $mathfrak S_{n+3}$-representation. In particular, we obtain that this web has maximal rank for any $ngeq 2$. Finally, we consider `Euler's abelian relation $boldsymbol{mathcal E}_n$', a particular AR for $boldsymbol{mathcal W}_{0,n+3}$ constructed by Damiano from a characteristic class on the grassmannian of 2-planes in $mathbf R^{n+3}$ by means of Gelfand-MacPherson theory of polylogarithmic forms. We give an explicit conjectural formula for the components of $boldsymbol{mathcal E}_n$ that we prove to be correct for $nleq 12$.
{"title":"On the $(n+3)$-webs by rational curves induced by the forgetful maps on the moduli spaces $mathcal M_{0,n+3}$","authors":"Luc Pirio","doi":"10.4064/dm866-2-2023","DOIUrl":"https://doi.org/10.4064/dm866-2-2023","url":null,"abstract":"We discuss the curvilinear web $boldsymbol{mathcal W}_{0,n+3}$ on the moduli space $mathcal M_{0,n+3}$ of projective configurations of $n+3$ points on $mathbf P^1$ defined by the $n+3$ forgetful maps $mathcal M_{0,n+3}rightarrow mathcal M_{0,n+2}$. We recall classical results which show that this web is linearizable when $n$ is odd, or is equivalent to a web by conics when $n$ is even. We then turn to the abelian relations (ARs) of these webs. After recalling the well-known case when $n=2$ (related to the 5-terms functional identity of the dilogarithm), we focus on the case of the 6-web $boldsymbol{mathcal W}_{{0,6}}$. We show that this web is isomorphic to the web formed by the lines contained in Segre's cubic primal $boldsymbol{S}subset mathbf P^4$ and that a kind of `Abel's theorem' allows to describe the ARs of $boldsymbol{mathcal W}_{{0,6}}$ by means of the abelian 2-forms on the Fano surface $F_1(boldsymbol{S})subset G_1(mathbf P^4)$ of lines contained in $boldsymbol{S}$. We deduce from this that $boldsymbol{mathcal W}_{{0,6}}$ has maximal rank with all its ARs rational, and that these span a space which is an irreducible $mathfrak S_6$-module. Then we take up an approach due to Damiano that we correct in the case when $n$ is odd: it leads to an abstract description of the space of ARs of $boldsymbol{mathcal W}_{0,n+3}$ as a $mathfrak S_{n+3}$-representation. In particular, we obtain that this web has maximal rank for any $ngeq 2$. Finally, we consider `Euler's abelian relation $boldsymbol{mathcal E}_n$', a particular AR for $boldsymbol{mathcal W}_{0,n+3}$ constructed by Damiano from a characteristic class on the grassmannian of 2-planes in $mathbf R^{n+3}$ by means of Gelfand-MacPherson theory of polylogarithmic forms. We give an explicit conjectural formula for the components of $boldsymbol{mathcal E}_n$ that we prove to be correct for $nleq 12$.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42156852","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"$C(betamathbb{N}setminusmathbb{N})$ among the Archimedean $ell$-groups with strong unit","authors":"P. Scowcroft","doi":"10.4064/dm854-7-2022","DOIUrl":"https://doi.org/10.4064/dm854-7-2022","url":null,"abstract":"","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70158574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Isolated points of spaces of homomorphisms from ordered AL-algebras","authors":"A. Bobrowski, W. Chojnacki","doi":"10.4064/dm845-11-2021","DOIUrl":"https://doi.org/10.4064/dm845-11-2021","url":null,"abstract":"","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70158430","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a simplified definition of Quillen’s fibration sequences in a pointed model category that fully captures the theory, although it is completely independent of the concept of action. This advantage arises from the understanding that the homotopy theory of the model category’s arrow category contains all homotopical information about its long fibration sequences. MSC 2020: 18E35, 18N40, 14A30
{"title":"A new approach to model categorical homotopy fiber sequences","authors":"Alisa Govzmann, Damjan Pivstalo, N. Poncin","doi":"10.4064/dm858-5-2022","DOIUrl":"https://doi.org/10.4064/dm858-5-2022","url":null,"abstract":"We propose a simplified definition of Quillen’s fibration sequences in a pointed model category that fully captures the theory, although it is completely independent of the concept of action. This advantage arises from the understanding that the homotopy theory of the model category’s arrow category contains all homotopical information about its long fibration sequences. MSC 2020: 18E35, 18N40, 14A30","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2021-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46065125","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Type I locally compact quantum groups: integral characters and coamenability","authors":"Jacek Krajczok","doi":"10.4064/DM818-9-2020","DOIUrl":"https://doi.org/10.4064/DM818-9-2020","url":null,"abstract":"","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70158147","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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