Pub Date : 2023-11-22DOI: 10.1007/s11083-023-09653-7
Milan Haiman
The Dushnik–Miller dimension of a poset P is the least d for which P can be embedded into a product of d chains. Lewis and Souza isibility order on the interval of integers ([N/kappa , N]) is bounded above by (kappa (log kappa )^{1+o(1)}) and below by (Omega ((log kappa /log log kappa )^2)). We improve the upper bound to (O((log kappa )^3/(log log kappa )^2).) We deduce this bound from a more general result on posets of multisets ordered by inclusion. We also consider other divisibility orders and give a bound for polynomials ordered by divisibility.
{"title":"The Dimension of Divisibility Orders and Multiset Posets","authors":"Milan Haiman","doi":"10.1007/s11083-023-09653-7","DOIUrl":"https://doi.org/10.1007/s11083-023-09653-7","url":null,"abstract":"<p>The Dushnik–Miller dimension of a poset <i>P</i> is the least <i>d</i> for which <i>P</i> can be embedded into a product of <i>d</i> chains. Lewis and Souza isibility order on the interval of integers <span>([N/kappa , N])</span> is bounded above by <span>(kappa (log kappa )^{1+o(1)})</span> and below by <span>(Omega ((log kappa /log log kappa )^2))</span>. We improve the upper bound to <span>(O((log kappa )^3/(log log kappa )^2).)</span> We deduce this bound from a more general result on posets of multisets ordered by inclusion. We also consider other divisibility orders and give a bound for polynomials ordered by divisibility.</p>","PeriodicalId":501237,"journal":{"name":"Order","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138511733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-22DOI: 10.1007/s11083-023-09655-5
Takanobu Aoyama
A vector topology on a vector space over a topological field is a (not necessarily Hausdorff) topology by which the addition and the scalar multiplication are continuous. We prove that, if an isomorphism between the lattices of topologies of two vector spaces preserves vector topologies, then the isomorphism is induced by a translation, a semilinear isomorphism and the complement map. As a consequence, if such an isomorphism exists, the coefficient fields are isomorphic as topological fields and these vector spaces have the same dimension. We also prove a similar rigidity result for an isomorphism between the lattice of vector topologies which preserves Hausdorff vector topologies. To obtain these results, we construct a Galois connection between a lattice of vector topologies and a lattice of subspaces and use the fundamental theorems of affine and projective geometries.
{"title":"On the Rigidity of Lattices of Topologies on Vector Spaces","authors":"Takanobu Aoyama","doi":"10.1007/s11083-023-09655-5","DOIUrl":"https://doi.org/10.1007/s11083-023-09655-5","url":null,"abstract":"<p>A vector topology on a vector space over a topological field is a (not necessarily Hausdorff) topology by which the addition and the scalar multiplication are continuous. We prove that, if an isomorphism between the lattices of topologies of two vector spaces preserves vector topologies, then the isomorphism is induced by a translation, a semilinear isomorphism and the complement map. As a consequence, if such an isomorphism exists, the coefficient fields are isomorphic as topological fields and these vector spaces have the same dimension. We also prove a similar rigidity result for an isomorphism between the lattice of vector topologies which preserves Hausdorff vector topologies. To obtain these results, we construct a Galois connection between a lattice of vector topologies and a lattice of subspaces and use the fundamental theorems of affine and projective geometries.</p>","PeriodicalId":501237,"journal":{"name":"Order","volume":"14 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138511735","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-20DOI: 10.1007/s11083-023-09650-w
Davoud Abdi
An N-free poset is a poset whose comparability graph does not embed an induced path with four vertices. We use the well-quasi-order property of the class of countable N-free posets and some labelled ordered trees to show that a countable N-free poset has one or infinitely many siblings, up to isomorphism. This, partially proves a conjecture stated by Thomassé for this class.
{"title":"A Proof of the Alternate Thomassé Conjecture for Countable N-Free Posets","authors":"Davoud Abdi","doi":"10.1007/s11083-023-09650-w","DOIUrl":"https://doi.org/10.1007/s11083-023-09650-w","url":null,"abstract":"<p>An <i>N</i>-free poset is a poset whose comparability graph does not embed an induced path with four vertices. We use the well-quasi-order property of the class of countable <i>N</i>-free posets and some labelled ordered trees to show that a countable <i>N</i>-free poset has one or infinitely many siblings, up to isomorphism. This, partially proves a conjecture stated by Thomassé for this class.</p>","PeriodicalId":501237,"journal":{"name":"Order","volume":"14 6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138511734","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-20DOI: 10.1007/s11083-022-09600-y
Taiga Yoshida, Masahiko Yoshinaga
In the context of combinatorial reciprocity, it is a natural question to ask what “(-Q)” is for a poset Q. In a previous work, the definition “(-Q:=Qtimes mathbb {R}) with lexicographic order” was proposed based on the notion of Euler characteristic of semialgebraic sets. In fact, by using this definition, Stanley’s reciprocity for order polynomials was generalized to an equality for the Euler characteristics of certain spaces of increasing maps between posets. The purpose of this paper is to refine this result, that is, to show that these spaces are homeomorphic if the topology of Q is metrizable.
{"title":"What is $$-Q$$ - Q for a poset Q?","authors":"Taiga Yoshida, Masahiko Yoshinaga","doi":"10.1007/s11083-022-09600-y","DOIUrl":"https://doi.org/10.1007/s11083-022-09600-y","url":null,"abstract":"<p>In the context of combinatorial reciprocity, it is a natural question to ask what “<span>(-Q)</span>” is for a poset <i>Q</i>. In a previous work, the definition “<span>(-Q:=Qtimes mathbb {R})</span> with lexicographic order” was proposed based on the notion of Euler characteristic of semialgebraic sets. In fact, by using this definition, Stanley’s reciprocity for order polynomials was generalized to an equality for the Euler characteristics of certain spaces of increasing maps between posets. The purpose of this paper is to refine this result, that is, to show that these spaces are homeomorphic if the topology of <i>Q</i> is metrizable.</p>","PeriodicalId":501237,"journal":{"name":"Order","volume":"36 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138511731","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-06DOI: 10.1007/s11083-020-09530-7
M. Can, Tien Le
{"title":"Diagonal Orbits in a Type A Double Flag Variety of Complexity One","authors":"M. Can, Tien Le","doi":"10.1007/s11083-020-09530-7","DOIUrl":"https://doi.org/10.1007/s11083-020-09530-7","url":null,"abstract":"","PeriodicalId":501237,"journal":{"name":"Order","volume":"39 6","pages":"97 - 110"},"PeriodicalIF":0.0,"publicationDate":"2020-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141206526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-04-28DOI: 10.1007/s11083-020-09529-0
J. P. Gollin, Jakob Kneip
{"title":"Representations of Infinite Tree Sets","authors":"J. P. Gollin, Jakob Kneip","doi":"10.1007/s11083-020-09529-0","DOIUrl":"https://doi.org/10.1007/s11083-020-09529-0","url":null,"abstract":"","PeriodicalId":501237,"journal":{"name":"Order","volume":"79 1","pages":"79 - 96"},"PeriodicalIF":0.0,"publicationDate":"2020-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141209681","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-04-27DOI: 10.1007/s11083-020-09526-3
M. Golumbic, V. Limouzy
{"title":"Containment Graphs and Posets of Paths in a Tree: Wheels and Partial Wheels","authors":"M. Golumbic, V. Limouzy","doi":"10.1007/s11083-020-09526-3","DOIUrl":"https://doi.org/10.1007/s11083-020-09526-3","url":null,"abstract":"","PeriodicalId":501237,"journal":{"name":"Order","volume":"142 35","pages":"37 - 48"},"PeriodicalIF":0.0,"publicationDate":"2020-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141209785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-04-27DOI: 10.1007/s11083-020-09526-3
M. Golumbic, V. Limouzy
{"title":"Containment Graphs and Posets of Paths in a Tree: Wheels and Partial Wheels","authors":"M. Golumbic, V. Limouzy","doi":"10.1007/s11083-020-09526-3","DOIUrl":"https://doi.org/10.1007/s11083-020-09526-3","url":null,"abstract":"","PeriodicalId":501237,"journal":{"name":"Order","volume":"139 25","pages":"37 - 48"},"PeriodicalIF":0.0,"publicationDate":"2020-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141210136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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