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The Dimension of Divisibility Orders and Multiset Posets 可除阶与多集序的维数
Order Pub Date : 2023-11-22 DOI: 10.1007/s11083-023-09653-7
Milan Haiman
{"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":null,"pages":null},"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}
引用次数: 2
On the Rigidity of Lattices of Topologies on Vector Spaces 论向量空间上拓扑格的刚性
Order Pub Date : 2023-11-22 DOI: 10.1007/s11083-023-09655-5
Takanobu Aoyama
{"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":null,"pages":null},"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}
引用次数: 0
A Proof of the Alternate Thomassé Conjecture for Countable N-Free Posets 可数n自由序集的交替托马斯猜想的证明
Order Pub Date : 2023-11-20 DOI: 10.1007/s11083-023-09650-w
Davoud Abdi
{"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":null,"pages":null},"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}
引用次数: 0
What is $$-Q$$ - Q for a poset Q? 对于偏序集Q, $$-Q$$ - Q是什么?
Order Pub Date : 2022-04-20 DOI: 10.1007/s11083-022-09600-y
Taiga Yoshida, Masahiko Yoshinaga
{"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":null,"pages":null},"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}
引用次数: 0
Diagonal Orbits in a Type A Double Flag Variety of Complexity One 复杂度为一的 A 型双旗变化中的对角轨道
Order Pub Date : 2020-05-06 DOI: 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":null,"pages":null},"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}
引用次数: 0
Representations of Infinite Tree Sets 无限树集的表示
Order Pub Date : 2020-04-28 DOI: 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":null,"pages":null},"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}
引用次数: 0
Containment Graphs and Posets of Paths in a Tree: Wheels and Partial Wheels 树中路径的包含图和 Posets：轮和部分轮
Order Pub Date : 2020-04-27 DOI: 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":null,"pages":null},"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}
引用次数: 0
Containment Graphs and Posets of Paths in a Tree: Wheels and Partial Wheels 树中路径的包含图和 Posets：轮和部分轮
Order Pub Date : 2020-04-27 DOI: 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":null,"pages":null},"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}
引用次数: 0
The Lattice of Functional Alexandroff Topologies 功能性亚历山德罗夫拓扑晶格
Order Pub Date : 2020-04-20 DOI: 10.1007/s11083-020-09523-6
Jacob Menix, T. Richmond
{"title":"The Lattice of Functional Alexandroff Topologies","authors":"Jacob Menix, T. Richmond","doi":"10.1007/s11083-020-09523-6","DOIUrl":"https://doi.org/10.1007/s11083-020-09523-6","url":null,"abstract":"","PeriodicalId":501237,"journal":{"name":"Order","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141210918","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}
引用次数: 0
The Lattice of Functional Alexandroff Topologies 功能性亚历山德罗夫拓扑晶格
Order Pub Date : 2020-04-20 DOI: 10.1007/s11083-020-09523-6
Jacob Menix, T. Richmond
{"title":"The Lattice of Functional Alexandroff Topologies","authors":"Jacob Menix, T. Richmond","doi":"10.1007/s11083-020-09523-6","DOIUrl":"https://doi.org/10.1007/s11083-020-09523-6","url":null,"abstract":"","PeriodicalId":501237,"journal":{"name":"Order","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141210969","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}
引用次数: 0
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