{"title":"Constructing New Geometries: A Generalized Approach to Halving for Hypertopes","authors":"Claudio Alexandre Piedade, Philippe Tranchida","doi":"10.1007/s00493-024-00134-y","DOIUrl":null,"url":null,"abstract":"<p>Given a residually connected incidence geometry <span>\\(\\Gamma \\)</span> that satisfies two conditions, denoted <span>\\((B_1)\\)</span> and <span>\\((B_2)\\)</span>, we construct a new geometry <span>\\(H(\\Gamma )\\)</span> with properties similar to those of <span>\\(\\Gamma \\)</span>. This new geometry <span>\\(H(\\Gamma )\\)</span> is inspired by a construction of Lefèvre-Percsy, Percsy and Leemans (Bull Belg Math Soc Simon Stevin 7(4):583–610, 2000). We show how <span>\\(H(\\Gamma )\\)</span> relates to the classical halving operation on polytopes, allowing us to generalize the halving operation to a broader class of geometries, that we call non-degenerate leaf hypertopes. Finally, we apply this generalization to cubic toroids in order to generate new examples of regular hypertopes.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"95 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Combinatorica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00493-024-00134-y","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Given a residually connected incidence geometry \(\Gamma \) that satisfies two conditions, denoted \((B_1)\) and \((B_2)\), we construct a new geometry \(H(\Gamma )\) with properties similar to those of \(\Gamma \). This new geometry \(H(\Gamma )\) is inspired by a construction of Lefèvre-Percsy, Percsy and Leemans (Bull Belg Math Soc Simon Stevin 7(4):583–610, 2000). We show how \(H(\Gamma )\) relates to the classical halving operation on polytopes, allowing us to generalize the halving operation to a broader class of geometries, that we call non-degenerate leaf hypertopes. Finally, we apply this generalization to cubic toroids in order to generate new examples of regular hypertopes.
期刊介绍：
COMBINATORICA publishes research papers in English in a variety of areas of combinatorics and the theory of computing, with particular emphasis on general techniques and unifying principles. Typical but not exclusive topics covered by COMBINATORICA are
- Combinatorial structures (graphs, hypergraphs, matroids, designs, permutation groups).
- Combinatorial optimization.
- Combinatorial aspects of geometry and number theory.
- Algorithms in combinatorics and related fields.
- Computational complexity theory.
- Randomization and explicit construction in combinatorics and algorithms.
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