Forgetful polygons as generalizations of semi-affine planes
We generalize the notion of a semi-affine plane to structures with higher girth n. We prove that, in the finite case, for n odd, and with an additional assumption also for n even, these geometries, which we call forgetful n-gons, always arise from (finite) generalized n-gons by ‘forgetting’ lines....
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| Veröffentlicht in: | Discrete mathematics Jg. 271; H. 1; S. 71 - 100 |
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| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Amsterdam
Elsevier B.V
28.09.2003
Elsevier |
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| ISSN: | 0012-365X, 1872-681X |
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| Abstract | We generalize the notion of a semi-affine plane to structures with higher girth
n. We prove that, in the finite case, for
n odd, and with an additional assumption also for
n even, these geometries, which we call forgetful
n-gons, always arise from (finite) generalized
n-gons by ‘forgetting’ lines. |
|---|---|
| AbstractList | We generalize the notion of a semi-affine plane to structures with higher girth
n. We prove that, in the finite case, for
n odd, and with an additional assumption also for
n even, these geometries, which we call forgetful
n-gons, always arise from (finite) generalized
n-gons by ‘forgetting’ lines. |
| Author | Govaert, E. Van Maldeghem, H. |
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| Cites_doi | 10.1007/BF01650055 10.1016/0021-8693(64)90028-6 10.1007/BF02684706 10.1007/BF01221058 10.1007/BF01390082 10.2140/pjm.1963.13.389 10.1007/978-3-0348-8827-1 |
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| Keywords | 51E12 Semiplane Finite generalized quadrangles Generalized polygon Semi-affine plane Geometry Projective plane Affine plane Classification Affine geometry Semi affine plane Projective geometry |
| Language | English |
| License | http://www.elsevier.com/open-access/userlicense/1.0 https://www.elsevier.com/tdm/userlicense/1.0 https://www.elsevier.com/open-access/userlicense/1.0 CC BY 4.0 |
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| References | Tits (BIB12) 1977; 43 H. Van Maldeghem, Generalized Polygons, Monographs in Mathematics, Vol. 93, Birkhäuser Verlag, Basel, Boston, Berlin, 1998. Bose (BIB1) 1963; 13 E. Govaert, Some contributions to the (geometric) theory of generalized polygons, Ph.D. Thesis, Ghent University, 2001. Tits (BIB11) 1959; 2 S.E. Payne, M.A. Miller, Collineations and characterizations of generalized quadrangles of order Feit, Higman (BIB7) 1964; 1 1 De Clerck, Van Maldeghem (BIB4) 1995 Brouwer, Cohen, Neumaier (BIB2) 1989 1) (Combinatorics ’98 (Mondello)), Rend. Circ. Mat. Palermo (2) Suppl. No. 53 (1998) 137–166. Dembowski (BIB5) 1962; 10 Payne, Thas (BIB10) 1984; Vol. 104 B. De Bruyn, Constructions and Characterizations of Near Polygons, Ph.D. Thesis, Ghent University, 2000. Dembowski (BIB6) 1968 10.1016/S0012-365X(02)00871-3_BIB9 10.1016/S0012-365X(02)00871-3_BIB8 Tits (10.1016/S0012-365X(02)00871-3_BIB11) 1959; 2 Dembowski (10.1016/S0012-365X(02)00871-3_BIB5) 1962; 10 Brouwer (10.1016/S0012-365X(02)00871-3_BIB2) 1989 10.1016/S0012-365X(02)00871-3_BIB3 10.1016/S0012-365X(02)00871-3_BIB13 De Clerck (10.1016/S0012-365X(02)00871-3_BIB4) 1995 Feit (10.1016/S0012-365X(02)00871-3_BIB7) 1964; 1 Tits (10.1016/S0012-365X(02)00871-3_BIB12) 1977; 43 Bose (10.1016/S0012-365X(02)00871-3_BIB1) 1963; 13 Dembowski (10.1016/S0012-365X(02)00871-3_BIB6) 1968 Payne (10.1016/S0012-365X(02)00871-3_BIB10) 1984; Vol. 104 |
| References_xml | – start-page: 443 year: 1995 end-page: 475 ident: BIB4 article-title: Some classes of rank 2 geometries publication-title: Handbook of Incidence Geometry, Buildings and Foundations – reference: S.E. Payne, M.A. Miller, Collineations and characterizations of generalized quadrangles of order ( – volume: 10 start-page: 120 year: 1962 end-page: 131 ident: BIB5 article-title: Semiaffine Ebenen publication-title: Arch. Math. – reference: E. Govaert, Some contributions to the (geometric) theory of generalized polygons, Ph.D. Thesis, Ghent University, 2001. – reference: −1) (Combinatorics ’98 (Mondello)), Rend. Circ. Mat. Palermo (2) Suppl. No. 53 (1998) 137–166. – volume: 43 start-page: 283 year: 1977 end-page: 295 ident: BIB12 article-title: Endliche Spiegelungsgruppen, die als Weylgruppen auftreten publication-title: Invent. Math. – reference: +1, – volume: Vol. 104 year: 1984 ident: BIB10 article-title: Finite Generalized Quadrangles publication-title: Research Notes in Mathematics – reference: B. De Bruyn, Constructions and Characterizations of Near Polygons, Ph.D. Thesis, Ghent University, 2000. – volume: 2 start-page: 14 year: 1959 end-page: 60 ident: BIB11 article-title: Sur la trialité et certains groupes qui s’en déduisent publication-title: Inst. Hautes Etudes Sci. Publ. Math. – volume: 13 start-page: 389 year: 1963 end-page: 419 ident: BIB1 article-title: Strongly regular graphs, partial geometries and partially balanced designs publication-title: Pacific J. Math. – year: 1989 ident: BIB2 publication-title: Distance-Regular Graphs – volume: 1 start-page: 114 year: 1964 end-page: 131 ident: BIB7 article-title: The nonexistence of certain generalized polygons publication-title: J. Algebra – reference: H. Van Maldeghem, Generalized Polygons, Monographs in Mathematics, Vol. 93, Birkhäuser Verlag, Basel, Boston, Berlin, 1998. – year: 1968 ident: BIB6 publication-title: Finite Geometries – volume: 10 start-page: 120 year: 1962 ident: 10.1016/S0012-365X(02)00871-3_BIB5 article-title: Semiaffine Ebenen publication-title: Arch. Math. doi: 10.1007/BF01650055 – start-page: 443 year: 1995 ident: 10.1016/S0012-365X(02)00871-3_BIB4 article-title: Some classes of rank 2 geometries – volume: Vol. 104 year: 1984 ident: 10.1016/S0012-365X(02)00871-3_BIB10 article-title: Finite Generalized Quadrangles – year: 1968 ident: 10.1016/S0012-365X(02)00871-3_BIB6 – volume: 1 start-page: 114 year: 1964 ident: 10.1016/S0012-365X(02)00871-3_BIB7 article-title: The nonexistence of certain generalized polygons publication-title: J. Algebra doi: 10.1016/0021-8693(64)90028-6 – volume: 2 start-page: 14 year: 1959 ident: 10.1016/S0012-365X(02)00871-3_BIB11 article-title: Sur la trialité et certains groupes qui s’en déduisent publication-title: Inst. Hautes Etudes Sci. Publ. Math. doi: 10.1007/BF02684706 – year: 1989 ident: 10.1016/S0012-365X(02)00871-3_BIB2 – ident: 10.1016/S0012-365X(02)00871-3_BIB9 – ident: 10.1016/S0012-365X(02)00871-3_BIB3 doi: 10.1007/BF01221058 – ident: 10.1016/S0012-365X(02)00871-3_BIB8 – volume: 43 start-page: 283 year: 1977 ident: 10.1016/S0012-365X(02)00871-3_BIB12 article-title: Endliche Spiegelungsgruppen, die als Weylgruppen auftreten publication-title: Invent. Math. doi: 10.1007/BF01390082 – volume: 13 start-page: 389 year: 1963 ident: 10.1016/S0012-365X(02)00871-3_BIB1 article-title: Strongly regular graphs, partial geometries and partially balanced designs publication-title: Pacific J. Math. doi: 10.2140/pjm.1963.13.389 – ident: 10.1016/S0012-365X(02)00871-3_BIB13 doi: 10.1007/978-3-0348-8827-1 |
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| Snippet | We generalize the notion of a semi-affine plane to structures with higher girth
n. We prove that, in the finite case, for
n odd, and with an additional... |
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| SubjectTerms | Exact sciences and technology Finite generalized quadrangles Generalized polygon Geometry Mathematics Sciences and techniques of general use Semi-affine plane Semiplane |
| Title | Forgetful polygons as generalizations of semi-affine planes |
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