Classification of general absolute planes by quasi-ends
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| 024 | 7 | 0 | |a 10.1007/s00010-014-0283-5 |2 doi |
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| 245 | 0 | 0 | |a Classification of general absolute planes by quasi-ends |h [Elektronische Daten] |c [Helmut Karzel, Silvia Pianta, Mahfouz Rostamzadeh, Sayed-Ghahreman Taherian] |
| 520 | 3 | |a General (i.e. including non-continuous and non-Archimedean) absolute planes have been classified in different ways, e.g. by using Lambert-Saccheri quadrangles (cf. Greenberg, J Geom 12/1:45-64, 1979; Hartshorne, Geometry; Euclid and beyond, Springer, Berlin, 2000; Karzel and Marchi, Le Matematiche LXI:27-36, 2006; Rostamzadeh and Taherian, Results Math 63:171-182, 2013) or coordinate systems (cf. Pejas, Math Ann 143:212-235, 1961 and, for planes over Euclidean fields, Greenberg, J Geom 12/1:45-64, 1979). Here we consider the notion of quasi-end, a pencil determined by two lines which neither intersect nor have a common perpendicular (an ideal point of Greenberg, J Geom 12/1:45-64, 1979). The cardinality ω of the quasi-ends which are incident with a line is the same for all lines hence it is an invariant $${\omega_\mathcal{A}}$$ ω A of the plane $${\mathcal{A}}$$ A and can be used to classify absolute planes. We consider the case $${\omega_\mathcal{A}=0}$$ ω A = 0 and, for $${\omega_\mathcal{A} \geq 2}$$ ω A ≥ 2 (it cannot be 1) we prove that in the singular case $${\omega_\mathcal{A}}$$ ω A must be infinite. Finally we prove that for hyperbolic planes, ends and quasi-ends are the same, so $${\omega_\mathcal{A}=2}$$ ω A = 2 . | |
| 540 | |a Springer Basel, 2014 | ||
| 690 | 7 | |a Absolute Plane |2 nationallicence | |
| 690 | 7 | |a Quasi-Parallel Line |2 nationallicence | |
| 690 | 7 | |a Quasi-End |2 nationallicence | |
| 700 | 1 | |a Karzel |D Helmut |u Zentrum Mathematik, T.U. München, 80290, München, Germany |4 aut | |
| 700 | 1 | |a Pianta |D Silvia |u Dipartimento di Matematica e Fisica, Università Cattolica, Via Trieste, 17, 25121, Brescia, Italy |4 aut | |
| 700 | 1 | |a Rostamzadeh |D Mahfouz |u Department of Mathematical Sciences, Isfahan University of Technology, 84156, Isfahan, Iran |4 aut | |
| 700 | 1 | |a Taherian |D Sayed-Ghahreman |u Department of Mathematical Sciences, Isfahan University of Technology, 84156, Isfahan, Iran |4 aut | |
| 773 | 0 | |t Aequationes mathematicae |d Springer Basel |g 89/3(2015-06-01), 863-872 |x 0001-9054 |q 89:3<863 |1 2015 |2 89 |o 10 | |
| 856 | 4 | 0 | |u https://doi.org/10.1007/s00010-014-0283-5 |q text/html |z Onlinezugriff via DOI |
| 898 | |a BK010053 |b XK010053 |c XK010000 | ||
| 900 | 7 | |a Metadata rights reserved |b Springer special CC-BY-NC licence |2 nationallicence | |
| 908 | |D 1 |a research-article |2 jats | ||
| 949 | |B NATIONALLICENCE |F NATIONALLICENCE |b NL-springer | ||
| 950 | |B NATIONALLICENCE |P 856 |E 40 |u https://doi.org/10.1007/s00010-014-0283-5 |q text/html |z Onlinezugriff via DOI | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Karzel |D Helmut |u Zentrum Mathematik, T.U. München, 80290, München, Germany |4 aut | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Pianta |D Silvia |u Dipartimento di Matematica e Fisica, Università Cattolica, Via Trieste, 17, 25121, Brescia, Italy |4 aut | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Rostamzadeh |D Mahfouz |u Department of Mathematical Sciences, Isfahan University of Technology, 84156, Isfahan, Iran |4 aut | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Taherian |D Sayed-Ghahreman |u Department of Mathematical Sciences, Isfahan University of Technology, 84156, Isfahan, Iran |4 aut | ||
| 950 | |B NATIONALLICENCE |P 773 |E 0- |t Aequationes mathematicae |d Springer Basel |g 89/3(2015-06-01), 863-872 |x 0001-9054 |q 89:3<863 |1 2015 |2 89 |o 10 | ||