caa a22 4500 605508577 CHVBK 20210128100637.0 cr unu---uuuuu 210128e20150601xx s 000 0 eng 10.1007/s00010-014-0283-5 doi (NATIONALLICENCE)springer-10.1007/s00010-014-0283-5 Classification of general absolute planes by quasi-ends [Elektronische Daten] [Helmut Karzel, Silvia Pianta, Mahfouz Rostamzadeh, Sayed-Ghahreman Taherian] 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 . Springer Basel, 2014 Absolute Plane nationallicence Quasi-Parallel Line nationallicence Quasi-End nationallicence Karzel Helmut Zentrum Mathematik, T.U. München, 80290, München, Germany aut Pianta Silvia Dipartimento di Matematica e Fisica, Università Cattolica, Via Trieste, 17, 25121, Brescia, Italy aut Rostamzadeh Mahfouz Department of Mathematical Sciences, Isfahan University of Technology, 84156, Isfahan, Iran aut Taherian Sayed-Ghahreman Department of Mathematical Sciences, Isfahan University of Technology, 84156, Isfahan, Iran aut Aequationes mathematicae Springer Basel 89/3(2015-06-01), 863-872 0001-9054 89:3<863 2015 89 10 https://doi.org/10.1007/s00010-014-0283-5 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00010-014-0283-5 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Karzel Helmut Zentrum Mathematik, T.U. München, 80290, München, Germany aut NATIONALLICENCE 700 1- Pianta Silvia Dipartimento di Matematica e Fisica, Università Cattolica, Via Trieste, 17, 25121, Brescia, Italy aut NATIONALLICENCE 700 1- Rostamzadeh Mahfouz Department of Mathematical Sciences, Isfahan University of Technology, 84156, Isfahan, Iran aut NATIONALLICENCE 700 1- Taherian Sayed-Ghahreman Department of Mathematical Sciences, Isfahan University of Technology, 84156, Isfahan, Iran aut NATIONALLICENCE 773 0- Aequationes mathematicae Springer Basel 89/3(2015-06-01), 863-872 0001-9054 89:3<863 2015 89 10