TY - JOUR

T1 - Classification of general absolute planes by quasi-ends

AU - Karzel, Helmut

AU - Pianta, Silvia

AU - Rostamzadeh, Mahfouz

AU - Taherian, Sayed-Ghahreman

PY - 2015

Y1 - 2015

N2 - 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 ω_A of the plane A and can be used to classify absolute planes. We consider the case ω_A=0 and, for ω_A≥2 (it cannot be 1) we prove that in the singular case ω_A must be infinite. Finally we prove that for hyperbolic planes, ends and quasi-ends are the same, so ω_A=2 .

AB - 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 ω_A of the plane A and can be used to classify absolute planes. We consider the case ω_A=0 and, for ω_A≥2 (it cannot be 1) we prove that in the singular case ω_A must be infinite. Finally we prove that for hyperbolic planes, ends and quasi-ends are the same, so ω_A=2 .

KW - absolute plane

KW - quasi-end

KW - absolute plane

KW - quasi-end

UR - http://hdl.handle.net/10807/65248

U2 - 10.1007/s00010-014-0283-5

DO - 10.1007/s00010-014-0283-5

M3 - Article

SN - 1420-8903

SP - 863

EP - 872

JO - Aequationes Mathematicae

JF - Aequationes Mathematicae

ER -