Lambert or Saccheri quadrilaterals as single primitive notions for plane hyperbolic geometry
type: Item
instance of: scholarly article
title: Lambert or Saccheri quadrilaterals as single primitive notions for plane hyperbolic geometry
zbMATH Open document ID: 1156.51015
author: Victor V. Pambuccian
published in: Journal of Mathematical Analysis and Applications
publication date: 2008-08-19T00:00:00Z
review text: In this short article (it has only two pages!) the author shows new systems of axioms of the plane hyperbolic geometry. We consider Lambert relation \(\lambda(abcd)\) (resp. Saccheri relation \(\sigma(abcd)\)) instead of colinearlity \(L(abc)\). Here \(\lambda(abcd)\) (resp. \(\sigma(abcd)\)) is true if and only if the four points \(a,b,c,d\) form the vertices of a Lambert quadrilateral (resp. Saccheri quadrilateral). To show these systems are axioms of the plane hyperbolic geometry, we need to show that the following problem, that is, `can we positively (i.e. with only quantifiers, \(\vee\) and \(\wedge\) as logical symbols) define \(L(abc)\) and \(\neg L(abc)\) in terms of \(\lambda\) or \(\sigma\)?' The author shows the problem in a simple way. We might supppose that it is not easy to show that \(\neg L(abc)\) can be defined positively in terms of \(\lambda\) (or \(\sigma\)), but the auther uses a proposition, which is in his previous paper [\textit{V. Pambuccian}, Acta Math. Hung. 100, 63--67 (2003; Zbl 1027.51021)], that says that \textsl{\(\neg L(abc)\) is positively definable in terms of \(L\) and \(\neq\)}.
reviewed by: Kazushi Ahara
Mathematics Subject Classification ID: 51M10
Mathematics Subject Classification ID: 51M09
zbMATH DE Number: 5313904
zbMATH Keywords: hyperolic geometry
zbMATH Keywords: axiomatization
zbMATH Keywords: Lambert quadrilateral
zbMATH Keywords: Saccheri quadrilateral
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.05.076
OpenAlex ID: W2005679796
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DOI: 10.1016/J.JMAA.2008.05.076
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description: scientific article; zbMATH DE number 5313904