Lambert or Saccheri quadrilaterals as single primitive notions for plane hyperbolic geometry (Q936611)

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 \(\neq\)}.