Bonetti, F.; Civolani, N. Binary relations and incidence structures. (English) Zbl 0542.08001 Rend. Sem. Mat. Brescia 7, 137-150 (1984). We consider such structures as (\({\mathcal G},I,{\mathfrak P})\), where I is a tolerance (i.e. a reflexive and symmetric binary relation) on the set \({\mathcal G}\) (whose elements are called lines), and \({\mathfrak P}\) is a family of I-criss-crosses (i.e. sets of mutually I-related lines), and we call them incidence structures with tolerance (briefly: IST). We study certain properties of these structures, and the behaviour of a family of IST’s with respect to intersections. For instance, let two IST’s be defined on the same set of lines; if in the former any two criss-crosses have exactly one common line, we give a sufficient condition in order that the same property be ”inherited” by the latter. MSC: 08A02 Relational systems, laws of composition 08A30 Subalgebras, congruence relations 51A99 Linear incidence geometry Keywords:incidence structures with tolerance; lines; criss-crosses PDFBibTeX XMLCite \textit{F. Bonetti} and \textit{N. Civolani}, Rend. Semin. Mat. Brescia 7, 137--150 (1984; Zbl 0542.08001)