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Binary relations and incidence structures. (English) Zbl 0542.08001

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
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