Spaces of type n on partially ordered sets
On any partially ordered set M one can define an incidence relation by xIy if and only if \(x\leq y\). If M possesses a subset E, the elements of which will be considered as points, such that M and E have a common lower bound, that any n (or less) elements of E have a least upper bound in M, and that each element of M occurs as such a bound, then the pair (M,E) is called a space of type n. The author presents some lattice theoretic characterizations of certain spaces of type n. In particular, he shows that a generalization of the geometric Veblen-Young axiom corresponds to a so-called n-generation property, which is a generalized version of the lattice theoretic ``Verbindungssatz''. He studies modularity and distributivity in the underlying lattices and, finally, discusses some specializations and examples.
incidence structures on partially ordered sets
scientific article; zbMATH DE number 4103852