Ruled systems in combinatorial spaces

n a generalized quadragon (S,R), if x and y are two distinct points of S, denote by tr(x,y) the subset of S consisting of points collinear to both x and y. The family A of such subsets defines in S a line space (S,A), because two distinct points of S belong to only one element of A. The degenerated blocks of (S,R), that is, the subsets of S consisting of points collinear to a fixed point x, are subspaces of (S,A). In this work for the first time projective generalized quadragons are defined, namely, quadragons such that the elements of A and the degenerated blocks are closed sets of a combinatorial geometry G on S. It is proved that, like the classical examples, the dimension of G is always less than or equal to five. Moreover it is proved that if a projective generalized quadragon (S,R), with q+1=|r|, r∈R, is not embeddable in a projective space PG(n,q), necessarily each point is regular and the parameters of (S,R) are those of a Hermitian nonsingular form of PG(3,q) and then q is a square. It follows that, except for the previous case, a projective generalized quadragon is necessarily isomorphic with one of the classical examples.