On representing contexts in line arrangements

Acontext is defined to be a triple (G, M, J) of setsG, M and an incidence relationJ G×M.A finite set ofn oriented lines in general position in the euclidean plane induces a cell decomposition of the plane. For a givenk-element subset of cells of dimension 2, we define an incidence relationJ × as follows:t _i andl _j are incident if and only ift _i lies on the positive side with respect tol _j .We call a context (G, M, J)represented in a line arrangement if and only if there are relation preserving bijections betweenG and ,M and , respectively. We study conditions for a context to be representable in a line arrangement.Especially, we provide a non-trivial infinite class of contexts which can not be represented in a line arrangement.