Finitary Prelinear and Linear Orthosets

An orthoset is a set equipped with a symmetric and irreflexive binary relation. A linear orthoset is an orthoset such that for any two distinct elements e , f there is a third element g such that exactly one of f and g is orthogonal to e and the pairs e , f and e , g have the same orthogonal complement. Linear orthosets naturally arise from anisotropic Hermitian spaces. We moreover define an orthoset to be prelinear by assuming the above-mentioned property for non-orthogonal pairs e , f only. In this paper, we establish some structural properties of prelinear and linear orthosets under the assumption of finiteness or finite rank.