Existence of states on quantum structures

Orthomodular lattices occurred as generalized event structures in the models of probability for quantum mechanics. Here we contribute to the question of existence of states (=probability measures) on orthomodular lattices. We prove that known techniques do not allow to find examples with less than 19 blocks (=maximal Boolean subalgebras). This bound is achieved by the example by Mayet [R. Mayet, Personal communication, 1993]. Although we do not finally exclude the existence of other techniques breaking this bound, existence of smaller examples is highly unexpected.