  In the present paper we study the following problem: how to construct a
coherent orthoalgebra which has only a finite number of elements, but at the
same time does not admit a bivaluation (i.e. a morphism with a codomain being
an orthoalgebra with just two elements). This problem is important in the
perspective of Bell-Kochen-Specker theory, since one can associate such an
orthoalgebra to every saturated non-colorable finite configuration of
projective lines. The first result obtained in this paper provides a general
method for constructing finite orthoalgebras. This method is then applied to
obtain a new infinite family of finite coherent orthoalgebras that do not admit
bivaluations. The corresponding proof is combinatorial and yields a description
of the groups of symmetries for these orthoalgebras.
