Characterization of spaces of filtering states (Q616808)

The set of states of an orthomodular lattice (OML) can be empty. If an OML \(L\) admits at least one state, the state space, \(\Omega(L),\) is a convex compact set whose set of \(\sigma\)-additive states forms a face. In the present paper, the author answers the question, which faces correspond to the set of filtering states. The main result is Theorem 2 which says: Let \(C\) be a compact convex sets and \(F\) a semi-exposed face of \(C.\) Then there are an OML \(L\) and an affine homeomorphism \(\phi: C \to \Omega(L)\) such that \(\phi(F) = \Omega_f(L),\) where \(\Omega_f(L)\) is the set of filtering states on \(L.\)