Classification of states on certain orthomodular structures

We define various types of states on implicative involutive BE-algebras (Jauch-Piron state, (P)-state, (B)-state, subadditive  state, valuation), and we investigate the relationships between these states. Moreover, we introduce the unital, full, and  rich sets of states, and we prove certain properties involving these notions. In the case when an implicative involutive BE- algebra possesses a rich or a full set of states, we prove that it is an implicative-orthomodular lattice. If an implicative  involutive BE-algebra possesses a rich set of (P)-states or a full set of valuations, then it is an implicative-Boolean algebra.  Additionally, based on their deductive systems, we give characterizations of implicative-orthomodular lattices and  implicative-Boolean algebras.