l.c. ciungu

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.

In this paper, we define and study the modal operators on pseudo-BE algebras as special cases of closure operators on these structures.We prove that the composition of two modal operators is a modal operator if and only if they commute.For the particular case of a good pseudo-BCK algebra an equivalent definition of the modal operators is given, and the notion of a strong modal operator is introduced and studied.We also define the notions of modal deductive systems and modal homomorphisms on pseudo-BE algebras and we investigate their properties. It is proved that, if two modal operators have the same image, then they coincide.Also, given a normal modal deductive system H of a distributive modal pseudo-BE algebra $(A,f)$ we construct a modal operator on the quotient pseudo-BE algebra A/H.