Modal operators on pseudo-BE 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.