Modal operators on pseudo-BE algebras

Document Type : Research Paper


1 Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, Iowa 52242-1419, USA

2 Department of Pure Mathematics, Faculty of Math. & Comput., Shahid Bahonar University of Kerman, Kerman, Iran

3 Department of Mathematics, Payame Noor University, P.O.Box. 19395-3697, Tehran, Iran


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.