Some Remarks on Idempotent Nullnorms on Bounded Lattices

Abstract

Nullnorms are generalizations of triangular norms (t-norms) and triangular conorms (t-conorms) with a zero element to be an arbitrary point from an arbitrary bounded lattice. In this paper, we study on the existence of idempotent nullnorms on bounded lattices. We show that there exists unique idempotent nullnorm on an arbitrary distributive bounded lattice. We prove that an idempotent nullnorm may not always exist on every bounded lattice. Furthermore, we propose the construction method to obtain idempotent nullnorms on a bounded lattice under additional assumptions on given zero element. As by-product of this method, we see that it is in existence an idempotent nullnorm on non-distributive bounded lattices.