Idempotent nullnorms on bounded lattices

Abstract

Nullnorms are generalizations of triangular norms and triangular conorms with a zero element to be an arbitrary point from a bounded lattice. In this paper, we study and discuss the existence of idempotent nullnorms on bounded lattices. Considering an arbitrary distributive bounded lattice L, we show that there exists a unique idempotent nullnorm on L. We prove that an idempotent nullnorm may not always exist on an arbitrary bounded lattice. Furthermore, we propose a construction method to obtain idempotent nullnorms on a bounded lattice L with an additional constraint on a for the given zero element a ∈ L\\{0, 1}.