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Nullnorms with a zero element being at any point of a bounded lattice are an important generalization of triangular norms and triangular conorms. This paper obtains an equivalent characterization for the existence of idempotent nullnorms with the zero element a on any bounded lattice containing only two distinct elements incomparable with a. Furthermore, some basic properties for the bounded lattice containing only two distinct elements incomparable with a are presented.

This paper continues to study the construction of uni-nullnorms on bounded lattices. At first, we introduce a new method for constructing the smallest uni-nullnorm on an arbitrary bounded lattice L having the elements e , a ∈ L , based on the existence of a uninorm on [ 0 , a ] 2 with the neutral element e and a triangular norm on [ a , 1 ] 2 . And then, we propose another new approach to obtain the largest uni-nullnorm on L via a uninorm on [ 0 , a ) 2 with the neutral element e and a triangular norm on [ a , 1 ] 2 . Furthermore, we provide some corresponding examples to illustrate that our construction methods differ from the existing ones.