Idempotent uninorms on bounded lattices with at most single point incomparable with the neutral element: Part I

We discuss the structure of idempotent uninorms defined on bounded lattices such that the set of all points incomparable with the neutral element contains at most one point. We show that the structure of idempotent uninorms on lattices with one point incomparable with the neutral element is much more complex than the structure of those defined on lattices where all points are comparable with the neutral element. In Part I of this two-part paper, we completely characterize idempotent uninorms defined on bounded lattices such that all points are comparable with the neutral element. Moreover, we show some basic properties and observations on idempotent uninorms on bounded lattices with one point incomparable with the neutral element and we completely characterize the class of internal uninorms on such lattices. In Part II, we will characterize two special classes of idempotent uninorms defined on bounded lattices with one point incomparable with the neutral element, including idempotent uninorms with annihilator incomparable with the neutral element, and by the composition of these special cases, we will show the complete characterization of idempotent uninorms on such lattices.