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.本文探讨定义于满足「与中性元（neutral element）不可比的点集至多包含一个元素」条件的有界格（bounded lattices）上的幂等一致模（idempotent uninorms）的结构。本文证明，相较于所有点均与中性元可比的有界格上的幂等一致模结构，仅含一个与中性元不可比点的有界格上的幂等一致模结构要复杂得多。在本系列论文的第一部分中，我们对所有点均与中性元可比的有界格上的幂等一致模进行了完整刻画。此外，我们还针对仅含一个与中性元不可比点的有界格上的幂等一致模，给出了若干基本性质与研究发现，并对这类格上的内部一致模（internal uninorms）类进行了完整刻画。在第二部分中，我们将对仅含一个与中性元不可比点的有界格上的两类特殊幂等一致模进行刻画，其中包括零元（annihilator）与中性元不可比的幂等一致模；并通过上述特殊情形的复合，完成这类格上幂等一致模的完整刻画。