Abstract Uninorms on the unit interval are a common extension of triangular norms (t-norms) and triangular conorms (t-conorms). As important aggregation operators, uninorms play a very important role in fuzzy logic and expert systems. Recently, several researchers have studied constructions of uninorms on more general bounded lattices. In particular, Cayli (2019) gave two methods for constructing uninorms on a bounded lattice L with e ∈ L ∖ { 0 , 1 } , which is based on a t-norm T e on [ 0 , e ] and a t-conorms S e on [ e , 1 ] that satisfy strict boundary conditions. In this paper, we propose two new methods for constructing uninorms on bounded lattices. Our constructed uninorms are indeed the largest and the smallest among all uninorms on L that have the same restrictions T e and S e on [ 0 , e ] and, respectively, [ e , 1 ] . Moreover, our constructions does not require the boundary condition, and thus completely solved an open problem raised by Cayli.

中文翻译：

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在有界格上构造最大和最小单范式

摘要 单位区间上的单范数是三角范数（t-norms）和三角范数（t-conorms）的共同推广。单范式作为重要的聚合算子，在模糊逻辑和专家系统中扮演着非常重要的角色。最近，一些研究人员研究了更一般的有界格上的单范式的构造。特别是，Cayli (2019) 给出了两种在 e ∈ L ∖ { 0 , 1 } 的有界格 L 上构造单范式的方法，它基于 [ 0 , e ] 上的 t-范数 T e 和 t-conorms S e on [ e , 1 ] 满足严格的边界条件。在本文中，我们提出了两种在有界格上构造单范式的新方法。我们构造的单项式确实是 L 上所有单项式中最大和最小的，这些单项式分别在 [ 0 , e ] 和 [ e , 1 ] 上具有相同的限制 T e 和 Se 。而且，