It is known that bisymmetry generalizes the simultaneous commutativity and associativity in the framework of the unit interval. In this work, we will completely characterize two classes of bisymmetric aggregation operators: one with a neutral element and the other with the vertical and horizontal sections of the idempotent elements being smooth on a finite chain, but not necessarily smooth and commutative. Thus, the previous results, based on the smoothness that is known as a very restrictive condition, are improved. For example, there is only one smooth Archimedean t-norm on a finite chain. In this paper, the discrete bisymmetric aggregation operators are explored without the limit of the smoothness. As a by-product, it is deduced that for smooth aggregation operators on a finite chain, the bisymmetry is equivalent to the commutativity and associativity, which improves the conclusion obtained by Mas et al. that associativity and bisymmetry are equivalent for commutative smooth aggregation operators on a finite chain.

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关于离散双对称

众所周知，双对称性概括了单位区间框架内的同时交换性和结合性。在这项工作中，我们将完全刻画两类双对称聚合算子：一类具有中性元素，另一类具有幂等元素的垂直和水平部分在有限链上是平滑的，但不一定是平滑和可交换的。因此，先前基于平滑度（已知为非常严格的条件）的结果得到了改进。例如，在有限链上只有一个平滑的阿基米德 t 范数。在本文中，在没有平滑度限制的情况下探索了离散双对称聚合算子。作为副产品，推导出对于有限链上的平滑聚合算子，双对称性等价于交换性和结合性，改进了 Mas 等人的结论。对于有限链上的可交换平滑聚合算子，结合性和双对称性是等效的。