有界格上零范数的完全表示定理

中文摘要 AI 辅助

我们建立了有界格上二元运算成为真零范数的充分必要条件。该表示结合了下区间上的三角余范、上区间上的三角范、两个保序映射，以及一个在$I_a^3\times I_a^3$上交换且递增的函数，其中$I_a^3$由与吸收元$a$不可比较且既不达到$a$也不被$a$达到的元素组成。与早期限制值域的结构不同，此函数的值可取格中的任意位置。为保持此类无限制值的结合性，我们引入了一个混合交互函数，该函数对至少一个分量在$I_a^3$中的每一对进行评估。我们还推导了该区域为空或仅含单个元素时的特化情形，包括单元素情形下可取值的确切描述。一个五元素格实例表明混合结合性条件独立于其余假设，而一个十四元素非传递格实例则证明了允许无限制值域的必要性。最后，主要的值域受限子类和有界格情形作为一般表示的特例被恢复。

英文摘要

We establish necessary and sufficient conditions under which a binary operation on a bounded trellis is a proper nullnorm. The representation combines a t-conorm on the lower interval, a t-norm on the upper interval, two order-preserving maps, and a commutative, increasing function on $I_a^3\times I_a^3$, where $I_a^3$ consists of the elements incomparable with the absorbing element $a$ that neither reach $a$ nor are reachable from $a$. Unlike earlier range-restricted constructions, this function may take values anywhere in the trellis. To preserve associativity for such unrestricted values, we introduce a mixed interaction function that evaluates every pair with at least one component in $I_a^3$. We also derive the specializations in which this region is empty or consists of a single element, including an exact description of the admissible value in the singleton case. A five-element lattice example shows that the mixed associativity condition is independent of the remaining hypotheses, while a fourteen-element nontransitive trellis example demonstrates the necessity of allowing the unrestricted range. Finally, the principal range-restricted subclasses and the bounded-lattice case are recovered as specializations of the general representation.

发表机构

• School of Mathematical Science, Guangxi Minzu University（广西民族大学数学科学学院）

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