Abstract The concept of ordinal sums in the sense of Clifford have long been blamed for their limitations in constructing new t-norms including inability to cope with general bounded lattices. Motivated by this observation, and based on the lattice-based sum of lattices that has been recently introduced by El-Zekey et al., we propose a new sum-type construction of t-norms, called a lattice-based sum of t-norms, for building new t-norms on bounded lattices from given ones. The proposed sum is generalizing the well-known ordinal and horizontal sum constructions of t-norms by allowing for lattice ordered index sets. We demonstrate that, like the ordinal sum of t-norms, the lattice-based sum of t-norms can be generalized using as summands so-called t-subnorms, still leading to a t-norm. Subsequently, we apply the results for constructing several new families of t-norms and t-subnorms on bounded lattices. In the same spirit, by the duality, we will also introduce lattice-based sums of t-conorms and t-subconorms.

中文翻译：

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有界晶格上基于晶格的 t 范数总和

摘要 长期以来，Clifford 意义上的序数和概念因其在构建新 t 范数方面的局限性而受到指责，包括无法处理一般有界格。受此观察启发，并基于 El-Zekey 等人最近引入的基于格的格总和，我们提出了一种新的 t-范数的和类型构造，称为基于格的 t-范数，用于从给定的有界格子上构建新的 t 范数。提议的总和通过允许格有序索引集来推广众所周知的 t 范数的有序和水平总和结构。我们证明，就像 t 范数的序数和一样，基于格的 t 范数和可以使用所谓的 t 子范数作为被加数进行推广，仍然导致 t 范数。随后，我们应用这些结果在有界格上构建几个新的 t-范数和 t-子范数族。本着同样的精神，通过对偶性，我们还将介绍基于格的 t-conorms 和 t-subconorms 和。