Abstract The ordinal sum construction provides a very effective way to generate a new triangular norm on the real unit interval from existing ones. One of the most prominent theorems concerning the ordinal sum of triangular norms on the real unit interval states that a triangular norm is continuous if and only if it is uniquely representable as an ordinal sum of continuous Archimedean triangular norms. However, the ordinal sum of triangular norms on subintervals of a bounded lattice is not always a triangular norm (even if only one summand is involved), if one just extends the ordinal sum construction to a bounded lattice in a naive way. In the present paper, appropriately dealing with those elements that are incomparable with the endpoints of the given subintervals, we propose an alternative definition of ordinal sum of countably many (finite or countably infinite) triangular norms on subintervals of a complete lattice, where the endpoints of the subintervals constitute a chain. The completeness requirement for the lattice is not needed when considering finitely many triangular norms. The newly proposed ordinal sum is shown to be always a triangular norm. Several illustrative examples are given.

中文翻译：

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有界格上三角范数的序数和

摘要 序数和构造提供了一种非常有效的方法，可以从现有的真实单位区间上生成新的三角范数。关于实单位区间上三角范数的序数和的最突出定理之一指出，三角范数是连续的，当且仅当它唯一可表示为连续阿基米德三角范数的序数和。然而，有界格子的子区间上的三角范数的序数和并不总是三角范数（即使只涉及一个被加数），如果只是以一种朴素的方式将序数和构造扩展到有界格。在本文中，适当处理那些与给定子区间的端点不可比的元素，我们提出了一个完整格子的子区间上的可数多个（有限或可数无限）三角范数的序数和的替代定义，其中子区间的端点构成一个链。当考虑有限多个三角范数时，不需要格子的完整性要求。新提出的序数和总是一个三角范数。给出了几个说明性的例子。