Triangular norms on bounded trellises

In this paper, we introduce the notion of a t-norm on bounded pseudo-ordered sets and in particular on bounded trellises (also known as weakly associative lattices), and provide some basic examples. The impact of abandoning transitivity is considerable: on a proper bounded trellis, the meet operation is not a t-norm, and there might actually exist no or even multiple maximal t-norms. We provide a first generic construction method that allows to extend a t-norm on an interior range of a given perpendicular to-semi-trellis to the entire perpendicular to-semi-trellis. Also, we discuss at length an instantiation of this method based on a particular interior range, namely a finite sub-trellis of the set of right-transitive elements of a given trellis. We pay specific attention to bounded pseudo-chains and modular trellises. (c) 2023 Elsevier B.V. All rights reserved.

Automated summary

In this paper, the concept of a t-norm on bounded pseudo-ordered sets and bounded trellises is introduced, along with some basic examples. The impact of abandoning transitivity is discussed, highlighting that the meet operation is not a t-norm on a proper bounded trellis, and there may be no or multiple maximal t-norms. A generic construction method is provided to extend a t-norm on an interior range of a given perpendicular to-semi-trellis to the entire trellis, with a specific instantiation based on a finite sub-trellis of right-transitive elements. The focus is also on bounded pseudo-chains and modular trellises. (c) 2023 Elsevier B.V. All rights reserved.