Triangular norm decompositions through methods using congruence relations. Issue 3 (3rd April 2022)

• Record Type: Journal Article
• Authors: Karaçal, Funda; Arpacı, Samet; Karacair, Kübra
• Is Part Of: International journal of general systems. Volume 51:Issue 3(2022)
• Page Start: 239
• Page End: 261
• Publication Date: 2022-04-03
• Subjects: Triangular norm -- bounded lattice -- congruence -- T-distributive element -- factor congruence; System theory -- Periodicals; 003
• DOI: 10.1080/03081079.2021.1999238
• Languages: English
• ISSNs: 0308-1079

Abstract

Abstract : In this paper, we introduce the concepts of T -distributive element, factor congruence, simple triangular norm and indecomposable t-norm on a bounded lattice through T -compatible congruence and study their related properties. We observe that the smallest t-norm on Boolean algebra is simple. We obtain a congruence by means of T -distributive element and investigate some properties of this type of congruence. A relation between the direct decomposability of any t-norm T and T -distributive elements whose supremum is T -∨-decomposable is obtained. Further, we show that any t-norm on a bounded lattice is directly indecomposable if and only if its underlying bounded lattice is directly indecomposable. We show that any t-norm T on a finite bounded lattice is isomorphic to a direct product of directly indecomposable t-norms. Finally, some illustrative examples are added.