Triangular norm decompositions through methods using congruence relations

KARAÇAL F., Arpacı S., Karacair K.

INTERNATIONAL JOURNAL OF GENERAL SYSTEMS, vol.51, no.3, pp.239-261, 2022 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 51 Issue: 3
  • Publication Date: 2022
  • Doi Number: 10.1080/03081079.2021.1999238
  • Journal Name: INTERNATIONAL JOURNAL OF GENERAL SYSTEMS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Applied Science & Technology Source, Communication Abstracts, Compendex, Computer & Applied Sciences, INSPEC, Metadex, zbMATH, Civil Engineering Abstracts
  • Page Numbers: pp.239-261
  • Keywords: Triangular norm, bounded lattice, congruence, T-distributive element, factor congruence, ORDINAL SUMS, T-NORMS
  • Karadeniz Technical University Affiliated: Yes

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-proves-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.