Constructions of triangular norms on lattices by means of irreducible elements

Yilmaz S., Kazanci O.

INFORMATION SCIENCES, vol.397, pp.110-117, 2017 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 397
  • Publication Date: 2017
  • Doi Number: 10.1016/j.ins.2017.02.041
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.110-117
  • Keywords: boolean AND-semilattice, boolean OR-semilattice, Lattice, Triangular norm, Irreducible element, ORDINAL SUMS
  • Karadeniz Technical University Affiliated: Yes


Constructions and characterizations of triangular norms(t-norms) have been discussed in many different contexts. In this paper, we present two methods to construct t-norms on lattices from given partial information. Since the idea is to follow from part-to-whole, we especially consider the lattices with boolean OR-decompositions. Firstly, we investigate the structure of the partially ordered set of boolean OR-irreducible elements. Then, for a given t-norm T on the complete distributive lattice [0, 1](2), we study the restriction of T to the poset of boolean OR-irreducible elements of [0, 1[(2). Furthermore, we give a method for generating t-norms on a finite distributive lattice L by means of boolean OR-irreducible elements in L. We show that a t-norm on a finite distributive lattice is idempotent if and only if it is idempotent on the set of boolean OR-irreducible elements. Finally, we introduce a formula to obtain t-norms on L-[n] from given t-norms on boolean OR-irreducible elements of L-[n]. We present the dual statements for t-conorms. (C) 2017 Elsevier Inc. All rights reserved.