Triangular Norm-Based Elements on Bounded Lattices
AXIOMS, vol.23, no.1, 2024 (SCI-Expanded)
- Publication Type: Article / Article
- Volume: 23 Issue: 1
- Publication Date: 2024
- Doi Number: 10.3390/axioms13010023
- Journal Name: AXIOMS
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED)
- Karadeniz Technical University Affiliated: Yes
Abstract
In this study, we introduce the notion of the T-irreducible element as a generalization of the notion of the meet-irreducible element in complete lattices. We derive some related properties of these elements and T-prime elements. We prove that T-irreducible elements and T-prime elements are preserved under the isomorphism that is generated by the same t-norm. We discuss the relationship between the sets of T-prime elements and co-atoms under some conditions. We illustrate this discussion with some examples. We also give some characterizations for the sets of T-irreducible elements and T-prime elements on the direct product of lattices. Then, we show that Theorem 2 given by Karacal and Sagiroglu is false by giving some counterexamples. We present a necessary and sufficient condition for the mentioned theorem to be correct.