Triangular Norm-Based Elements on Bounded Lattices

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 preserve...

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Bibliographic Details
Published inAxioms Vol. 13; no. 1; p. 23
Main Authors Yılmaz, Şerife, İşçi, Rabia
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.01.2024
Subjects
Decision theory
direct product
Fuzzy sets
irreducible
Isomorphism
lattice
Lattice theory
Lattices
Mathematical research
Norms
prime
Theorems
triangular norm
Turkey
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Summary: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 Karaçal and Sağıroğlu is false by giving some counterexamples. We present a necessary and sufficient condition for the mentioned theorem to be correct.
ISSN:2075-1680
2075-1680
DOI:10.3390/axioms13010023