Triangular Norm-Based Elements on Bounded Lattices
In this study, we introduce the notion of the <i>T</i>-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 <i>T</i>-prime elements. We prove that <i>T</i&g...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-12-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/13/1/23 |
| Summary: | In this study, we introduce the notion of the <i>T</i>-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 <i>T</i>-prime elements. We prove that <i>T</i>-irreducible elements and <i>T</i>-prime elements are preserved under the isomorphism that is generated by the same t-norm. We discuss the relationship between the sets of <i>T</i>-prime elements and co-atoms under some conditions. We illustrate this discussion with some examples. We also give some characterizations for the sets of <i>T</i>-irreducible elements and <i>T</i>-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. |
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| ISSN: | 2075-1680 |