\(\alpha\)-cuts and models of fuzzy logic. (English) Zbl 1301.03052
In a category-theoretic context the characterization of fuzzy sets via cut systems is lifted to a corresponding characterization of fuzzy sets with graded identities, i.e.with fuzzy similarities, via suitable cut systems. Formally, this means that the author gives natural transformations between suitable categories.
Finally, these ideas are applied to interpretations for first-order mathematical fuzzy logics to give such interpretations the form of cut systems.
Finally, these ideas are applied to interpretations for first-order mathematical fuzzy logics to give such interpretations the form of cut systems.
Reviewer: Siegfried J. Gottwald (Leipzig)
MSC:
| 03E72 | Theory of fuzzy sets, etc. |
| 03B52 | Fuzzy logic; logic of vagueness |
| 18A23 | Natural morphisms, dinatural morphisms |
| 18B05 | Categories of sets, characterizations |
Keywords:
fuzzy sets; similarity relations; cut representation; categories of sets; graded identities; fuzzy similaritiesReferences:
| [1] | DOI: 10.1007/978-1-4615-0633-1 · doi:10.1007/978-1-4615-0633-1 |
| [2] | Bělohlávek R., Fuzzy Equational Logic (2005) · doi:10.1007/11376422_3 |
| [3] | DOI: 10.1007/978-94-011-2616-8_3 · doi:10.1007/978-94-011-2616-8_3 |
| [4] | MacLane S., Categories for the Working Mathematician (1971) · Zbl 0705.18001 |
| [5] | Močkoř J., Advances in Fuzzy Sets and Systems 1 (2) pp 83– (2006) |
| [6] | DOI: 10.1007/s00500-008-0336-6 · Zbl 1183.03026 · doi:10.1007/s00500-008-0336-6 |
| [7] | DOI: 10.1016/j.fss.2010.07.009 · Zbl 1225.03070 · doi:10.1016/j.fss.2010.07.009 |
| [8] | DOI: 10.1080/03081070903552809 · Zbl 1205.03037 · doi:10.1080/03081070903552809 |
| [9] | DOI: 10.1007/s00500-011-0737-9 · Zbl 1260.03096 · doi:10.1007/s00500-011-0737-9 |
| [10] | DOI: 10.1007/978-1-4615-5217-8 · doi:10.1007/978-1-4615-5217-8 |
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