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Erschienen in:
01.01.2012 | Original Paper
Fuzzy sets and cut systems in a category of sets with similarity relations
Erschienen in: Soft Computing | Ausgabe 1/2012
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Abstract
Let \(\Upomega\) be a complete residuated lattice. Let \({\mathbf{SetR}}(\Upomega)\) be the category of sets with similarity relations with values in \(\Upomega\) (called \(\Upomega\)-sets), which is an analogy of the category of classical sets with relations as morphisms. A fuzzy set in an \(\Upomega\)-set in the category \({\mathbf{SetR}}(\Upomega)\) is a morphism from \(\Upomega\)-set to a special \(\Upomega\)-set \((\Upomega,\leftrightarrow),\) where \(\leftrightarrow\) is the biresiduation operation in \(\Upomega.\) In the paper, we prove that fuzzy sets in \(\Upomega\)-sets in the category \({\mathbf{SetR}}(\Upomega)\) can be expressed equivalently as special cut systems \((C_{\alpha})_{\alpha\in\Upomega}.\)