Images and preimages of subobjects under the morphisms in a new category of fuzzy sets. I (Q1937934)

Some properties of the category \(\mathcal S\) of fuzzy sets are investigated, with objects all pairs \((X,\mu)\), where \(X\) is a set and \(\mu\) is a fuzzy set in \(X\), and with morphisms \(f:(X,\mu)\rightarrow (Y,\theta)\) such that \(f\) is not crisp, but more ``fuzzy'', i.e., \(f=(\{f_t\}_{t\in\operatorname{Im}\mu},\alpha)\), where \(\alpha:\operatorname{Im }\mu\rightarrow \operatorname{Im }\theta\) and \(f_t\) is a map from the \(t\)-cut of \(\mu\) into an \(\alpha(t)\)-cut of \(\theta\), satisfying some natural conditions. The authors define a fuzzy set \((X',\mu')\) to be a fuzzy subset of a fuzzy set \((X,\mu)\) in \(\mathcal S\) if \(X'\subseteq X\), \(\operatorname{Im }\mu'\subseteq\operatorname{Im }\mu\), and \(\mu'_t\subseteq\mu_t\) for all \(t\in\operatorname{Im }\mu'\), and they investigate images and preimages of such fuzzy subsets under morphisms from \(\mathcal S\). Finally, they investigate properties of images and preimages in the subcategory of \(\mathcal S\) that consists of all fuzzy sets with finite range sets.