Fuzzy nominal sets

In this paper, we use two approaches to define the concept of fuzzy nominal sets: classic and universal algebraic. We see that the fuzzy nominal sets obtained using the universal algebraic approach (so-called fuzzy nominal sets) are within finitely supported mathematics, whereas the fuzzy nominal sets derived using the classical approach (so-called fuzzy nominal \(\nu _{_{\textrm{supp}}}\) ${\nu }_{{}_{\text{supp}}}$ -sets) are within ordinary mathematics and each fuzzy nominal set can be considered as a fuzzy nominal \(\nu _{\textrm{supp}}\) ${\nu }_{\text{supp}}$ -set. We also go over the presheaf representation of fuzzy nominal sets and some other properties of these various types of fuzzy nominal sets.