Andrei Alexandru
Romanian Academy, IIT, Iaşi, Romania

Group-valued fuzzy sets are mappings from a crisp set to a group. In particular, classical hybrid sets (defined as multisets with possibly negative multiplicities) are examples of such group-valued fuzzy sets. The theory of atomic finitely supported structures (which has historical roots in the permutative models of set theory with atoms) allows a discrete representation of infinite structures by analyzing the properties of their finite supports. This atomic framework actually generalizes the classical Zermelo-Fraenkel set theory by allowing infinitely many basic elements having no internal structures (called atoms) instead of a single basic element represented by the empty set. Here we describe the groups in the framework of finitely supported sets, and then we introduce and analyze the finitely supported group-valued fuzzy sets. In this way we generalize the properties of hybrid sets and other classical group-valued fuzzy sets known in the Zermelo-Fraenkel framework.