The World of Structures with Finite Supports

Abstract

We introduce the theory of atomic finitely supported algebraic structures (that are finitely supported sets equipped with finitely supported internal operations or with finitely supported relations), and describe topics related to this theory such as permutation models of Zermelo-Fraenkel set theory with atoms, Fraenkel- Mostowski set theory, the theory of nominal sets, the theory of orbit-finite sets, and the theory of admissible sets. The motivation for developing such a theory comes from both experimental sciences (by modelling infinite algebraic structures hierarchically defined by involving some basic elements called atoms in a finitary manner, by analyzing their finite supports) and computer science (where finitely supported sets are used in various areas such as semantics, domain theory, automata theory and software verification). We describe the methods of translating the results from the non-atomic framework of Zermelo-Fraenkel sets into the atomic framework of sets with finite supports, focusing on the S-finite support principle and on the constructive method of defining supports. We also emphasize the limits of translating non-atomic results into an atomic set theory by presenting examples of valid Zermelo-Fraenkel results that cannot be formulated using atomic sets.