The Theory of Finitely Supported Structures and Choice Forms

The theory of finitely supported algebraic structures provides a first step in computing infinite algebraic structures that are finitely supported modulo certain atomic permutation actions. The motivation for developing such a theory comes from both mathematics (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 foundation, automata theory, domain theory, proof theory and software verification). The results presented in this paper include the meta-theoretical presentation of finitely supported structures, the study of the consistency of choice principles (and of results requiring choice principles) within this framework, and the presentation of several connections with other topics.