Algebraic Structures in Finitely Supported Mathematics

Abstract

Finitely Supported Mathematics (FSM) is the mathematics developed in the framework of invariant/finitely supported structures. The aim of this chapter is to translate into FSM several algebraic concepts which were initially described using the Zermelo-Fraenkel axioms of set theory. We focus on multisets, generalized multisets, partially ordered sets and groups because these concepts are particularly relevant for experimental science. Moreover, we provide the main principles of translating a given algebraic concept into FSM. These principles are based on the remark that only finitely supported objects are allowed in FSM. We present in detail some FSM properties of the related algebraic structures, emphasizing the analogy between the results obtained in the framework of invariant sets and those obtained in the usual Zermelo-Fraenkel framework. This chapter may be read without using notions from higher-order logic, category theory, or the general equivariance principles for formulas in classical higher-order logic.