Algebraic Properties of Generalized Multisets

We present generalized multisets in the Zermelo-Fraenkel framework, in Reverse Mathematics, and in the Fraenkel-Mostowski framework. In the Zermelo-Fraenkel framework, we prove that the set of all generalized multisets over a certain finite set is a finitely-generated, lattice-ordered, free abelian group. Similar properties are then discussed in Reverse Mathematics. Finally, we study the generalized multisets in the Fraenkel-Mostowski framework, and present their nominal properties. Several Zermelo-Fraenkel algebraic properties of generalized multisets are translated into the Fraenkel-Mostowski framework by using the finite support axiom of the Fraenkel-Mostowski set theory.