Generalized Multisets Over Infinite Alphabets with Atoms

Hybrid sets are defined as multisets having also negative multiplicities, i.e. as functions from a crisp set to the group <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{Z}$</tex> of all integers. In this short presentation, we consider a generalization of the hybrid sets by defining the group-valued multisets over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$X$</tex> as functions from <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$X$</tex> to an arbitrary group <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$G$</tex>. By involving the theory of finitely supported sets, we present finiteness properties of infinite group-valued multisets.