Numerical expressive power of logical languages with cardinality comparison

In this paper, we investigate the numerical expressive power of various logical languages, encompassing fragments of Presburger Arithmetic (PbA), monadic second-order logic with counting with respect to finite domains (MSOφ(#)) and shallow second-order graded modal logic with counting with respect to image-finite frames (SOGMLs,φ(#)). We show that in their respective existential fragments, the $1$-free fragment of PbA, the =-free fragment of MSOφ(#) and the graded modality-free fragment of SOGMLs,φ(#) possess equivalent numerical expressive power, specifically defining strongly semilinear sets. When adding universal quantifiers or adding $1$, = and graded modality to these three languages, the resulting definable sets become semilinear sets.