Logics of Finite Hankel Rank

TL;DR

This paper explores the connection between the Feferman-Vaught Theorem and Hankel matrices in finite structures, identifying the maximal logics satisfying the theorem for sum-like and product-like operations.

Contribution

It establishes the relationship between Hankel matrices and the Feferman-Vaught Theorem, identifying CFOL and CMSOL as maximal logics for product-like and sum-like operations respectively.

Findings

CFOL satisfies Feferman-Vaught for product-like operations

CMSOL satisfies Feferman-Vaught for sum-like operations

Discussion on the maximality of logics for finite structures

Abstract

We discuss the Feferman-Vaught Theorem in the setting of abstract model theory for finite structures. We look at sum-like and product-like binary operations on finite structures and their Hankel matrices. We show the connection between Hankel matrices and the Feferman-Vaught Theorem. The largest logic known to satisfy a Feferman-Vaught Theorem for product-like operations is CFOL, first order logic with modular counting quantifiers. For sum-like operations it is CMSOL, the corresponding monadic second order logic. We discuss whether there are maximal logics satisfying Feferman-Vaught Theorems for finite structures.