Extending the Ginsburg-Spanier Theorem to Functions and Mixed Arithmetic

2026-07-06

AI Summary

This study investigates algebraic characterizations of definable functions and sets in the additive structures FO(ℤ,+,≤), FO(ℝ,+,≤), and the mixed structure FO(ℝ,ℤ,+,≤). Employing purely algebraic methods, it extends the Ginsburg–Spanier theorem from definable sets to definable functions for the first time, introducing novel notions such as semi-multilinear sets and piecewise-simple functions. A key technical tool is a stability lemma that enables a unified treatment across all three theories. The main results establish that definable functions in both the integer and real additive theories are precisely the piecewise-linear functions, while in the mixed theory, definable sets coincide with semi-multilinear sets and definable functions are exactly the piecewise-simple functions. Notably, this framework avoids reliance on automata or computational models, offering a fully algebraic characterization.

Abstract

We study sets and functions definable in the three additive theories $\FO(\Z,+,\leq)$, $\FO(\R,+,\leq)$, and $\FO(\R,\Z,+,\leq)$. The Ginsburg--Spanier theorem~\cite{GS66} characterizes $\FO(\Z,+,\leq)$-definable sets as exactly the semi-linear sets. We extend this characterization in two directions. First, we show that $\FO(\Z,+,\leq)$-definable \emph{functions} are exactly the piecewise linear functions (Theorem~\ref{thm:integer}), and that $\FO(\R,+,\leq)$-definable functions are also exactly the piecewise linear functions (Theorem~\ref{thm:real}). The proofs are direct algebraic arguments using only a stability lemma and the Ginsburg--Spanier theorem. Second, we introduce \emph{semi-polinear sets} as the $\FO(\R,\Z,+,\leq)$ analogue of semi-linear sets, and prove that the class of mixed-linear sets and the class of semi-polinear sets coincide (Theorem~\ref{thm:mixed-sets}). We further show that $\FO(\R,\Z,+,\leq)$-definable functions are exactly the \emph{piecewise-simple} functions (Theorem~\ref{thm:mixed-functions}), a new class of functions that are linear in the integer part and in the fractional part of the argument, but with potentially different linear coefficients for each. These algebraic characterizations unify the three theories in a single framework, and the proofs are purely algebraic, without reference to automata or machines.

Problem

• definable functions
• semi-linear sets
• piecewise linear functions
• mixed arithmetic
• additive theories

Innovation

• piecewise-linear functions
• semi-polinear sets
• mixed arithmetic
• definable functions
• Ginsburg-Spanier theorem