Shepherdson's theorems for fragments of open induction

TL;DR

This paper explores algebraic characterizations of various fragments of the first-order arithmetic IOpen, extending Shepherdson's classical results to broader logical fragments and their models.

Contribution

It establishes new algebraic equivalences for different fragments of IOpen, generalizing Shepherdson's theorem beyond the original scope.

Findings

Models of certain IOpen fragments correspond to specific algebraic structures.

New algebraic characterizations for fragments of IOpen are proved.

Extensions of Shepherdson's theorem to broader logical fragments are demonstrated.

Abstract

By a well-known result of Shepherdson, models of the theory IOpen (a first order arithmetic containing the scheme of induction for all quantifier free formulas) are exactly all the discretely ordered semirings that are integer parts of their real closures. In this paper we prove several analogous results that provide algebraic equivalents to various fragments of IOpen.