On Undecidability of Finite Subsets Theory for Torsion Abelian Groups

Sergey Mikhailovich Dudakov

doi:10.3390/math10030533

Abstract

Let M be a commutative cancellative monoid with an element of infinite order. The binary operation can be extended to all finite subsets of M by the pointwise definition. So, we can consider the theory of finite subsets of M. Earlier, we have proved the following result: in the theory of finite subsets of M elementary arithmetic can be interpreted. In particular, this theory is undecidable. For example, the free monoid (the sets of all words with concatenation) has this property, the corresponding algebra of finite subsets is the theory of all finite languages with concatenation. Another example is an arbitrary Abelian group that is not a torsion group. But the method of proof significantly used an element of infinite order, hence, it can’t be immediately generalized to torsion groups. In this paper we prove the given theorem for Abelian torsion groups that have elements of unbounded order: for such group, the theory of finite subsets allows interpreting the elementary arithmetic.

Highlights

Among the central subjects of mathematical logic, there are formal theories investigations
The method of proof significantly used an element of infinite order, it can’t be immediately generalized to torsion groups
In this paper we prove the given theorem for Abelian torsion groups that have elements of unbounded order: for such group, the theory of finite subsets allows interpreting the elementary arithmetic

Summary

Introduction

Among the central subjects of mathematical logic, there are formal theories investigations. Arrays, maps, or sets (of numbers, words, Boolean values, etc.) can be declared as new data types and can be used in databases These constructions are finite due to “natural”. Let A be a commutative cancellative monoid with an element of infinite order, the theory of finite subsets of A allows to interpret elementary arithmetic, this theory is undecidable. Examples of such monoids are well-known: numbers with addition or multiplication, polynomials and spaces over a field of characteristic zero, and so on. The result holds for any direct sums of unbounded cyclic groups and infinite subgroups of the unity roots group

Basic Definitions and Notation

Finite Subsets of G

Interpretation of Elementary Arithmetic

Multiple Relations

The Equivalence Relation

Conclusions