ON UNDECIDABILITY OF FINITE SUBSETS THEORY FOR TORSION ABELIAN GROUPS

Mathematics. 2022. Vol. 10. No. 3. Article 533.

Sergey Mikhailovich Dudakov

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. The compactness is not applicable here because the theories of subsets can differ for elementary equivalent source algebras. In this paper we prove the given theorem for Abelian torsion groups those have elements of unbounded order.

Keywords: torsion, algebra of subsets, arithmetic, abelian groups