On Undecidability of Subset Theory for Some Monoids

Abstract

Early we (with B. N. Karlov) have proved the following claim for the infinite cyclic monoid ℋ. Let exp ℋ be an algebra of finite subsets of ℋ with the same operation, exp ℋ must be a monoid again. So the theory of exp ℋ is equivalent to elementary arithmetic. Thus, the theory of the monoid exp ℋ is undecidable. Here we consider an arbitrary commutative cancellative monoid ℋ with an element of infinite order, and generalize the previous claims to the corresponding monoid exp ℋ.