Word problems and ceers

TL;DR

This paper explores which computably enumerable equivalence relations (ceers) can be realized as word problems of computably enumerable structures, revealing that all ceers can be represented in some form, but not all match specific subclasses like finitely presented semigroups.

Contribution

It demonstrates that every ceer is isomorphic to a word problem of some c.e. semigroup and clarifies limitations regarding finitely presented and non-periodic semigroups.

Findings

Every ceer is isomorphic to a c.e. semigroup's word problem.

Not all ceers are in the same reducibility degree as finitely presented semigroup word problems.

The provable equivalence of Peano Arithmetic shares strong isomorphism with a c.e. ring's word problem.

Abstract

This note addresses the issue as to which ceers can be realized by word problems of computably enumerable (or, simply, c.e.) structures (such as c.e. semigroups, groups, and rings), where being realized means to fall in the same reducibility degree (under the notion of reducibility for equivalence relations usually called "computable reducibility"), or in the same isomorphism type (with the isomorphism induced by a computable function), or in the same strong isomorphism type (with the isomorphism induced by a computable permutation of the natural numbers). We observe for instance that every ceer is isomorphic to the word problem of some c.e. semigroup, but (answering a question of Gao and Gerdes) not every ceer is in the same reducibility degree of the word problem of some finitely presented semigroup, nor is it in the same reducibility degree of some non-periodic semigroup. We also show that the ceer provided by provable equivalence of Peano Arithmetic is in the same strong isomorphism type as the word problem of some non-commutative and non-Boolean c.e. ring.