On countable isotypic structures

TL;DR

This paper explores isotypic structures, proving fields of finite transcendence degree are type-definable and constructing countable isotypic but non-isomorphic structures across various algebraic systems, answering a longstanding question.

Contribution

It demonstrates that certain algebraic structures can be isotypic without being isomorphic, and establishes that fields of finite transcendence degree are definable by types.

Findings

Fields of finite transcendence degree are type-definable.

Existence of countable isotypic but non-isomorphic structures in various algebraic systems.

Answering B. Plotkin's question for groups.

Abstract

We obtain several results concerning the concept of isotypic structures. Namely we prove that any field of finite transcendence degree over a prime subfield is defined by types; then we construct isotypic but not isomorphic structures with countable underlying sets: totally ordered sets, fields, and groups. This answers an old question by B. Plotkin for groups.