aleph_0-categorical Structures: Endomorphisms and Interpretations

TL;DR

This paper extends the analysis of interpretability in countably categorical structures, showing how various types of interpretations are governed by monoids of self-embeddings and endomorphisms, generalizing previous automorphism-based results.

Contribution

It generalizes the Ahlbrandt--Ziegler analysis by linking existential and positive existential interpretations to monoids of self-embeddings and endomorphisms.

Findings

Existential interpretation is controlled by the monoid of self-embeddings.

Positive existential interpretation is governed by the monoid of endomorphisms.

General interpretability is controlled by the automorphism group.

Abstract

We extend the Ahlbrandt--Ziegler analysis of interpretability in aleph_0-categorical structures by showing that existential interpretation is controlled by the monoid of self--embeddings and positive existential interpretation of structures without constant endomorphisms is controlled by the monoid of endomorphisms in the same way as general interpretability is controlled by the automorphism group.