The reducts of equality up to primitive positive interdefinability

TL;DR

This paper studies the classification of reducts of the logic of equality up to primitive positive interdefinability, revealing a continuum of such reducts and providing tools for their analysis.

Contribution

It introduces methods for analyzing reducts up to primitive positive interdefinability and classifies all such reducts of the logic of equality.

Findings

Existence of a continuum of reducts of the logic of equality.

Classification of locally closed clones containing all permutations.

Development of tools for studying reducts up to primitive positive interdefinability.

Abstract

We initiate the study of reducts of relational structures up to primitive positive interdefinability: After providing the tools for such a study, we apply these tools in order to obtain a classification of the reducts of the logic of equality. It turns out that there exists a continuum of such reducts. Equivalently, expressed in the language of universal algebra, we classify those locally closed clones over a countable domain which contain all permutations of the domain.