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.
"The reducts of equality up to primitive positive interdefinability." J. Symbolic Logic 75 (4) 1249 - 1292, December 2010. https://doi.org/10.2178/jsl/1286198146