It is known that a countable $\omega $ -categorical structure interprets all finite structures primitively positively if and only if its polymorphism clone maps to the clone of projections on a two-element set via a continuous clone homomorphism. We investigate the relationship between the existence of a clone homomorphism to the projection clone, and the existence of such a homomorphism which is continuous and thus meets the above criterion.

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 continum 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.

We study the computational complexity of deciding whether a given set of term equalities and inequalities has a solution in an ω-categorical algebra ????. There are ω-categorical groups where this pro...

We study the computational complexity of deciding whether a given set of term equalities and inequalities has a solution in an ω-categorical algebra ????. There are ω-categorical groups where this pro...