The reducts of equality 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.

The complexity of counting quantifiers on equality languages

“…See Theorem 8 Part 1 in [21]. It is also a consequence of Lemma 1 since the existental positive closure of {x = y ∨ u = v} includes all definitions in positive (quantifier-free) logic.…”

“…Against this we set the nice aesthetic of our results and the way in which they complement [11] and [21]. For example, the specially negative languages, play an important role in our classifications, but where do they sit in the context of [21]? Do they even form a class of relations closed under pp-definitions (a so-called coclone)?…”

“…We may abuse notation and write R pp when properly we mean {R} pp . A great project was launched in [21] to identify sets of the form Γ pp , charting their inclusion relations in a lattice. The lattice is mostly identified through a dually-isomorphic algebraic lattice of local clones, but here we will only be interested in the co-clones.…”

“…The lattice is mostly identified through a dually-isomorphic algebraic lattice of local clones, but here we will only be interested in the co-clones. In line with [21] and [11], we will consider the co-clone ∅ pp = = pp to be at the top of the lattice, with the co-clone of all equality-definable relations at the bottom.…”

“…We resolve this by considering all references to Γ as a language inside X-CSP(Γ) to be restricting of Γ to the relationally finite. 1 We now recall some basic results from [11,21]. We typically write = to indicate the binary relation of disequality (in terms of our notation this abbreviates x = y).…”