The wonderland of reflections

• Published in: Israel journal of mathematics Vol. 223; no. 1; pp. 363 - 398
• Main Authors: Barto, Libor, Opršal, Jakub, Pinsker, Michael
• Format: Journal Article
• Language: English
• Published: Jerusalem The Hebrew University Magnes Press 01.02.2018; Springer Nature B.V
• Subjects: Algebra; Analysis; Applications of Mathematics; Construction; Group Theory and Generalizations; Homomorphisms; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Polymorphism; Semantics; Theoretical; Theory of constraints
• ISSN: 0021-2172; 1565-8511
• DOI: 10.1007/s11856-017-1621-9

A fundamental fact for the algebraic theory of constraint satisfaction problems (CSPs) over a fixed template is that pp-interpretations between at most countable ω -categorical relational structures have two algebraic counterparts for their polymorphism clones: a semantic one via the standard algebraic operators H , S , P , and a syntactic one via clone homomorphisms (capturing identities). We provide a similar characterization which incorporates all relational constructions relevant for CSPs, that is, homomorphic equivalence and adding singletons to cores in addition to ppinterpretations. For the semantic part we introduce a new construction, called reflection, and for the syntactic part we find an appropriate weakening of clone homomorphisms, called h1 clone homomorphisms (capturing identities of height 1). As a consequence, the complexity of the CSP of an at most countable ω-categorical structure depends only on the identities of height 1 satisfied in its polymorphism clone as well as the natural uniformity thereon. This allows us in turn to formulate a new elegant dichotomy conjecture for the CSPs of reducts of finitely bounded homogeneous structures. Finally, we reveal a close connection between h1 clone homomorphisms and the notion of compatibility with projections used in the study of the lattice of interpretability types of varieties.