The wonderland of reflections

A fundamental fact for the algebraic theory of constraint satisfaction problems (CSPs) over a fixed template is that pp-interpretations between at most countable \omega-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 pp-interpretations. 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 $\omega$-categorical structure depends only on the identities of height 1 satisfied in its polymorphism clone as well as the 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.Comment: 24 page

Digraphs and homomorphisms: Cores, colorings, and constructions

A natural digraph analogue of the graph-theoretic concept of an `independent set\u27 is that of an acyclic set, namely a set of vertices not spanning a directed cycle. Hence a digraph analogue of a graph coloring is a decomposition of the vertex set into acyclic sets

Origami constraints on the initial-conditions arrangement of dark-matter caustics and streams

In a cold-dark-matter universe, cosmological structure formation proceeds in rough analogy to origami folding. Dark matter occupies a three-dimensional 'sheet' of free- fall observers, non-intersecting in six-dimensional velocity-position phase space. At early times, the sheet was flat like an origami sheet, i.e. velocities were essentially zero, but as time passes, the sheet folds up to form cosmic structure. The present paper further illustrates this analogy, and clarifies a Lagrangian definition of caustics and streams: caustics are two-dimensional surfaces in this initial sheet along which it folds, tessellating Lagrangian space into a set of three-dimensional regions, i.e. streams. The main scientific result of the paper is that streams may be colored by only two colors, with no two neighbouring streams (i.e. streams on either side of a caustic surface) colored the same. The two colors correspond to positive and negative parities of local Lagrangian volumes. This is a severe restriction on the connectivity and therefore arrangement of streams in Lagrangian space, since arbitrarily many colors can be necessary to color a general arrangement of three-dimensional regions. This stream two-colorability has consequences from graph theory, which we explain. Then, using N-body simulations, we test how these caustics correspond in Lagrangian space to the boundaries of haloes, filaments and walls. We also test how well outer caustics correspond to a Zel'dovich-approximation prediction.Comment: Clarifications and slight changes to match version accepted to MNRAS. 9 pages, 5 figure

On First-Order Definable Colorings

We address the problem of characterizing $H$-coloring problems that are first-order definable on a fixed class of relational structures. In this context, we give several characterizations of a homomorphism dualities arising in a class of structure