Searching for an algebra on CSP solutions

The Constraint Satisfaction Problem (CSP) is a problem of computing a homomorphism R → Γ between two relational structures, where R is defined over a domain V and Γ is defined over a domain D. In fixed template CSPs, denoted CSP(Γ), the right side structure Γ is fixed and the left side structure R is unconstrained. In the last 2 decades it was impressively revealed that reasons that make fixed template CSPs polynomially solvable are of algebraic nature, namely, those templates Γ are tractable that are preserved under certain polymorphisms p1, ..., pk. Moreover, the set of all solutions (homomorphisms from R to Γ) are preserved under the same p1, ..., pk. I.e. p1, ..., pk not only guide us in the decision problem, but also define algebraic properties of the solutions set. This view makes of interest the following formulation: given a prespecified finite set of algebras B whose domain is D, is it possible to present the solutions set of a CSP instance as a subalgebra of A1 × ...× A|V | where Ai ∈ B? We study this formulation and prove that the latter problem itself is an instance of a certain fixed-template CSP, over another template ΓB. We prove that CSP(ΓB) can be reduced to a certain fragment of CSP(Γ), under natural assumptions on B. We also study the conditions under which CSP(Γ) can be reduced to CSP(ΓB). Since the complexity of CSP(ΓB) is defined by Pol(ΓB), we study the relationship between Pol(Γ) and Pol(ΓB). It turns out that if B is preserved by some p ∈ Pol(Γ), then p can be extended to a polymorphism of ΓB. In the end we demonstrate our theory on an example.