Functors on relational structures which admit both left and right adjoints
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This paper describes several cases of adjunction in the homomorphism order of relational structures. For these purposes, we say that two functors Γ and Δ between categories of relational structures are adjoint if for all structures A and B, we have that Γ(A) maps homomorphically to B if and only if A maps homomorphically to Δ(B). If this is the case Γ is called the left adjoint to Δ and Δ the right adjoint to Γ. In 2015, Foniok and Tardif described some functors category of digraphs that allow both left and right adjoints. The main contribution of Foniok and Tardif is a construction of right adjoints to some of the functors identified as right adjoints by Pultr in 1970. We generalise results of Foniok and Tardif to arbitrary relational relational structures, and coincidentely, we also provide more right adjoints on digraphs, and since these constructions are connected to finite duality, we also provide a new construction of duals to trees. Our results are motivated by the application in promise constraint satisfaction – it has been shown that such functors can be used as efficient reductions between these problems.
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