Functors on Relational Structures Which Admit Both Left and Right Adjoints

This paper describes several cases of adjunction in the homomorphism preorder of relational structures. We say that two functors Λ and Γ between thin categories of relational structures are adjoint if for all structures 𝐀 and 𝐁, we have that Λ⁡(𝐀) maps homomorphically to 𝐁 if and only if 𝐀 maps homomorphically to Γ⁡(𝐁). If this is the case, Λ is called the left adjoint to Γ and Γ the right adjoint to Λ. Foniok and Tardif [Discrete Math., 338 (2015), pp. 527–535] described some functors on the 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 [Reports of the Midwest Category Seminar IV, Lecture Notes in Math. 137, Springer, 1970, pp. 100–113]. We generalize results of Foniok and Tardif to arbitrary relational structures, and coincidently, 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 inspired by an application in promise constraint satisfaction—it has been shown that such functors can be used as efficient reductions between these problems.