# Functors on relational structures which admit both left and right adjoints

## TL;DR

This paper generalizes the concept of adjoint functors in the homomorphism preorder of relational structures, extending previous work on digraphs, and introduces new constructions relevant to finite duality and promise constraint satisfaction.

## Contribution

It extends the theory of adjoint functors to arbitrary relational structures, providing new right adjoints and dual constructions, and connects these to applications in promise constraint satisfaction.

## Key findings

- Generalized adjoint functors to relational structures
- Constructed new right adjoints and duals for trees
- Linked functor constructions to promise constraint satisfaction

## Abstract

This paper describes several cases of adjunction in the homomorphism preorder of relational structures. We say that two functors $\Lambda$ and $\Gamma$ between thin categories of relational structures are adjoint if for all structures $\mathbf A$ and $\mathbf B$, we have that $\Lambda(\mathbf A)$ maps homomorphically to $\mathbf B$ if and only if $\mathbf A$ maps homomorphically to $\Gamma(\mathbf B)$. If this is the case, $\Lambda$ is called the left adjoint to $\Gamma$ and $\Gamma$ the right adjoint to $\Lambda$. In 2015, Foniok and Tardif 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 in 1970. We generalise 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.