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Functors on relational structures which admit both left and right adjoints

Víctor Dalmau, Andrei Krokhin, Jakub Opršal

TL;DR

The paper studies when central Pultr functors $\Gamma$ on categories of relational structures admit right adjoints $\Omega$, extending Foniok–Tardif's digraph results to arbitrary signatures. It provides two explicit, constructive adjoint frameworks: (i) when $\mathbf P = \mathbf V_1$ and each $\mathbf Q_R$ is a $\sigma$-tree, yielding an explicit $\Omega$ via tuples of hom-sets and a proof of the adjunction, and (ii) when $\mathbf P = \check S_1$ (edge-based) with two cases depending on $S$ and $\check S$, again yielding an explicit $\Omega$ and a detailed edge-adjoint proof. The work also shows how composing such adjoints yields additional adjunctions and connects adjunction to finite duality, yielding new duals for trees and informing promise CSP reductions. These results broaden the toolbox for normal forms and reductions in (promise) CSPs and deepen the understanding of adjunction phenomena in relational structures beyond digraphs. The constructions emphasize an inductive, tree-driven intuition and relate right adjoints to duals of trees, with practical implications for efficient reductions in CSPs.

Abstract

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 $\mathbf A$ and $\mathbf B$, we have that $Λ(\mathbf A)$ maps homomorphically to $\mathbf B$ if and only if $\mathbf A$ maps homomorphically to $Γ(\mathbf B)$. If this is the case, $Λ$ is called the left adjoint to $Γ$ and $Γ$ the right adjoint to $Λ$. 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.

Functors on relational structures which admit both left and right adjoints

TL;DR

The paper studies when central Pultr functors on categories of relational structures admit right adjoints , extending Foniok–Tardif's digraph results to arbitrary signatures. It provides two explicit, constructive adjoint frameworks: (i) when and each is a -tree, yielding an explicit via tuples of hom-sets and a proof of the adjunction, and (ii) when (edge-based) with two cases depending on and , again yielding an explicit and a detailed edge-adjoint proof. The work also shows how composing such adjoints yields additional adjunctions and connects adjunction to finite duality, yielding new duals for trees and informing promise CSP reductions. These results broaden the toolbox for normal forms and reductions in (promise) CSPs and deepen the understanding of adjunction phenomena in relational structures beyond digraphs. The constructions emphasize an inductive, tree-driven intuition and relate right adjoints to duals of trees, with practical implications for efficient reductions in CSPs.

Abstract

This paper describes several cases of adjunction in the homomorphism preorder of relational structures. We say that two functors and between thin categories of relational structures are adjoint if for all structures and , we have that maps homomorphically to if and only if maps homomorphically to . If this is the case, is called the left adjoint to and the right adjoint to . 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.
Paper Structure (19 sections, 17 theorems, 40 equations, 2 figures)

This paper contains 19 sections, 17 theorems, 40 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Structures and homomorphisms
  4. Pultr functors: gadget replacement and pp-constructions
  5. An inductive construction of trees
  6. Prelude: Duals to trees
  7. Example: Dual to a directed path
  8. Adjoints to functors not changing the domain
  9. Example: An oriented path
  10. Example: A 4-ary relation defined by an oriented path
  11. Duals from adjoints
  12. Adjoints to functors with domains defined by a relation
  13. Case 1: $S = \check S$
  14. Case 2: $S \neq \check S$
  15. Example: The arc graph construction
  16. ...and 4 more sections

Key Result

Theorem 2.10

Let $\Lambda$ and $\Gamma$ be a pair of left and central Pultr functors defined by a $(\sigma, \tau)$-Pultr template. If $\Gamma$ has a right adjoint, then, for each $\tau$-tree $\mathbf T$, $\Lambda(\mathbf T)$ is homomorphically equivalent to a $\sigma$-tree.

Figures (2)

  • Figure 1: Examples of terms and trees they represent.
  • Figure 2: A digraph, its arc graph and its arc structure. The $O$ and $I$ relations of $\partial(\mathbf G)$ are symmetric, and $O$ and $I$ loops on all vertices of $\partial(\mathbf G)$ are omitted for readability.

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Example 2.9
  • Theorem 2.10: FT15
  • ...and 24 more