In praise of homomorphisms

Abstract

In this brief paper we survey some of the recent development related to the notion of homomorphism. We briefly survey the many faceted development of the homomorphism concept in the last 50 years with emphasizes on recent developments.

Introduction

Given two (undirected) graphs $G_1=(V_1,E_1),G_2=V_2,E_2$) a homomorphism $f$ of $G_1$ to $G_2$ is a mapping $f:V_1⟶V_2$ satisfying $\{f\left(u\right),f\left(v\right)\}\in E_2$ for every $\{u,v\}\in E_1$.

Thus homomorphisms are just mapping which preserve all edges (and in general nothing else). This notion present a natural habitat to study graphs particularly from the algebraic point of view. Homomorphisms generalize isomorphisms, subgraphs, embeddings, but also colorings and various other special partitions.

Being in nature algebraic, homomorphisms generalize to other structures such as relational structures or even more finite models which are specified by a language $L$. This can be done as follows: Let $L$ be a set containing relational symbols (like $R,S,\dots$) and function symbols (like $F,f,g,\dots$). Each relational symbol $R$ comes with arity $a\left(R\right)$ which is a natural number. Each function symbol $F$ comes with domain arity $d\left(F\right)$ (in this paper the range arity $r\left(F\right)$ can be assumed without loss of generality equal to $1$). An $L$-$\mathit{structure}$ $\mathbf{A}$ is then a set A together with relations $R_{\mathbf{A}}\subset A^{a\left(R\right)}$ for every relation symbol $R\in L$ and together with functions $F_{\mathbf{A}}:A^{d\left(F\right)}\to A$ for every function symbol $F\in L$. Such $L$-structures are also called $\mathit{models}$. We are interested in the case that all sets are finite. If $L$ contains no function symbols then we speak about $\mathit{relational}$ structures.

Given two $L$-structures $\mathbf{A}$ and $\mathbf{B}$ a homomorphism $\mathbf{A}$ to $\mathbf{B}$ is any mapping $f:A\to B$ satisfying:

$$(f\left(x_1\right),\dots ,f\left(x_{a\left(R\right)}\right))\in R_{\mathbf{B}}\mathrm{for every}R\in L\mathrm{and}(x_1\dots ,x_{a\left(R\right)})\in R_{\mathbf{A}},$$

and

$$F_{\mathbf{B}}f\left(x_1\right),\dots ,f\left(x_{d\left(F\right)}\right)=f\left(F_{\mathbf{A}}(x_1,\dots ,x_{d\left(F\right)})\right)\mathrm{for every}F\in Land(x_1,\dots ,x_{d\left(F\right)})\in A^{d\left(F\right)}.$$

Such notion is very flexible and it is domain of universal algebra, theory of categories and model theory (to name just a few). This was also the original context in which homomorphisms were studied (see e.g. [1], [2]). But

at several occasions the attention turned to combinatorial side thus displaying the emerging maturity of graph theory and (early) theoretical computer science.

One should mention here pioneering works of Sabidussi, Hedrlín and Pultr. These papers had not only a direct influence but perhaps more importantly (and more indirectly) encouraged more algebraical (or categorical) reasoning. This history is outlined in the first book devoted to this area Hell, Nešetřil [3] (and partly also in Godsil, Roy [4]). Here we complement this by providing a personal outline of some more recent results and problems which seem to be motivating today research.

The development could be described along the following key words:

Invariants, Constructions, Model Theory, Complexity, Data Science.

We have to be selective and thus we cover here just three of those areas and we have to be brief even then. Note that presently authors prepare the second edition of [3] and that Lovász [5] contains a very nice chapter on homomorphisms. The homomorphisms are also treated in [6] from different perspective of sparsity. Among other things which are not covered here are Ramsey theory [7], [8] and e.g. Rossman’s Homomorphism preservation theorem [9].