Two Fra\"iss\'e-style theorems for homomorphism-homogeneous relational structures

TL;DR

This paper establishes two Fraïssé-style theorems for various notions of homomorphism-homogeneity in relational structures, advancing understanding of their existence, uniqueness, and classification, especially for countable graphs.

Contribution

It proves two key theorems for twelve notions of homomorphism-homogeneity and clarifies their implications for the classification of countable homogeneous graphs.

Findings

Complete classification of homomorphism-homogeneous undirected graphs

Two Fraïssé-style theorems for twelve notions of homomorphism-homogeneity

Insights into directed graph cases

Abstract

In this paper, we state and prove two Fra\"{i}ss\'{e}-style results that cover existence and uniqueness properties for twelve of the eighteen different notions of homomorphism-homogeneity as introduced by Lockett and Truss, and provide forward directions and implications for the remaining six cases. Following these results, we completely determine the extent to which the countable homogeneous undirected graphs (as classified by Lachlan and Woodrow) are homomorphism-homogeneous; we also provide some insight into the directed graph case.

Tables

Table 1. Table 1: Table of XY-homogeneity: ℳ ℳ \mathcal{M} is XY-homogeneous if a finite partial map of type X (column) extends to a map of type Y (row) in the associated monoid. The collection of all notions of homomorphism-homogeneity is denoted by ℌ ℌ \mathfrak{H} .

	isomorphism (I)	monomorphism (M)	homomorphism (H)
End$\left(ℳ\right)$ (H)	IH	MH	HH
Epi$\left(ℳ\right)$ (E)	IE	ME	HE
Mon$\left(ℳ\right)$ (M)	IM	MM	HM
Bi$\left(ℳ\right)$ (B)	IB	MB	HB
Emb$\left(ℳ\right)$ (I)	II	MI	HI
Aut$\left(ℳ\right)$ (A)	IA	MA	HA

Table 2. Table 2: Maximal homomorphism-homogeneity classes for countable homogeneous graphs.

mhh-classes	countable IA graph	reference
$\{$HA$\}$	$K^{{\aleph }_0}$	(5.1)
$\{$MA, HE$\}$	${\overline{K}}^{{\aleph }_0}$	(5.1)
$\{$IA, MB, HE$\}$	$R$, ${⨆}_{i\in \mathbb{N}}{K}_i^{{\aleph }_0}$, $\overline{{⨆}_{i\in \mathbb{N}}{K}_i^{{\aleph }_0}}$	(5.1, 5.1 (3), 5.1 (3))
$\{$IA, HE$\}$	${⨆}_{i\in \mathbb{N}}{K}_i^n$	(5.1 (2))
$\{$IA, MM, HH$\}$	${\overline{K}}^n$-free ($n\ge 3$), ${⨆}_{i=1}^n{K}_i^{{\aleph }_0}$, $\overline{{⨆}_{i\in \mathbb{N}}{K}_i^n}$	(5.1, 5.1 (1), 5.1 (2))
$\{$IA$\}$	$K^n$-free ($n\ge 3$), $\overline{{⨆}_{i=1}^n{K}_i^{{\aleph }_0}}$	(5, 5.1 (1))

Equations

$$af=a\theta {\theta }^{-1}f=a\theta \alpha$$

$$af=afe{e}^{-1}=a{h}^{\prime }\alpha .$$

$${\rho }^c=\{(y,x):(x,y)\in \rho \}\subseteq Y\times X$$

$$\underline{y}\bar{f}=\{(x_1,\dots ,x_n)\text{mbox}:x_i\in y_i\bar{f}\text{mbox}forall1\le i\le n\}.$$

$$W\bar{f}=\{x\in X\text{mbox}:(w,x)\in \bar{f}\text{mbox}forsomew\in W\}=\bigcup _{w\in W}w\bar{f}.$$

$$s(x)=\{\begin{array}{ll}-1& \text{if }x<0\\ 0& \text{if }x=0\\ 1& \text{if }x>0\end{array}$$

$$\mathcal{S}=\{\text{mbox}(E,H),(B,M),(A,I)\}.$$

$$a\bar{f}=a\theta {\theta }^{-1}\bar{f}=a\theta \bar{h}=a\theta \bar{\beta }$$

$$a\bar{f}=a\bar{f}e{e}^{-1}=a\bar{f}e\bar{\alpha }=a{\bar{h}}^{\prime }\bar{\alpha }$$

$$T_j^{\to }(b)=\{c\in T_j:c\to b\}and{T}_j^{\leftarrow }(b)=\{d\in T_j:b\to d\}.$$

Full text

Keywords: homomorphism-homogeneous, relational structures, Fraïssé theory, infinite graph theory.

2010 Mathematics Subject Classification: 03C15 (primary), 05C63

1 Introduction

A relational first-order structure $\mathcal{M}$ is homogeneous (or ultrahomogeneous, depending on source) if every isomorphism between finite substructures of $\mathcal{M}$ extends to an automorphism of $\mathcal{M}$. Examples of homogeneous structures include the countable dense linear order without endpoints $(\mathbb{Q},<)$, the random graph $R$ and the universal tournament $\mathbb{T}$ [26]. The celebrated theorem of Fraïssé [16] states that a structure $\mathcal{M}$ is homogeneous if and only if the class $\mathscr{C}$ of finite substructures of $\mathcal{M}$ (often called the age of $\mathcal{M}$) satisfies four distinct conditions; furthermore, $\mathcal{M}$ is unique up to isomorphism if this is the case. A large body of literature, in a range of subjects across mathematics, is devoted to the study of homogeneous structures. Particular areas of interest include classification results [36] [23] [10], combinatorial aspects [6] and model-theoretic properties, both as objects in their own right and via a link to $\aleph_{0}$-categorical structures [17]. Through the link to $\aleph_{0}$-categorical structures and the famous Ryll-Nardzewski theorem (also Engeler, Svenonius [17]) there is a well-established connection between automorphisms of countable homogeneous structures and interesting infinite permutation groups [3]. Many properties of automorphism groups of homogeneous structures have been studied, such as topological dynamics [20] and applications to constraint satisfaction problems on infinite domains [5]. Furthermore, the groups themselves have lent themselves to research into properties such as: simplicity [27]; generation [15]; reconstruction of the structure from the automorphism group [2]; the existence of generic elements [37]; and the small index property [18] [21]. An excellent overview of this vast subject is [26].

The idea of homogeneity has been extended to other types of finite partial maps between relational structures. This is the concept of homomorphism-homogeneity; originally developed by Cameron and Nešetřil in 2006 [9] and Mašulović in 2007 [29]. This definition was taken to its logical conclusion in the papers of Lockett and Truss [24] [25], in which they detailed eighteen different notions of homomorphism-homogeneity based on both the finite partial map and the type of endomorphism (see Section 2). An example of this is MB-homogeneity; a structure $\mathcal{M}$ is MB-homogeneous if every monomorphism between finite substructures of $\mathcal{M}$ extends to a bijective endomorphism of $\mathcal{M}$. The development of this subject has been rapid, culminating in detailed accounts of homomorphism-homogeneous graphs [35] [12], finite tournaments with loops [19], and posets [25]. Aside from these combinatorial studies, finding analogues of results about automorphism groups for endomorphism monoids is a motivating factor in this subject; examples of these include the development of oligomorphic transformation monoids [30] [12] and the idea of generic endomorphisms [24].

Throughout this article, let $\sigma$ be a countable relational signature. Suppose that $\mathscr{C}$ is a class of finite $\sigma$-structures. The proof of one direction of Fraïssé’s theorem uses conditions on $\mathscr{C}$ to inductively construct a homogeneous structure $\mathcal{M}$ whose age is $\mathscr{C}$. One of these conditions is the joint embedding property (JEP); this property (along with two others) ensures that we can construct a countable $\sigma$-structure $\mathcal{M}$ with age $\mathscr{C}$. The second is the amalgamation property (AP); this ensures that $\mathcal{M}$ is homogeneous. This final claim is verified by showing that $\mathcal{M}$ has the extension property, a necessary and sufficient condition for a countable $\sigma$-structure $\mathcal{M}$ to be homogeneous. Fraïssé’s theorem also states that any two homogeneous $\sigma$-structures with the same age are isomorphic; this is shown using a back-and-forth argument between two similarly constructed structures that builds the desired isomorphism. The forth part of the argument ensures that the extended map is totally defined; the back part ensures that the map is eventually surjective.

Cameron and Nešetřil [9] proved an analogue of Fraïssé’s theorem for MM-homogeneity, where every monomorphism between finite substructures of some structure $\mathcal{M}$ extends to a monomorphism of $\mathcal{M}$ (see Section 2). This proof necessitated modification of the amalgamation property to ensure MM-homogeneity; resulting in the mono-amalgamation property (MAP). In a slight departure to the technique used to prove Fraïssé’s theorem, the proof of the analogous theorem for MM-homogeneity in [9] utilised a forth alone argument; this is because the extended map need not be surjective. On the uniqueness side, the same article also showed that two MM-structures with the same age may be non-isomorphic; instead detailing that two MM-homogeneous structures were unique up to a weaker notion called mono-equivalence. The proof of this again used a forth alone argument. Finally, work has been done on the case of HH-homogeneity, where every finite partial homomorphisms of $\mathcal{M}$ extends to an endomorphism of $\mathcal{M}$. The notion of a homo-amalgamation property (HAP) first appeared in the preprint of Pech and Pech [33], in which they used this to prove a version of Fraïssé’s theorem for HH-homogeneity by using a forth alone argument. In addition, they also showed that two HH-homogeneous structures with the same age are homomorphism-equivalent. These results were later published in their following paper [34]. Further insights were made by Dolinka [14], who used the HAP (and the equivalent one-point homomorphism extension property (1PHEP)) to determine which structures were both homogeneous and HH-homogeneous.

In the case of MB-homogeneity, a forth alone approach does not suffice. As the extended map must be surjective, we are required to use a back-and-forth argument. The fact that monomorphisms are not invertible in general necessitates the use of a second amalgamation property alongside the MAP of [9]; this was defined by Coleman, Evans and Gray [12] using antimonomorphisms in the bi-amalgamation property (BAP). In a similar situation to [9], two MB-homogeneous structures with the same age may not be isomorphic but instead are unique up to bi-equivalence; the proof of this also requires a back-and-forth argument.

In light of these previous generalisations of Fraïssé’s theorem and the multitude of types of homomorphism-homogeneity (see Section 2), the natural aim would be to find an “umbrella” version of Fraïssé’s theorem; one that encapsulates all possible notions of homomorphism-homogeneity. This result would supply Fraïssé’s theorem, and the versions of [9] and [12], as corollaries. Such a theorem could help to determine the extent to which a structure is homomorphism-homogeneous based on structural properties. In turn, this will provide a rich source of oligomorphic transformation monoids [12, Theorem 1.7].

However, a compromise must be reached between idealism and practicality for two reasons. First, as discussed above, differing approaches are required if the extended map is surjective; see the contrast between analogues of Fraïssé’s theorem for MM-homogeneity [9] and MB-homogeneity [12] for a case in point. In the forth alone case, we can utilise a single modified amalgamation property in order to construct the structure and extend the map. The issue is that monomorphisms and homomorphisms are not “invertible” in general. This is particularly problematic in the homomorphism case; what could you use to accurately describe the ‘back’ condition for homomorphisms, given that the underlying function may not be invertible? As evidenced in [12], we need two modified amalgamation properties in the back-and-forth case; one for the forth part to ensure the extended map is totally defined, and one for the back part to ensure the resulting map is surjective. Second, some kinds of homomorphism-homogeneity are easier to deal with than others. There is a distinct dichotomy in the set of notions of homomorphism-homogeneity, split between those whose extended maps are not necessarily the same “type” as the partial map (such as MH-homogeneity, in that a homomorphism is not necessarily an monomorphism), and those whose extended maps are definitely of the same type than the partial map (such as MM, or MI-homogeneity). The former case causes issues in inductively constructing a structure due to the lack of certainty about the extended map; this is discussed in further detail in Section 2.

The first of these reasons therefore necessitate two similar but markedly different theorems (Theorem 1.1 and Theorem 1.2) based on whether or not the proof uses forth alone or a back-and-forth argument; these form the central theorems of the paper. The second only allows the two theorems to cover twelve of the eighteen different notions of homomorphism-homogeneity. In the statement of the theorems below, what constitutes the “relevant” amalgamation property and notion of equivalence will be explained in Sections 2 and 4.

Theorem 1.1**.**

Let XY $\in\{$II, MI, MM, HI, HM, HH$\}$.

1. (1)

If $\mathcal{M}$ is an XY-homogeneous $\sigma$-structure, then Age$(\mathcal{M})$* has the relevant amalgamation property.* 2. (2)

If $\mathscr{C}$ is a class of finite $\sigma$-structures with countably many isomorphism types, is closed under isomorphisms and substructures, has the JEP and the relevant amalgamation property, then there exists a XY-homogeneous $\sigma$-structure $\mathcal{M}$ with age $\mathscr{C}$. 3. (3)

Any two XY-homogeneous $\sigma$-structures with the same age are equivalent up to a relevant notion of equivalence.

Theorem 1.2**.**

Let XZ $\in\{$IA, MA, MB, HA, HB, HE$\}$.

1. (1)

If $\mathcal{M}$ is an XZ-homogeneous $\sigma$-structure, then Age$(\mathcal{M})$* has the two relevant amalgamation properties.* 2. (2)

If $\mathscr{C}$ is a class of finite $\sigma$-structures with countably many isomorphism types, is closed under isomorphisms and substructures, has the JEP and the two relevant amalgamation properties, then there exists a XZ-homogeneous $\sigma$-structure $\mathcal{M}$ with age $\mathscr{C}$. 3. (3)

Any two XZ-homogeneous $\sigma$-structures with the same age are equivalent up to a relevant notion of equivalence.

While not the ideal “umbrella” theorem, these two results are still useful in determining the extent to which a structure is homomorphism-homogeneous; thus providing interesting examples of oligomorphic transformation monoids. To that end, this article is dedicated to the proof of these two theorems; as well as determining a complete picture of homomorphism-homogeneity for some well-known structures.

Section 2 begins by defining the eighteen different notions of homomorphism-homogeneity, and contains the proof of Theorem 1.1 split into three propositions (2, 2, 2) that correspond to the three points of the theorem. Section 3 utilises the idea of the converse of a function to introduce the concept of an antihomomorphism between two $\sigma$-structures; essentially, this is the preimage of a homomorphism that preserves non-relations. This machinery underpins the ‘back’ condition that is used to extend a map between finite substructures of a $\sigma$-structure $\mathcal{M}$ to a surjective endomorphism of $\mathcal{M}$. Following this, Section 4 is dedicated to the proof of Theorem 1.2. In Section 5 we introduce the idea of a maximal homomorphism-homogeneity class (mhh-class), and determine mhh-classes for every countable homogeneous undirected graph in the classification of Lachlan and Woodrow [23], as well as turning our attention to the directed graph case.

Throughout the article, we will write $\underline{x}$ to mean an $n$-tuple of some set; this non-standard notation is motivated by the use of barred notation to mean converses of functions, which appear more regularly. The notation -1 is reserved exclusively for the inverse of a function. For some countable indexing set $I$, we define $\sigma=\{R_{i}\;:\;i\in I\}$ to be a relational signature. Usually, $\mathcal{M}$ will denote a countable $\sigma$-structure on domain $M$. The age Age$(\mathcal{M})$ of $\mathcal{M}$ is the class of all finite structures that can be embedded in $\mathcal{M}$. For more on the introductory concepts of model theory, [17] is a good place to start.

2 Proof of Theorem 1.1

We recall the eighteen different notions of homomorphism-homogeneity as developed in the two papers of Lockett and Truss [24] [25]. Following their lead, we denote each type of endomorphism by a symbol: H for endomorphism, E for epimorphism, M for monomorphism, B for bimorphism, I for embedding and A for automorphism. We cannot assert that a finite partial map is surjective; there is no well defined notion of a finite partial epimorphism, for instance. Therefore, there are only three types of finite partial map of a structure: H for homomorphism, M for monomorphism, and I for embedding. Without loss of generality, maps between finite substructures can be taken to be surjective.

Definition \thedefinition.

Let $\mathcal{M}$ be a first-order structure, and take X $\in\{\text{H},\text{M},\text{I}\}$ and Y $\in\{\text{H},\text{E},\text{M},\text{B},\text{I},\text{A}\}$. Say that $\mathcal{M}$ is XY-homogeneous if every finite partial map of type X of $\mathcal{M}$ extends to a map of type Y of $\mathcal{M}$. We denote the collection of all notions of homomorphism-homogeneity by $\mathfrak{H}$. Furthermore, we denote the class of all XY-homogeneous structures by XY, and say that $\mathbb{H}$ is the set of all classes of XY-homogeneous structures.

For example, a structure $\mathcal{M}$ is HE-homogeneous if every finite partial homomorphism (H) of $\mathcal{M}$ extends to a epimorphism (E) of $\mathcal{M}$. Regular homogeneity (as in [26], for instance) corresponds to IA-homogeneity using this notation. All possible types of homomorphism-homogeneity given in Section 2 are outlined in Table 1.

It is important to make the distinction between a notion of homomorphism-homogeneity and the associated class of homomorphism-homogeneous structures. For example, II-homogeneity and IA-homogeneity represent two different notions of homomorphism-homogeneity under consideration. As outlined in Table 1, II-homogeneity is where every finite partial isomorphism extends to an embedding; IA-homogeneity is where every finite partial isomorphism extends to an automorphism. However, the classes II and IA of homomorphism-homogeneous structures coincide. For countable structures, it was shown by Lockett and Truss [25] that a structure is II (MI, HI)-homogeneous if and only if it is IA (MA, HA)-homogeneous; that is, II = IA, MI = MA and HI = HA. This difference between notions of homomorphism-homogeneity and classes of homomorphism-homogeneous structures is apparent in Section 4, where we re-prove this result of [25] from a Fraïssé-theoretic perspective.

It follows that some notions of homomorphism-homogeneity are stronger than others. For instance, as every bimorphism is a monomorphism, it follows that every MB-homogeneous structure is also MM-homogeneous. Similarly, as an isomorphism is both a monomorphism and a homomorphism, it follows that if a structure $\mathcal{M}$ is XY-homogeneous then it is also IY-homogeneous. This natural concept inversely corresponds to a natural containment order on the set $\mathbb{H}$ of homomorphism-homogeneity classes; see Figure 1 for a diagram of this order. Notice that the stronger the notion of homomorphism-homogeneity, the class of countable structures that satisfy that notion is smaller. This difference is explained in more detail in Section 5.

As discussed in the introduction, it is necessary to partition $\mathfrak{H}$ into two pieces based on whether or not the extended map is surjective. This represents the division between cases where a forth alone argument will suffice and the other when we require a back-and-forth construction. Furthermore, there are some elements of $\mathfrak{H}$ that are weaker notions of homogeneity than others. These are of the form XY where a map of type Y does not necessarily imply that it is a map of type X; for instance, a homomorphism is not necessarily a monomorphism. These phenomena motivate the division of $\mathfrak{H}$ into the following subsets:

• •

forth alone $\mathfrak{F}=\left\{\text{XY }\in\mathfrak{H}\;:\;\text{X, Y}\in\{\text{H, M, I}\}\right\}$;

• •

back-and-forth $\mathfrak{B}=\{$XZ $\in\mathfrak{H}$ : X $\in\{$H, M, I$\}$, Z $\in\{$E, B, A$\}\}$;

• •

no implication $\mathfrak{N}=\{$IH, IE, IM, IB, MH, ME$\}$;

• •

implication $\mathfrak{I}=\mathfrak{H}\smallsetminus\mathfrak{N}$.

This partitions $\mathfrak{H}$ into four parts based on the intersections of $\mathfrak{B},\mathfrak{F}$ with $\mathfrak{N},\mathfrak{I}$ (see Figure 2, where the boxes represent intersections).

We move on to establish the machinery required for the proof of Theorem 1.1. This result deals with types of homomorphism-homogeneity in $\mathfrak{F}$ (see Figure 2); those that only require a forth construction to prove. Consequently, we have that X,Y $\in\{$H, M, I$\}$ throughout this section. When we say a map of type X, we are referring to this instance; so if $\alpha$ is a map of type H, it is a homomorphism. Notice that I $\subseteq$ M $\subseteq$ H.

A critical step in Fraïssé’s proof is the establishment that the inductively constructed structure $\mathcal{M}$ is homogeneous; nominally by showing that $\mathcal{M}$ satisfies the extension property, a necessary and sufficient condition for homogeneity. Since different kinds of homomorphism-homogeneity rely on extending different kinds of maps, this property needs to be generalised and then shown to be an equivalent condition to the relevant notion of homomorphism-homogeneity. To that end, we define the XY-extension property (XYEP), where X,Y $\in\{$H, M, I$\}$:

(XYEP) A structure $\mathcal{M}$ with age $\mathscr{C}$ has the XYEP if for all $A\subseteq B\in\mathscr{C}$ and maps $f:A\xrightarrow{}\mathcal{M}$ of type X, there exists a map $g:B\xrightarrow{}\mathcal{M}$ of type Y extending $f$.

For example, the standard extension property in Fraïssé’s theorem is the IIAP, and the mono-extension property of [9] is the MMAP.

Ideally, we would like to say a structure $\mathcal{M}$ is XY-homogeneous if and only if $\mathcal{M}$ has the XYEP, generalising the observation of Fraïssé. However, complications occur in the proof of the converse direction of this statement; this is due to the inductive construction of the extended map. For example, suppose that $\mathcal{M}$ has the IMEP and that $f:A\xrightarrow{}B$ is an isomorphism. Extending this using the IMEP gives a monomorphism $g:A^{\prime}\to B^{\prime}$ where $A\subseteq A^{\prime}$ and $B\subseteq B^{\prime}$. However, $g$ is a monomorphism between finite substructures; and so in general, we cannot extend $g$ to another monomorphism $h$ between finite substructures. The only way we can continue extending is if the map of type Y is also of type X. This behaviour is the motivating factor in splitting $\mathfrak{H}$ into $\mathfrak{I}$ and $\mathfrak{N}$ (see • ‣ 2). In light of this, we show that the XYEP is a necessary condition for XY-homogeneity in general, and that it is also sufficient when the extended map of type Y is also a map of type X.

Proposition \theproposition.

Let $\mathcal{M}$ be a countable $\sigma$-structure with age $\mathscr{C}$.

1. (1)

Suppose that XY $\in\mathfrak{F}$. If $\mathcal{M}$ is XY-homogeneous, then $\mathcal{M}$ has the XYEP. 2. (2)

Suppose that XY $\in\mathfrak{F}\cap\mathfrak{I}$. If $\mathcal{M}$ has the XYEP, then $\mathcal{M}$ is XY-homogeneous.

Proof.

(1) Let $A\subseteq B\in\mathscr{C}$ and $f:A\to\mathcal{M}$ be a map of type X. As Age$(\mathcal{M})=\mathscr{C}$, there exists an isomorphism $\theta:B\to B\theta\subseteq\mathcal{M}$. Therefore, $\theta^{-1}f:B\theta\to Af$ is a map of type X between finite substructures of $\mathcal{M}$. As $\mathcal{M}$ is XY-homogeneous, extend $\theta^{-1}f$ to a map $\alpha:\mathcal{M}\to\mathcal{M}$ of type Y. Hence, $\theta\alpha:B\to\mathcal{M}$ is a map of type Y. It remains to show that $\theta\alpha$ extends $f$; for any $a\in A$ it follows that

[TABLE]

as $\alpha$ extends $\theta^{-1}f$. Therefore $\mathcal{M}$ has the XYEP.

(2) Suppose that $f:A\xrightarrow{}B$ is a map of type X between finite substructures of $\mathcal{M}$. We use a forth argument to extend $f$ to a map $\alpha$ of type Y. As $\mathcal{M}$ is countable, we enumerate the points of $M=\{m_{0},m_{1},\ldots\}$. Set $A=A_{0},B=B_{0}$ and $f=f_{0}$ and assume that we have extended $f$ to a map $f_{k}:A_{k}\xrightarrow{}B_{k}$, where $A_{i}\subseteq A_{i+1}$ and $B_{i}\subseteq B_{i+1}$ for all $0\leq i\leq k-1$. At most, we can assume that $f_{k}$ is a map of type Y. Select $m_{i}\in\mathcal{M}\smallsetminus A_{k}$, where $i$ is the least natural number such that $m_{i}\notin A_{k}$. We can see that $A_{k}\cup\{m_{i}\}\subseteq\mathcal{M}$ belongs to $\mathscr{C}$. As XY $\in\mathfrak{I}$, the map $f_{k}$ of type Y is also of type X; so use the XYEP to find a map $f_{k+1}:A_{k}\cup\{m_{i}\}\xrightarrow{}\mathcal{M}$ of type Y extending $f_{k}$. Repeating this process infinitely many times, ensuring that each $m_{i}$ appears at some stage, extends $f$ to a map $\alpha:\mathcal{M}\xrightarrow{}\mathcal{M}$ of type Y; so $\mathcal{M}$ is XY-homogeneous. ∎

We now move to the proof of Theorem 1.1. Our eventual aim is to construct a countable structure $\mathcal{M}$ with age $\mathscr{C}$, where $\mathcal{M}$ is XY-homogeneous. Recall (from [26]) that a class $\mathscr{C}$ of finite $\sigma$-structures has the joint embedding property (JEP) if for all $A,B\in\mathscr{C}$ there exists a $D\in\mathscr{C}$ such that $D$ jointly embeds $A$ and $B$. This property, along with $\mathscr{C}$ being closed under substructures and isomorphisms, and having countably many isomorphism types, is required to construct a countable structure $\mathcal{M}$ with age $\mathscr{C}$; it has nothing to do with the homogeneity of the structure $\mathcal{M}$. In Fraïssé’s theorem, it is the amalgamation property that is central to ensuring that the constructed structure is homogeneous; following the lead of [9], this must be generalised in order to ensure XY-homogeneity. So to construct a countable, XY-homogeneous structure $\mathcal{M}$ with age $\mathscr{C}$, the class of finite structures $\mathscr{C}$ must have the JEP and some generalised amalgamation property.

Since different types of homogeneity require different amalgamation properties, it then makes sense to define an “umbrella” condition; one that encompasses every required amalgamation property. This is the XY-amalgamation property (XYAP), where X,Y $\in\{$H, M, I$\}$:

(XYAP) Let $\mathscr{C}$ be a class of finite $\sigma$-structures. Then $\mathscr{C}$ has the XYAP if for all $A,B_{1},B_{2}\in\mathscr{C}$, map $f_{1}:A\xrightarrow{}B_{1}$ of type X and embedding $f_{2}:A\xrightarrow{}B_{2}$, there exists a $D\in\mathscr{C}$, embedding $g_{1}:B_{1}\xrightarrow{}D$ and map $g_{2}:B_{2}\xrightarrow{}D$ of type Y such that $f_{1}g_{1}=f_{2}g_{2}$ (see Figure 3).

Based on choices for X and Y, the XYAP yields nine different amalgamation properties; one for each notion of XY-homogeneity in $\mathfrak{F}$. For instance, the IIAP is the standard amalgamation property, the MMAP is the MAP in [9] and the HHAP is the HAP from [14]. These are the relevant amalgamation properties as alluded to in Theorem 1.1. Our next result demonstrates Theorem 1.1 (1).

Proposition \theproposition (Theorem 1.1 (1)).

Suppose that XY $\in\mathfrak{F}$. If a countable $\sigma$-structure $\mathcal{M}$ is XY-homogeneous, then Age$(\mathcal{M})$ has the XYAP.

Proof.

Suppose that $A,B_{1},B_{2}\in$ Age$(\mathcal{M})$, $f_{1}:A\xrightarrow{}B_{1}$ is a map of type X and $f_{2}:A\xrightarrow{}B_{2}$ is an embedding. Without loss of generality, suppose that $f_{2}$ is the inclusion map and that $A,B_{1},B_{2}\subseteq\mathcal{M}$. Using XY-homogeneity of $\mathcal{M}$, extend $f_{1}:A\xrightarrow{}B$ to a map $\alpha:\mathcal{M}\xrightarrow{}\mathcal{M}$ of type Y. Set $D=B_{1}\cup B_{2}\alpha$ and induce the structure on $D$ with relations from $\mathcal{M}$. Finally, take $g_{1}:B_{1}\xrightarrow{}D$ to be the inclusion map and define $g_{2}$ to be the map $g_{2}=\alpha|_{B_{2}}:B_{2}\xrightarrow{}D$ of type Y. We can see that $f_{1}g_{1}=f_{2}g_{2}$ and so these choices verify the XYAP for Age$(\mathcal{M})$. ∎

Now, we proceed with the proof of Theorem 1.1 (2). As in [9], different stages of the inductive construction are achieved at even and odd steps.

Proposition \theproposition (Theorem 1.1 (2)).

Suppose that XY $\in\mathfrak{F}\cap\mathfrak{I}$. Let $\mathscr{C}$ be a class of finite $\sigma$-structures that is closed under isomorphism and substructures, has countably many isomorphism types, and has the JEP and XYAP. Then there exists a countable XY-homogeneous $\sigma$-structure $\mathcal{M}$ with age $\mathscr{C}$.

Proof.

Along the lines of similar proofs of Fraïssé’s theorem (see [3], [9]), the idea is to construct $\mathcal{M}$ over countably many stages, assuming that $M_{k}$ has been constructed at some stage $k\in\mathbb{N}$, with $M_{0}$ being some $A\in\mathscr{C}$. As the number of isomorphism types in $\mathscr{C}$ is countable, we can choose a countable set $S$ of pairs $(A,B)$ with $A\subseteq B\in\mathscr{C}$ such that every pair $A^{\prime}\subseteq B^{\prime}\in\mathscr{C}$ is represented by some pair $(A,B)\in S$. Define a bijection $\beta:\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ such that $\beta(i,j)\geq i$.

Assume first that $k$ is even. As $\mathscr{C}$ has countably many isomorphism types, we can enumerate the isomorphism types of $\mathscr{C}$ by $\{M_{0}=T_{0},T_{2},T_{4},\ldots\}$. Use the JEP to find a structure $D$ that contains both $M_{k}$ and a copy of $C\cong T_{k}\in\mathscr{C}$; define $M_{k+1}$ to be this structure $D$. Now suppose that $k$ is odd. Let $L=(A_{kj},B_{kj},f_{kj})_{j\in\mathbb{N}}$ be the list of all triples $(A,B,f)$ such that $(A,B)\in S$ and $f:A\to M_{k}$ is a map of type X. This list is countable as $S$ is and there are finitely many maps of type X from $A$ into $M_{k}$. Let $k=\beta(i,j)$. Then as $\beta(i,j)\geq i$, the map $f_{ij}:A_{ij}\to M_{i}\subseteq M_{k}$ exists. Therefore, we can use the XYAP to define $M_{k+1}$ such that $M_{k}\subseteq M_{k+1}$ and the map $f_{ij}:A_{ij}\to M_{k}$ of type X extends to some map $g_{ij}:B_{ij}\xrightarrow{}M_{k+1}$ of type Y. This ensures that every possible XY-amalgamation occurs.

Define $\mathcal{M}=\bigcup_{k\in\mathbb{N}}M_{k}$. Our construction ensures that every isomorphism type of $\mathscr{C}$ appears at a 0 mod 3 stage, so every structure in $\mathscr{C}$ embeds into $\mathcal{M}$. Conversely, we have that $M_{k}\in\mathscr{C}$ for all $k\in\mathbb{N}$. As $\mathscr{C}$ is closed under substructures, every structure that embeds into $\mathcal{M}$ is in $\mathscr{C}$, showing that Age$(\mathcal{M})=\mathscr{C}$.

It remains to show that $\mathcal{M}$ is XY-homogeneous. As XY $\in\mathfrak{F}\cap\mathfrak{I}$, it is enough to show that $\mathcal{M}$ has the XYEP by Section 2. So assume that $A\subseteq B\in\mathscr{C}$ and that $f:A\to\mathcal{M}$ is a map of type X. As $Af$ is finite, it follows that there exists $j\in\mathbb{N}$ such that $Af\subseteq M_{j}$. Furthermore, there exists a triple $(A_{j\ell},B_{j\ell},f_{j\ell})\in L$ such that there exists an isomorphism $\theta:B\to B_{j\ell}$ with $A\theta|_{A}=A_{j\ell}$ and $f=\theta|_{A}f_{j\ell}$. Define $n=\beta(j,\ell)$; as $n\geq j$, it follows that $Af=A_{j\ell}f_{j\ell}\subseteq M_{j}\subseteq M_{n}$. Here, $M_{n+1}$ is constructed by XY-amalgamating $M_{n}$ and $B_{j\ell}$ over $A_{j\ell}$; this provides an extension $g_{j\ell}:B_{j\ell}\to M_{n+1}$ of type Y to the map $f_{j\ell}$ of type X. It follows that $g=\theta g_{j\ell}:B\to\mathcal{M}$ is a map of type Y extending the map $f$ of type $X$ (see Figure 4 for a diagram). Therefore, $\mathcal{M}$ has the XYEP, proving that it is XY-homogeneous.∎

All that remains to show is part (3) of Theorem 1.1. It was previously mentioned in [9] that two MM-homogeneous structures with the same age need not be isomorphic, but are instead mono-equivalent. This inspires another collective definition; and is the relevant notion of equivalence as mentioned in the statement of Theorem 1.1.

Definition \thedefinition.

Let $\mathcal{M,N}$ be $\sigma$-structures and suppose that Y $\in\{$H, M, I$\}$. Say that $\mathcal{M}$ and $\mathcal{N}$ are Y-equivalent if Age$(\mathcal{M})=$ Age$(\mathcal{N})$ and every embedding $f:A\to\mathcal{N}$ from a finite substructure $A$ of $\mathcal{M}$ can be extended to a map $g:\mathcal{M}\to\mathcal{N}$ of type Y, and vice versa.

Note that if two structures $\mathcal{M,N}$ are M-equivalent, then they are mono-equivalent in the sense of [9]. If two structures $\mathcal{M,N}$ are I-equivalent, then they are mutually embeddable.

Proposition \theproposition (Theorem 1.1 (3)).

Let $\mathcal{M,N}$ be countable $\sigma$-structures, and suppose that XY $\in\mathfrak{F}\cap\mathfrak{I}$.

1. (1)

Suppose that $\mathcal{M,N}$ are Y-equivalent. Then $\mathcal{M}$ is XY-homogeneous if and only if $\mathcal{N}$ is. 2. (2)

If $\mathcal{M,N}$ are XY-homogeneous and Age$(\mathcal{M})$* = Age*$(\mathcal{N})$* then $\mathcal{M,N}$ are Y-equivalent.*

Proof.

(1) It suffices to show that $\mathcal{N}$ has the XYEP by Section 2 (2). Suppose then that $A\subseteq B\in$ Age$(\mathcal{N})$ and there exists a map $f:A\xrightarrow{}A^{\prime}\subseteq\mathcal{N}$ of type X. Note that $A$ need not be isomorphic to $A^{\prime}$. As Age$(\mathcal{M})=$ Age$(\mathcal{N})$, there exists a copy $A^{\prime\prime}$ of $A^{\prime}$ in $\mathcal{M}$; fix an embedding $e:A^{\prime}\to A^{\prime\prime}$ between the two. Therefore, $e^{-1}:A^{\prime\prime}\to A^{\prime}$ is a isomorphism from a finite substructure of $\mathcal{M}$ into $\mathcal{N}$; as the two are Y-equivalent, we extend this to a map $\alpha:\mathcal{M}\to\mathcal{N}$ of type Y. Now, define a map $h=fe:A\to A^{\prime\prime}$; this is a map of type X from $A$ into $\mathcal{M}$. Since $\mathcal{M}$ is XY-homogeneous, it has the XYEP by Section 2 (1) and so we extend $h$ to a map $h^{\prime}:B\to\mathcal{M}$ of type Y. Now, the map $h^{\prime}\alpha:B\xrightarrow{}\mathcal{N}$ is a map of type Y; we need to show it extends $f$. So using the facts that $\alpha$ extends $e^{-1}$ and $h^{\prime}$ extends $h=fe$, we have that for all $a\in A$:

[TABLE]

Therefore $\mathcal{N}$ has the XYEP. A diagram of this process can be found in Figure 5.

(2) Let $A\subseteq\mathcal{M},B\subseteq\mathcal{N}$ and suppose that $f:A\xrightarrow{}B$ is an embedding; trivially, $f$ is also a map of type X. We extend $f$ to a map $\alpha:\mathcal{M}\xrightarrow{}\mathcal{N}$ of type Y via an inductive argument. As $\mathcal{M}$ is countable, we enumerate its elements $M=\{m_{0},m_{1},\ldots\}$. Set $A=A_{0},B=B_{0}$ and $f=f_{0}$, and suppose that $f_{k}:A_{k}\xrightarrow{}B_{k}$ is a map of type Y where $A_{i}\subseteq A_{i+1}$ and $B_{i}\subseteq B_{i+1}$ for all $0\leq i\leq k-1$. As XY $\in\mathfrak{F}\cap\mathfrak{I}$, $f_{k}$ is also a map of type X. Select a point $m_{i}\in\mathcal{M}\smallsetminus A_{k}$, where $i$ is the least natural number such that $m_{i}\notin A_{k}$. We see that $A_{k}\cup\{m_{i}\}$ is a substructure of $\mathcal{M}$ and is therefore an element of Age$(\mathcal{N})$ by assumption. As $\mathcal{N}$ is XY-homogeneous, by Section 2 (1) it has the XYEP. Using this, extend $f_{k}:A_{k}\xrightarrow{}\mathcal{N}$ to a map $f_{k+1}:A_{k}\cup\{m_{i}\}\xrightarrow{}\mathcal{N}$ of type Y. As XY $\in\mathfrak{F}\cap\mathfrak{I}$, we can repeat this process infinitely many times; by ensuring that every $m_{i}\in\mathcal{M}$ is included at some stage, we can extend the map $f$ to a map $\alpha:\mathcal{M}\xrightarrow{}\mathcal{N}$ of type Y as required. We can use a similar argument to construct a map $\beta:\mathcal{N}\to\mathcal{M}$ of type Y; therefore $\mathcal{M}$ and $\mathcal{N}$ are Y-equivalent. ∎

3 Multifunctions and antihomomorphisms

As mentioned in the introduction, homomorphisms are not “invertible” in general. For instance, there could be a homomorphism between two relational $\sigma$-structures $\alpha:\mathcal{A}\to\mathcal{B}$ sending a non-relation of $\mathcal{A}$ to a relation in $\mathcal{B}$; that is, such that $\underline{a}\notin R_{i}^{\mathcal{A}}$ but $\underline{a}\alpha\in R_{i}^{\mathcal{B}}$. Furthermore, there is no guarantee that the homomorphism is even injective; so $\alpha$ could send two points in $\mathcal{A}$ to the same point in $\mathcal{B}$. In establishing a suitable ‘back’ amalgamation property for our Fraïssé-style theorem, both of these considerations must be taken into account. This is achieved by the use of the converse of a function.

For a relation $\rho\subseteq X\times Y$, define the converse of $\rho$ to be the set

[TABLE]

We say that a relation $\bar{f}\subseteq Y\times X$ is a partial multifunction if $(y,x),(z,x)\in\bar{f}$ implies that $y=z$; and that $\bar{f}$ is a multifunction if, in addition, for all $y\in Y$ there exists $x\in X$ such that $(y,x)\in\bar{f}$. It is easy to see that $\bar{f}$ is a partial multifunction if and only if it is the converse $f^{c}$ of a partial function $f$, and that $\bar{f}$ is a multifunction if and only if the partial function $f$ is surjective. A multifunction $\bar{f}\subseteq Y\times X$ is surjective if for all $x\in X$ there exists $y\in Y$ such that $(y,x)\in\bar{f}$. Consequently, $\bar{f}$ is a surjective multifunction if and only if it is the converse of a surjective function $f$. It is clear that a (partial) multifunction $\bar{f}$ is a (partial) function if and only if it is the converse $f^{c}$ of a (partial) injective function $f$. We adopt this barred notation throughout the rest of this article; if $f\subseteq X\times Y$ is a function, denote the partial multifunction given by the converse $f^{c}$ of $f$ by $\bar{f}\subseteq Y\times X$, and vice versa. Note that $\bar{\bar{f}}=f$ for any function $f$.

Example \theexample.

Let $X=\{1,\,2,\,3,\,4\}$ and $Y=\{a,\,b,\,c,\,d,\,e\}$ be two sets, and suppose that $f=\{(1,b),\,(2,b),\,(3,a),\,(4,c)\}$ is a function. Then the converse $\bar{f}$ of $f$ is a partial multifunction given by $\bar{f}=\{(b,1),\,(b,2),\,(a,3),\,(c,4)\}$ (see Figure 6). By restricting the codomain $Y$ of $f$ to its image $\text{im }f$, the resulting function $g:X\to\text{im }f$ that behaves like $f$ is surjective. In this case, the converse $\bar{g}:\text{im }f\to X$ of $g$ is a surjective and totally defined multifunction (see shaded portion of Figure 6). This technique will be used frequently in Section 4.

If $\bar{f}\subseteq Y\times X$ is a multifunction, we will abuse notation and write $\bar{f}:Y\xrightarrow{}X$ where the context is clear. If $y\in Y$, define the set $y\bar{f}=\{x\in X$ : $(y,x)\in\bar{f}\}$. In a contrast with a function, notice that $y\bar{f}$ is a set and not a single point; in fact, the set could be infinite (see Section 3). For a tuple $\underline{y}=(y_{1},\ldots,y_{n})\in Y^{n}$, define $\underline{y}\bar{f}$ to be the following set of tuples

[TABLE]

For a subset $W$ of $Y$, we write

[TABLE]

Abusing terminology, we say that $Y\bar{f}$ is the image of $\bar{f}$.

Example \theexample.

Recall that the sign function $s:\mathbb{R}\to\{-1,0,1\}$ is the surjective function defined by

[TABLE]

Let $\bar{s}:\{-1,0,1\}\to\mathbb{R}$ be the corresponding surjective multifunction. For example, $1\bar{s}=\mathbb{R}^{+}$ where $\mathbb{R}^{+}$ is the set of positive real numbers, $(0,-1)\bar{s}=\{(0,x)\in\mathbb{R}^{2}\;:\;x\in\mathbb{R}^{-}\}$ where $\mathbb{R}^{-}$ is the set of negative real numbers, and $\{-1,1\}\bar{s}=1\bar{s}\cup-1\bar{s}=\mathbb{R}\setminus\{0\}$.

Remark*.*

We note the equivalence between the image of the converse of a function $f$ and the preimage of $f$. The reason for the expression of these sets in terms of converses of functions, and not preimages, is for ease of use and notation.

For a multifunction $\bar{f}:Y\xrightarrow{}X$ and a subset $W\subseteq Y$, we say that the multifunction $\bar{f}|_{W}:W\to X$ that acts like $\bar{f}$ on $W$ is the restriction of $\bar{f}$ to $W$. If $Y\subseteq B$ and $X\subseteq A$ are sets, and $\bar{f}:Y\to X$ and $\bar{g}:B\to A$ are two multifunctions, then we say that $\bar{g}$ extends $\bar{f}$ if $y\bar{f}=y\bar{g}$ for all $y\in Y$.

Throughout, we would like to be able to compose functions with multifunctions and vice versa; we achieve this by composing them as relations.

Lemma \thelemma.

1. (1)

Suppose that $\bar{g}:C\xrightarrow{}B$ and $\bar{f}:B\xrightarrow{}A$ are multifunctions. Then $\bar{g}\circ\bar{f}=\bar{g}\bar{f}\subseteq C\times A$ is a multifunction. 2. (2)

Let $f:A\to B$ and $g:B\to C$ be two functions, and suppose that $fg:A\to C$ is their composition. Then the converse map $\overline{fg}:C\to A$ is equal to $\bar{g}\bar{f}:C\to A$, where $\bar{g}$ and $\bar{f}$ are composed as relations. ∎

Remark*.*

We previously noted that a function $g$ is also a multifunction if and only if it is injective; so by this lemma, the composition of a multifunction $\bar{f}$ with an injective function $g$ (or vice versa) is again a multifunction. Furthermore, the assumption that the function $g$ is injective in this case is necessary for the composition $\bar{f}\circ g$ to be a multifunction.

Now, we extend the theory of multifunctions into the setting of relational first-order structures.

Definition \thedefinition.

Suppose that $\mathcal{A},\mathcal{B}$ are two $\sigma$-structures and that $\bar{f}:\mathcal{B}\xrightarrow{}\mathcal{A}$ is a multifunction. We say that $\bar{f}$ is an antihomomorphism if $\neg R_{i}^{\mathcal{B}}(\underline{b})$ in $\mathcal{B}$ then $\neg R_{i}^{\mathcal{A}}(\underline{a})$ in $\mathcal{A}$ for all $\underline{a}\in\underline{b}\bar{f}$ and $R_{i}\in\sigma$.

Remark*.*

This definition is equivalent to saying that $\bar{f}:\mathcal{B}\to\mathcal{A}$ is an antihomomorphism if for all $R_{i}\in\sigma$ and for all $\underline{a}\in\underline{b}\bar{f}$, then $R_{i}^{\mathcal{A}}(\underline{a})$ implies that $R_{i}^{\mathcal{B}}(\underline{b})$.

Informally, an antihomomorphism is a multifunction that preserves non-relations. The motivation behind this definition is explained by the following alternate characterisation of antihomomorphisms.

Lemma \thelemma.

Let $\mathcal{A},\mathcal{B}$ be two $\sigma$-structures. Then $f^{c}:\mathcal{B}\xrightarrow{}\mathcal{A}$ is a surjective antihomomorphism if and only if $f:\mathcal{A}\xrightarrow{}\mathcal{B}$ is a surjective homomorphism.

Proof.

Assume that $f:\mathcal{A}\xrightarrow{}\mathcal{B}$ is a surjective homomorphism. As $f:A\xrightarrow{}B$ is a surjective function we have that $f^{c}:B\xrightarrow{}A$ is a surjective multifunction. Now suppose that $\neg R_{i}^{\mathcal{B}}(\underline{b})$. As $f$ must preserve relations, we have $\neg R_{i}^{\mathcal{A}}(\underline{a})$ whenever $\underline{a}f=\underline{b}$; this is precisely when $\underline{a}\in\underline{b}f^{c}$. Conversely, suppose that $f^{c}:\mathcal{B}\xrightarrow{}\mathcal{A}$ is a surjective antihomomorphism; therefore $f:A\xrightarrow{}B$ is a surjective function. Suppose also that $R_{i}^{\mathcal{A}}(\underline{a})$ holds. As $f^{c}$ is an antihomomorphism, it follows that $\underline{a}\notin\underline{b}f^{c}$ for every $\underline{b}$ such that $\neg R_{i}^{\mathcal{B}}(\underline{b})$. Since $f$ is a function, it must be that $\underline{a}\in\underline{b}f^{c}$ for some $\underline{b}$ such that $R_{i}^{\mathcal{B}}(\underline{b})$; so $f$ is a homomorphism. ∎

Remark*.*

Following this lemma, if $f:A\to B$ is a surjective homomorphism, we write $f^{c}=\bar{f}:B\to A$ in order to emphasise that the converse of $f$ is both a multifunction and antimonomorphism. Similarly, if $\bar{f}:B\to A$ is a surjective antihomomorphism, we write $\bar{f}^{c}=f:A\to B$ to emphasise that the converse of $\bar{f}$ is both a function and a homomorphism. The context for when we use this notation should be clear.

If $f:\mathcal{A}\xrightarrow{}\mathcal{B}$ is any homomorphism, we can restrict the codomain to the image to see that $f:\mathcal{A}\xrightarrow{}\mathcal{A}f$ is a surjective homomorphism; and hence $\bar{f}:\mathcal{A}f\xrightarrow{}\mathcal{A}$ is a surjective antihomomorphism by Section 3. This technique will be used regularly in Section 4.

This result leads to an immediate corollary; an analogue of Section 3 (2) for $\sigma$-structures.

Corollary \thecorollary.

Let $\mathcal{A,B,C}$ be $\sigma$-structures, and $f:\mathcal{A}\to\mathcal{B}$ and $g:\mathcal{B}\to\mathcal{C}$ are surjective homomorphisms. Then $\overline{fg}=\bar{g}\bar{f}$ is a surjective antihomomorphism. ∎

Note that if $f:\mathcal{A}\xrightarrow{}\mathcal{B}$ is a bijective homomorphism, then $\bar{f}:\mathcal{B}\xrightarrow{}\mathcal{A}$ is an bijective function from $B$ to $A$ that preserves non-relations; this is the definition of an antimonomorphism (see [12]). Furthermore, if $f:\mathcal{A}\xrightarrow{}\mathcal{B}$ is a isomorphism, then $\bar{f}:\mathcal{B}\xrightarrow{}\mathcal{A}$ is exactly $f^{-1}$, the inverse isomorphism of $f$. Having determined that the product of two multifunctions is again a multifunction in Section 3, an easy composition lemma for antihomomorphisms follows suit.

Lemma \thelemma.

Let $\mathcal{A,B,C}$ be $\sigma$-structures. Suppose that $\bar{f}:\mathcal{A}\xrightarrow{}\mathcal{B}$ and $\bar{g}:\mathcal{B}\xrightarrow{}\mathcal{C}$ are antihomomorphisms. Then their composition $\bar{f}\bar{g}:\mathcal{A}\xrightarrow{}\mathcal{C}$ is an antihomomorphism. ∎

Remark*.*

We note that as every antimonomorphism and isomorphism is also an antihomomorphism, the product $\bar{f}\bar{g}$ of any antihomomorphism $\bar{f}$ with any antimonomorphism or isomorphism $\bar{g}$ is again an antihomomorphism. This fact turns out to be crucial in the statement of a suitable amalgamation property for the back part of the back-and-forth argument. Furthermore, the product of two antimonomorphisms (or an antihomomorphism and an isomorphism) is again an antimonomorphism.

4 Proof of Theorem 1.2

We now move on to discussing extension of finite partial maps of a $\sigma$-structure to surjective endomorphisms; this is when XZ $\in\mathfrak{B}$ (see page • ‣ 2, item). Due to the lack of symmetry when working with homomorphisms as opposed to isomorphisms, we must provide a backwards condition to achieve the back part of the required back-and-forth argument. Similar to the more conventional amalgamation properties, this backwards condition is defined on finite structures. This will involve using the concept of antihomomorphisms outlined in Section 3 in three distinct cases; antihomomorphisms ($\overline{\text{H}}$) as converse homomorphisms (H), antimonomorphisms ($\overline{\text{M}}$) as the converse of monomorphisms (M), and inverse isomorphisms ($\overline{\text{I}}$) of isomorphisms (I). Note that the classes I and $\overline{\text{I}}$ coincide; we use the barred version when applicable throughout for notational consistency. It can be seen that $\overline{\text{I}}\subseteq\overline{\text{M}}\subseteq\overline{\text{H}}$. We will write $\bar{f}:B\xrightarrow{}A$ to mean some multifunction of type $\overline{\text{X}}\in\{\overline{\text{H}},\overline{\text{M}},\overline{\text{I}}\}$ from $B$ to $A$ (see Figure 7).

This notation is used in another manner: if $f:A\to B$ is a surjective homomorphism of type X, we write $\bar{f}:B\to A$ to be the corresponding surjective antihomomorphism of type $\overline{\text{X}}$. This is uniquely determined by Section 3; see Figure 7 for corresponding pairs. The context of when we use this will usually be clear. We also recall Section 3 and its following remarks; the composition of two multifunctions of type $\overline{\text{H}},\overline{\text{M}},\overline{\text{I}}$ is again a multifunction of type $\overline{\text{H}},\overline{\text{M}},\overline{\text{I}}$; a composition table is given in Figure 7.

We note that if Z = E then it is a surjective map of type Y = H; likewise, when Z = B we have that Y = M and when Z = A we have that Y = I. This relation is codified by the following set of pairs:

[TABLE]

It follows that any XZ-homogeneous structure $\mathcal{M}$ is also XY-homogeneous, where the two are related by the relevant pair (Z,Y) $\in\mathscr{S}$. Therefore, we need to ensure that any XZ-homogeneous structure $\mathcal{M}$ we construct is also XY-homogeneous for the appropriate Y; so results in Section 2 should be satisfied by $\mathcal{M}$.

As mentioned previously, new properties are required to take care of extension and amalgamation in the backwards direction to ensure the map is surjective. This is achieved by writing generalised conditions utilising the concept of antihomomorphisms. Throughout, we let X,Y $\in\{$H, M, I$\}$, $\overline{\text{X}},\overline{\text{Y}}\in\{\overline{\text{H}},\overline{\text{M}},\overline{\text{I}}\}$, and Z $\in\{$E, B, A$\}$. To avoid any potential confusion, whenever we refer to a map of type Z being a surjective map of type Y, the symbol Z is always related to Y in the manner illustrated in $\mathscr{S}$ (see Equation 1).

Motivated by the desire to take care of the ‘back’ part of a back-and-forth argument that would extend a finite partial map of $\mathcal{M}$ to a surjective endomorphism of $\mathcal{M}$, we can state the $\overline{\text{XY}}$-extension property ($\overline{\text{XY}}$EP) along similar lines to the XYEP in Section 2.

($\overline{\text{XY}}$EP) Suppose that $\mathcal{M}$ is a structure with age $\mathscr{C}$. For all $A\subseteq B\in\mathscr{C}$ and a multifunction $\bar{f}:A\xrightarrow{}\mathcal{M}$ of type $\overline{\text{X}}$ such that $A\bar{f}$ is finite, there exists a multifunction $\bar{g}:B\xrightarrow{}\mathcal{M}$ of type $\overline{\text{Y}}$ extending $\bar{f}$.

Notice that this property differs slightly from previous extension properties as it requires an extra finiteness condition on the image of the multifunction $\bar{f}$. There could exist multifunctions $\bar{f}:A\to\mathcal{M}$ of type $\overline{\text{H}}$ where the image $A\bar{f}$ is an infinite set (see Section 3); this would cause problems in the proofs of Section 4 and Section 4.

We turn our attention to finding necessary and sufficient conditions for XZ-homogeneity, to be used throughout the proof of Theorem 1.2. As stated above, we need to ensure that any XZ-homogeneous structure we construct is also XY-homogeneous for the appropriate Y. It follows that such a structure must satisfy all the conditions outlined in Section 2; in particular, XY must be in $\mathfrak{I}$ for part (2). With these restrictions in mind, and a desire to obtain the most general result possible, we show that both the XYEP and $\overline{\text{XY}}$EP are necessary conditions for XZ-homogeneity in general, and that it these are also sufficient when the extended map of type Y is also a map of type X.

Proposition \theproposition.

Let $\mathcal{M}$ be a $\sigma$-structure with age $\mathscr{C}$.

1. (1)

*Suppose that XZ $\in\mathfrak{B}$. If $\mathcal{M}$ is XZ-homogeneous, then $\mathcal{M}$ has both the XYEP and the *$\overline{\text{XY}}$EP. 2. (2)

*Suppose that XZ $\in\mathfrak{B}\cap\mathfrak{I}$. If $\mathcal{M}$ has the XYEP and the *$\overline{\text{XY}}$EP, then $\mathcal{M}$ is XZ-homogeneous.

Proof.

(1) As $\mathcal{M}$ is XZ-homogeneous, it is also XY-homogeneous and so it has the XYEP by Section 2 (1). Now, suppose that $A,B\in\mathscr{C}$ and $\bar{f}:A\xrightarrow{}\mathcal{M}$ is a multifunction of type $\overline{\text{X}}$ with $A\bar{f}$ finite. As $\mathscr{C}$ is the age of $\mathcal{M}$, it follows that $\mathcal{M}$ contains copies $A^{\prime}\subseteq B^{\prime}$ of $A$ and $B$ and there are isomorphisms $\theta:B\to B^{\prime}$ and $\theta^{-1}:B^{\prime}\to B$. Restrict the codomain of $\bar{f}$ to its image to find a map $\bar{f}^{\prime}:A\xrightarrow{}A\bar{f}$; as this is a surjective multifunction of type $\overline{\text{X}}$, we have that $\theta^{-1}\bar{f}^{\prime}=\bar{h}:A^{\prime}\xrightarrow{}A\bar{f}$ is also a surjective multifunction of type $\overline{\text{X}}$. By Section 3, the converse $h:A\bar{f}\to A^{\prime}$ of $\theta|_{A}^{-1}\bar{f}^{\prime}$ is a surjective map of type X with finite domain $A\bar{f}$; as $\mathcal{M}$ is XZ-homogeneous, extend $h$ to a map $\beta:\mathcal{M}\to\mathcal{M}$ of type Z. So $\beta\theta^{-1}:\mathcal{M}\to B$ is a surjective map of type Y; by Section 3, define $\bar{g}=\theta\bar{\beta}:B\to\mathcal{M}$ to be the corresponding surjective multifunction of type $\overline{\text{Y}}$. We need to show it extends $\bar{f}$. As $\beta$ extends $h$, then $\bar{\beta}$ extends $\bar{h}$. So for all $a\in A$:

[TABLE]

and hence $\mathcal{M}$ has the $\overline{\text{XY}}$EP.

(2) Now suppose that XZ $\in\mathfrak{B}\cap\mathfrak{I}$; so a multifunction of type $\overline{\text{Y}}$ implies that it is also a multifunction of type $\overline{\text{X}}$. Suppose also that $\mathcal{M}$ has the XYEP and the $\overline{\text{XY}}$EP, and that $f:A\xrightarrow{}B$ is a map of type X between substructures of $\mathcal{M}$. We use a back-and-forth argument to show that $\mathcal{M}$ is XZ-homogeneous.

Set $A=A_{0}$, $B=B_{0}$ and $f_{0}=f$, and assume that we have extended $f$ to a surjective map $f_{k}:A_{k}\xrightarrow{}B_{k}$ of type Y (and hence of type X, by assumption), where each $A_{i}\subseteq A_{i+1}$ and $B_{i}\subseteq B_{i+1}$ for all $0\leq i\leq k-1$. Note also that $A_{k}$ and $B_{k}$ are finite for all $k\in\mathbb{N}$. Furthermore, as $\mathcal{M}$ is countable we can enumerate the elements of $M=\{m_{0},m_{1},\ldots\}$.

If $k$ is even, select a point $m_{i}\in\mathcal{M}\smallsetminus A_{k}$ where $i$ is the smallest number such that $m_{i}\notin A_{k}$, so $A_{k}\cup\{m_{i}\}\subseteq\mathcal{M}$. Using the XYEP, extend $f_{k}$ to a map $f^{\prime}_{k+1}:A_{k}\cup\{m_{i}\}\xrightarrow{}B_{k}^{\prime}$ of type Y; by restricting the codomain of $f^{\prime}_{k+1}$ to its image, it follows that $f_{k+1}:A_{k}\cup\{m_{i}\}\xrightarrow{}B_{k}\cup\{m_{i}f^{\prime}_{k+1}\}$ is a surjective map of type Y extending $f_{k}$.

If $k$ is odd, choose a point $m_{i}\in\mathcal{M}\smallsetminus B_{k}$ where $i$ is the smallest number such that $m_{i}\notin B_{k}$; so $B_{k}\cup\{m_{i}\}\subseteq\mathcal{M}$. Note that as $f_{k}$ is a surjective map of type X, we have that $\bar{f}_{k}:B_{k}\xrightarrow{}A_{k}$ is a surjective multifunction of type $\overline{\text{X}}$. As $A_{k}$ is finite, we can use the $\overline{\text{XY}}$EP to extend $\bar{f}$ to a multifunction $\bar{f}_{k+1}^{\prime}:B_{k}\cup\{m_{i}\}\to\mathcal{M}$ of type $\overline{\text{Y}}$. Restricting the codomain of $\bar{f}_{k+1}$ to its image gives a surjective multifunction $\bar{f}_{k+1}:B_{k}\cup\{m_{i}\}\to A_{k}\cup m_{i}\bar{f}_{k+1}$ of type $\overline{\text{Y}}$, where $m_{i}\bar{f}_{k+1}=\{y\in\mathcal{M}:(y,m_{i})\in\bar{f}_{k+1}\}$ is a non-empty set. As $\bar{f}_{k+1}$ is a surjective multifunction of type $\overline{\text{Y}}$, we have that $f_{k+1}:A_{k}\cup m_{i}\bar{f}_{k+1}\to B_{k}\cup\{m_{i}\}$ is a surjective map of type Y extending $f_{k}$.

Since XZ $\in\mathfrak{B}\cap\mathfrak{I}$, a map of type Y is also a map of type X; so we can use the XYEP and $\overline{\text{XY}}$EP to repeat this process infinitely many times. By ensuring that each point of $\mathcal{M}$ appears at both an odd and even step, we extend $f$ to a surjective map $\beta$ of type Y; which is a map of type Z and so $\mathcal{M}$ is XZ-homogeneous. ∎

Remark*.*

Together, Section 2 and Section 4 re-prove [25, Lemma 1.1], which states that a countable structure $\mathcal{M}$ is II (MI, HI)-homogeneous if and only if it is IA (MA, HA)-homogeneous. For if a structure $\mathcal{M}$ is HI-homogeneous, then it has the HIEP by Section 2; this implies that every homomorphism between finite substructures of $\mathcal{M}$ is an isomorphism. Since this happens, it follows that every antihomomorphism between finite substructures of $\mathcal{M}$ is an isomorphism. Finally, as $\mathcal{M}$ has the HIEP it must have the $\overline{\text{HI}}$EP as well and so $\mathcal{M}$ is HA-homogeneous by Section 4. A similar argument works for the equality concerning MI-homogeneous structures. In the II case, the IIEP is the standard extension property (EP) from Fraïssé’s theorem, and so any structure $\mathcal{M}$ with the IIEP is homogeneous by the same result.

We now state our new amalgamation property to accommodate the back portion of a back-and-forth argument; this is the $\overline{\text{XY}}$-amalgamation property ($\overline{\text{XY}}$AP):

($\overline{\text{XY}}$AP) Let $\mathscr{C}$ be a class of finite $\sigma$-structures. We say that $\mathscr{C}$ has the $\overline{\text{XY}}$AP if for all $A,B_{1},B_{2}\in\mathscr{C}$, multifunction $\bar{f}_{1}:A\xrightarrow{}B_{1}$ of type $\overline{\text{X}}$ and embedding $f_{2}:A\xrightarrow{}B_{2}$, there exists a $D\in\mathscr{C}$, embedding $g_{1}:B_{1}\xrightarrow{}D$ and multifunction $\bar{g}_{2}:B_{2}\xrightarrow{}D$ of type $\overline{\text{Y}}$ such that $\bar{f}_{1}g_{1}=f_{2}\bar{g}_{2}$ (see Figure 8).

Note that this property represents nine different amalgamation conditions. This corresponds to one for each class XZ $\in\mathfrak{B}$, where (Z,Y) $\in\mathscr{S}$ (see Equation 1 on page 1) and X and $\overline{\text{X}}$ are related as in Figure 7. For examples, the $\overline{\text{II}}$AP is the standard amalgamation property, and the $\overline{\text{MM}}$AP is the BAP of [12].

We can now prove Theorem 1.2 (1). Before we do, we state a straightforward yet important fact about surjective endomorphisms of an infinite first-order structure $\mathcal{M}$.

Lemma \thelemma.

Let $\mathcal{M}$ be a $\sigma$-structure, with $\mathcal{A}$ a finite substructure of $\mathcal{M}$. Then for any $\alpha\in$ Epi$(\mathcal{M})$, there exists a finite structure $\mathcal{B}\subseteq\mathcal{M}$ such that $\mathcal{B}\alpha=\mathcal{A}$. ∎

Proposition \theproposition (Theorem 1.2 (1)).

Suppose that XZ $\in\mathfrak{B}$. If a structure $\mathcal{M}$ is XZ-homogeneous, then Age$(\mathcal{M})$ has the XYAP and the $\overline{\text{XY}}$AP.

Proof.

As $\mathcal{M}$ is XZ-homogeneous then it is XY-homogeneous and so has the XYAP by Section 2. To show that Age$(\mathcal{M})$ has the $\overline{\text{XY}}$AP, suppose that $A,B_{1},B_{2}\in$ Age$(\mathcal{M})$, $\bar{f}_{1}:A\xrightarrow{}B_{1}$ is a multifunction of type $\overline{\text{X}}$ and $f_{2}:A\xrightarrow{}B_{2}$ is an embedding. We can assume without loss of generality that $A,B_{1},B_{2}$ are actually substructures of $\mathcal{M}$ and that $f_{2}$ is the inclusion mapping.

By restricting the codomain of $\bar{f}_{1}$ to its image, $\bar{f}_{1}:A\xrightarrow{}A\bar{f}_{1}$ is a surjective multifunction of type $\overline{\text{X}}$; hence the converse $f_{1}:A\bar{f}_{1}\xrightarrow{}A$ of $\bar{f}_{1}$ is a surjective map of type X. Use XZ-homogeneity to extend $f_{1}$ to a map $\beta:\mathcal{M}\xrightarrow{}\mathcal{M}$ of type Z; and so a surjective map of type Y. We see that $B_{1}\beta$ is a structure containing $A$, and that $\beta|_{B_{1}}:B_{1}\xrightarrow{}B_{1}\beta$ extends $f_{1}$. Define $D=B_{1}\beta\cup B_{2}$. As $\beta$ is surjective, there exists a finite substructure $C$ such that $C\beta=D$ by Section 4. Now, define the map $g_{1}:B_{1}\xrightarrow{}C$ to be the inclusion map. Since $\beta$ is a surjective map of type Y, $\bar{\beta}:\mathcal{M}\xrightarrow{}\mathcal{M}$ is a surjective multifunction of type $\overline{\text{Y}}$ by Section 3. Therefore $\bar{\beta}|_{B_{2}}:B_{2}\xrightarrow{}B_{2}\bar{\beta}$ is a surjective multifunction of type $\overline{\text{Y}}$; furthermore, $B_{2}\bar{\beta}\subseteq C$ as $B_{2}\subseteq D$. Define $\bar{g}_{2}:B_{2}\xrightarrow{}C$ to be the multifunction $\bar{\beta}|_{B_{2}}$ of type $\overline{\text{Y}}$. It is easy to check that $\bar{f}_{1}g_{1}=f_{2}\bar{g}_{2}$ and so Age$(\mathcal{M})$ has the $\overline{\text{XY}}$AP. ∎

We now show the existence portion of Theorem 1.2. Note that the previously described inductive construction of an infinite structure in Section 2 used even and odd steps to achieve different stages of the construction at different times. Because we have two amalgamation properties, as well as the JEP to ensure a countable structure exists, we proceed using an inductive argument at steps congruent to 0, 1, 2 mod 3 to accommodate different stages of the construction.

Proposition \theproposition (Theorem 1.2 (2)).

Suppose that XZ $\in\mathfrak{B}\cap\mathfrak{I}$. Let $\mathscr{C}$ be a class of finite $\sigma$-structures that is closed under substructures and isomorphism, has countably many isomorphism types and has the JEP, XYAP and the $\overline{\text{XY}}$AP. Then there exists a XZ-homogeneous $\sigma$-structure $\mathcal{M}$ with age $\mathscr{C}$.

Proof.

We build $\mathcal{M}$ over countably many stages, assuming that $M_{k}$ has been constructed at some stage $k\in\mathbb{N}$, with $M_{0}$ being some $A\in\mathscr{C}$. As the number of isomorphism types in $\mathscr{C}$ is countable, we can choose a countable set $S$ of pairs $(A,B)$ with $A\subseteq B\in\mathscr{C}$ such that every pair $A^{\prime}\subseteq B^{\prime}\in\mathscr{C}$ is represented by some pair $(A,B)\in S$. Let $[\mathbf{m}]=\{n\in\mathbb{N}\;:\;n\equiv m\mod{3}\}$, where $m=1,2$. Define two bijections $\beta_{m}:[\mathbf{m}]\times\mathbb{N}\to[\mathbf{m}]$ such that $\beta_{m}(i,j)\geq i$ for $m=1,2$.

Assume first that $k\equiv 0\mod{3}$. As $\mathscr{C}$ has countably many isomorphism types, we can enumerate the isomorphism types of $\mathscr{C}$ by $\{T_{0}=M_{0},T_{3},T_{6},\ldots\}$. Use the JEP to find a structure $D$ that contains both $M_{k}$ and a copy of some $A\cong T_{k}\in\mathscr{C}$; define $M_{k+1}$ to be this structure $D$. Now suppose that $k\equiv 1\mod{3}$. Let $L_{1}=(A_{kj},B_{kj},f_{kj})_{j\in\mathbb{N}}$ be the list of all triples $(A,B,f)$ such that $(A,B)\in S$ and $f:A\to M_{k}$ is a map of type X. This list is countable as $S$ is and there are finitely many maps of type X from $A$ into $M_{k}$. Let $k=\beta_{1}(i,j)$. Then as $\beta_{1}(i,j)\geq i$, the map $f_{ij}:A_{ij}\to M_{i}\subseteq M_{k}$ exists. Therefore, we can use the XYAP to define $M_{k+1}$ such that $M_{k}\subseteq M_{k+1}$ and the map $f_{ij}:A_{ij}\to M_{k}$ of type X extends to some map $g_{ij}:B_{ij}\xrightarrow{}M_{k+1}$ of type Y (see Figure 9 (1)). This ensures that every possible XY-amalgamation occurs. If $k\equiv 2\mod{3}$, let $L_{2}=(P_{kj},Q_{kj},\bar{f}_{kj})_{j\in\mathbb{N}}$ be the list of all triples $(P,Q,\bar{f})$ such that $(P,Q)\in S$ and $\bar{f}:P\to M_{k}$ is a multifunction of type $\overline{\text{X}}$ with finite image $P\bar{f}$; again, this list is countable. Let $k=\beta_{2}(i,j)$; as $\beta_{2}(i,j)\geq i$ it follows that the multifunction $\bar{f}_{ij}:P_{ij}\to M_{i}\subseteq M_{k}$ of type $\overline{\text{X}}$ is well-defined. So we can use the XZAP to define $M_{k+1}$ such that $M_{k}\subseteq M_{k+1}$ and the multifunction $\bar{f}_{ij}:P_{ij}\to M_{k}$ extends to some multifunction $\bar{g}_{ij}:Q_{ij}\xrightarrow{}M_{k+1}$ (see Figure 9 (2)). This construction ensures that every possible XZ-amalgamation occurs.

Define $\mathcal{M}=\bigcup_{k\in\mathbb{N}}M_{k}$. Our construction ensures that every isomorphism type of $\mathscr{C}$ appears at a 0 mod 3 stage, so every structure in $\mathcal{C}$ embeds into $\mathcal{M}$. Conversely, we have that $M_{k}\in\mathscr{C}$ for all $k\in\mathbb{N}$. As $\mathscr{C}$ is closed under substructures, every structure that embeds into $\mathcal{M}$ is in $\mathscr{C}$, showing that Age$(\mathcal{M})=\mathscr{C}$.

It remains to show that $\mathcal{M}$ is XZ-homogeneous. By Section 4 and the fact that XZ $\in\mathfrak{B}\cap\mathfrak{I}$, it is enough to show that $\mathcal{M}$ has both the XYEP and the $\overline{\text{XY}}$EP. Assume that $A\subseteq B\in\mathscr{C}$ and that $f:A\to\mathcal{M}$ is a map of type X. Using a similar argument to that of Section 2 (with $j\in[\mathbf{1}]$ and $(A_{j\ell},B_{j\ell},f_{j\ell})\in L_{1}$), we can show that as $\mathscr{C}$ has the XYAP, then $\mathcal{M}$ has the XYEP.

Now suppose that $P\subseteq Q\in\mathscr{C}$ and $\bar{f}:P\xrightarrow{}\mathcal{M}$ is a multifunction of type $\overline{\text{X}}$ with finite image $P\bar{f}$. As $P\bar{f}$ is finite, it follows that there exists $u\in[\mathbf{2}]$ such that $P\bar{f}\subseteq M_{u}$. Furthermore, there exists a triple $(P_{uv},Q_{uv},\bar{f}_{uv})\in L_{2}$ such that there exists an isomorphism $\eta:Q\to Q_{uv}$ with $P\eta|_{P}=P_{uv}$ and $\bar{f}=\eta|_{P}\bar{f}_{uv}$. Define $w=\beta_{2}(u,v)$; as $w\geq u$, then $P\bar{f}=P_{uv}\bar{f}_{uv}\subseteq M_{u}\subseteq M_{w}$. Here, $M_{w+1}$ is constructed by $\overline{\text{XY}}$-amalgamating $M_{w}$ and $Q_{uv}$ over $P_{uv}$; this amalgamation extends the multifunction $\bar{f}_{uv}$ of type $\overline{\text{X}}$ to the multifunction $\bar{g}_{uv}:Q_{uv}\to M_{w+1}$ of type $\overline{\text{Y}}$. Consequently, the multifunction $\bar{g}=\eta\bar{g}_{uv}:Q\to\mathcal{M}$ of type $\overline{\text{Y}}$ extends the multifunction $\bar{f}$ of type $\overline{\text{X}}$ (see Figure 10 for a diagram). As $\mathcal{M}$ has both the XYEP and the $\overline{\text{XY}}$EP, it follows that $\mathcal{M}$ is XZ-homogeneous by Section 4. ∎

Finally, we show part (3) of Theorem 1.2. Using the fact that XZ-homogeneous structures have two extension properties, we can ensure that a map between two of them is surjective by using a back-and-forth argument. This motivates a new definition, building on that of Y-equivalence.

Definition \thedefinition.

Let $\mathcal{M,N}$ be $\sigma$-structures, and suppose that Z $\in\{$E, B, A$\}$ corresponds to the surjective map of type Y $\in\{$H, M, I$\}$ via the relation $\mathscr{S}$. Say that $\mathcal{M}$ and $\mathcal{N}$ are Z-equivalent if Age$(\mathcal{M})$ = Age$(\mathcal{N})$ and every embedding $f:A\xrightarrow{}\mathcal{N}$ from a finite substructure $A$ of $\mathcal{M}$ into $\mathcal{N}$ extends to a surjective map $g:\mathcal{M}\xrightarrow{}\mathcal{N}$ of type Y, and vice versa.

For an example, $\mathcal{M,N}$ are B-equivalent means that they are bi-equivalent in the sense of [12]. Note that if two structures $\mathcal{M}$ and $\mathcal{N}$ are Z-equivalent, then they are also Y-equivalent where (Z,Y) $\in\mathscr{S}$ (from Equation 1).

Proposition \theproposition (Theorem 1.2 (3)).

Suppose that XZ $\in\mathfrak{B}\cap\mathfrak{I}$.

1. (1)

Assume that $\mathcal{M,N}$ are Z-equivalent. Then $\mathcal{M}$ is XZ-homogeneous if and only if $\mathcal{N}$ is. 2. (2)

If $\mathcal{M,N}$ are XZ-homogeneous and Age$(\mathcal{M})$* = Age*$(\mathcal{N})$, then $\mathcal{M}$ and $\mathcal{N}$ are Z-equivalent.

Proof.

(1) As $\mathcal{M,N}$ are Z-equivalent they are also Y-equivalent; as $\mathcal{M}$ is also XY-homogeneous, so is $\mathcal{N}$ by Section 2. By Section 2, it follows that $\mathcal{N}$ has the XYEP. We show now that $\mathcal{N}$ has the $\overline{\text{XY}}$EP. Suppose that $A\subseteq B\in$ Age$(\mathcal{N})$ and there exists a multifunction $\bar{f}:A\xrightarrow{}A^{\prime}\subseteq\mathcal{N}$ of type $\overline{\text{X}}$ with $A^{\prime}$ finite. Note that $A$ need not be isomorphic to $A^{\prime}$. As Age$(\mathcal{M})=$ Age$(\mathcal{N})$ there exists a copy $A^{\prime\prime}$ of $A^{\prime}$ in $\mathcal{M}$; fix an isomorphism $e:A^{\prime}\to A^{\prime\prime}$ between the two. Therefore, $e$ is a isomorphism from a finite structure of $\mathcal{N}$ into $\mathcal{M}$; as the two are Z-equivalent, we extend this to a surjective map $\alpha:\mathcal{N}\to\mathcal{M}$ of type Y. This in turn induces a surjective map $\bar{\alpha}:\mathcal{M}\xrightarrow{}\mathcal{N}$ by Section 3. Note that $\bar{\alpha}$ extends the isomorphism $e^{-1}:A^{\prime\prime}\to A^{\prime}$. Now, define $\bar{h}=\bar{f}e:A\to A^{\prime\prime}$; this is a multifunction of type $\overline{\text{X}}$ from $A$ into $\mathcal{M}$ with $A\bar{h}=A^{\prime\prime}$ finite. Since $\mathcal{M}$ is XZ-homogeneous, it has the $\overline{\text{XY}}$EP by Section 4 and so we extend $\bar{h}$ to a multifunction $\bar{h}^{\prime}:B\to\mathcal{M}$ of type $\overline{\text{Y}}$. Here, the multifunction $\bar{h}^{\prime}\bar{\alpha}:B\xrightarrow{}\mathcal{N}$ is also of type $\overline{\text{Y}}$; we need to show it extends $\bar{f}$. As $\bar{h}^{\prime}$ extends $\bar{h}=\bar{f}e$, it follows that:

[TABLE]

for all $a\in A$. Therefore $\mathcal{N}$ has the XZEP.

(2) It is enough to show that $\mathcal{N}$ has the XYEP and the $\overline{\text{XY}}$EP by Section 4. We utilise a back-and-forth argument constructing the surjective map over infinitely many stages. Let $f:A\xrightarrow{}B$ be a bijective embedding from a finite structure $A\subseteq\mathcal{M}$ to a finite substructure $B\subseteq\mathcal{N}$. Set $A=A_{0}$, $B=B_{0}$ and $f=f_{0}$ and assume that $f_{k}:A_{k}\xrightarrow{}B_{k}$ is a surjective map of type Y (and so of type X by assumption) extending $f_{k}$. Note that as both $\mathcal{M}$ and $\mathcal{N}$ are countable, then there exists enumerations $\mathcal{M}=\{m_{0},m_{1},\ldots\}$ and $\mathcal{N}=\{n_{0},n_{1},\ldots\}$.

If $k$ is even, select a $m_{i}\in\mathcal{M}\smallsetminus A_{k}$, where $i$ is the smallest natural number such that $m_{i}\notin A_{k}$. So $A_{k}\cup\{m_{i}\}\subseteq\mathcal{M}$, and is also in Age$(\mathcal{N})$ by assumption. As $\mathcal{N}$ is XZ-homogeneous it has the XYEP by Section 2 and we use this to extend $f_{k}$ to a map $f_{k+1}^{\prime}:A_{k}\cup\{m_{i}\}\xrightarrow{}\mathcal{N}$ of type Y. Restricting the codomain of $f_{k+1}^{\prime}$ to its image yields a surjective map $f_{k+1}:A_{k}\cup\{m_{i}\}\xrightarrow{}B_{k}\cup\{m_{i}f_{k+1}\}$ of type Y. If $k$ is odd, select a $n_{i}\in\mathcal{N}\smallsetminus B_{k}$ such that $i$ is the smallest natural number such that $n_{i}\notin B_{k}$. Hence $B_{k}\cup\{n_{i}\}\subseteq\mathcal{N}$ and thus it is an element of Age$(\mathcal{M})$ by assumption. As $f_{k}$ is a surjective map of type Y, its converse $\bar{f}_{k}:B_{k}\xrightarrow{}A_{k}$ is a surjective multifunction of type $\overline{\text{Y}}$ by Section 3, and of type $\overline{\text{X}}$ by assumption. As $\mathcal{M}$ is XZ-homogeneous it has the $\overline{\text{XY}}$EP and so we can extend $\bar{f}_{k}$ to a multifunction $\bar{f}_{k+1}^{\prime}:B_{k}\cup\{n_{i}\}\xrightarrow{}\mathcal{M}$ of type $\overline{\text{Y}}$. By restricting the codomain of $\bar{f}_{k+1}^{\prime}$ to its image, we obtain a surjective multifunction $\bar{f}_{k+1}:B_{k}\cup\{n_{i}\}\xrightarrow{}A_{k}\cup n_{i}\bar{f}_{k+1}^{\prime}$ of type $\overline{\text{Y}}$, where $n_{i}\bar{f}_{k+1}^{\prime}=\{(n_{i},y)$ : $y\in\mathcal{M}\}$ is a non-empty set. So by Section 3, there exists a surjective map $f_{k+1}:A_{k}\cup n_{i}\bar{f}_{k+1}^{\prime}\to B_{k}\cup\{n_{i}\}$ of type Y extending $f_{k}$. By our earlier assumption, as a map of type Y is also a map of type X, we can repeat this process infinitely many times. By ensuring all points in $\mathcal{M}$ appear at even stages and all points in $\mathcal{N}$ appear at odd stages, we construct a surjective map $\alpha:\mathcal{M}\xrightarrow{}\mathcal{N}$ of type Y as required. We can use a similar method to show that we can extend any embedding $g:A\xrightarrow{}B$, where $A\in\mathcal{N}$ and $B\in\mathcal{M}$, to a surjective map of type Y; proving that $\mathcal{M}$ and $\mathcal{N}$ are Z-equivalent. ∎

Of course, the open problem that arises from Sections 2 and 4 is:

Question \thequestion.

Can we expand Theorem 1.1 and Theorem 1.2 to include those homomorphism-homogeneity classes in $\mathfrak{B}$?

5 Maximal homomorphism-homogeneity classes

This section is devoted to determining the extent to which well known examples of homogeneous structures are also homomorphism-homogeneous. In some cases, verifying that a structure $\mathcal{M}$ is homogeneous involves using a property of $\mathcal{M}$ to determine that $\mathcal{M}$ has the EP, and so is homogeneous. Good examples of such properties are the density of $(\mathbb{Q},<)$, and Alice’s restaurant property characteristic of $R$ (see Subsection 5.1). In the homomorphism-homogeneity case, this idea was used by Cameron and Lockett [8] and Lockett and Truss [25] to classify homomorphism-homogeneous posets and determine their position relative to the natural containment order on $\mathbb{H}$ (see Figure 1). In addition to this, Dolinka [14] used properties of known homogeneous structures to show that they satisfied the one-point homomorphism extension property (1PHEP), a necessary and sufficient condition for HH-homogeneity. Our approach in this section is similar to that of Section 3 of [14]; by defining necessary and sufficient conditions for XY and XZ-homogeneity and using properties of structures to show that these are satisfied or not satisfied. As in Section 4, we let X,Y $\in\{$H, M, I$\}$, $\overline{\text{X}},\overline{\text{Y}}\in\{\overline{\text{H}},\overline{\text{M}},\overline{\text{I}}\}$, and Z $\in\{$E, B, A$\}$ throughout this section. Furthermore, the pair (Z,Y)$\in\mathscr{S}$ is related as in Equation 1 on page 1.

So to begin this section, we define the one-point XY-extension property, and the one-point $\overline{\text{XY}}$-extension property:

(1PXYEP) We say that a $\sigma$-structure $\mathcal{M}$ with age $\mathscr{C}$ has the 1PXYEP if for all $A\subseteq B\in\mathscr{C}$ with $|B\smallsetminus A|=1$ and maps $f:A\xrightarrow{}\mathcal{M}$ of type X, there exists a map $g:B\xrightarrow{}\mathcal{M}$ of type Y extending $f$.

(1P$\overline{\text{XY}}$EP) Suppose that $\mathcal{M}$ is a $\sigma$-structure with age $\mathscr{C}$. Say that $\mathcal{M}$ has the 1P$\overline{\text{XY}}$EP if for all $A\subseteq B\in\mathscr{C}$ with $|B\smallsetminus A|=1$, and a multifunction $\bar{f}:A\xrightarrow{}\mathcal{M}$ of type $\overline{\text{X}}$ with $A\bar{f}$ finite, there exists a multifunction $\bar{g}:B\xrightarrow{}\mathcal{M}$ of type $\overline{\text{Y}}$ extending $\bar{f}$.

For an example, the 1PHHEP is the same thing as the 1PHEP of [14]. These properties, together with the next proposition, provide some of the theoretical basis for the examples that follow.

Proposition \theproposition.

Suppose that XY $\in\mathfrak{I}$. A countable $\sigma$-structure $\mathcal{M}$ has the XYEP / $\overline{\text{XY}}$EP if and only if it has the 1PXYEP / 1P$\overline{\text{XY}}$EP.

Proof.

The forward direction for both the XYEP and $\overline{\text{XY}}$EP cases is clear. We now aim to show that if $\mathcal{M}$ has the 1PXYEP then $\mathcal{M}$ has the XYEP. Assume that $A\subseteq B$ with $|B\smallsetminus A|=n$, and $f:A\to\mathcal{M}$ is a map of type X. We prove the result by induction on the size of this complement; the base case (where $n$ = 1) is true by the assumption that $\mathcal{M}$ has the 1PXYEP.

So suppose that for some $k\in\mathbb{N}$, for any $A\subseteq B\in\mathscr{C}$ where $|B\smallsetminus A|=k$ and any map $f:A\to\mathcal{M}$ of type X can be extended to a map $g:B\to\mathcal{M}$ of type Y. Take $P\subseteq Q\in\mathscr{C}$ where $|Q\smallsetminus P|=k+1$ and $f^{\prime}:P\to\mathcal{M}$ to be some map of type X. Now, there exists $S\in\mathscr{C}$ containing $P$ such that $|Q\smallsetminus S|=1$. By the inductive hypothesis, we can extend $f^{\prime}$ to a map $h:S\to\mathcal{M}$ of type Y. As XY $\in\mathfrak{I}$, it follows that $h$ is also a map of type X. Now, using the 1PXYEP, extend $h$ to a map $g^{\prime}:Q\to\mathcal{M}$ of type Y. Since $P\subseteq S\subseteq Q$ and $g^{\prime}$ extends $h$ which extends $f^{\prime}$, we have that $g^{\prime}$ extends $f^{\prime}$ and so we are done. Using a similar argument, we can show that if $\mathcal{M}$ has the 1P$\overline{\text{XY}}$EP then it has the $\overline{\text{XY}}$EP. ∎

Remark*.*

Let XY $\in\mathfrak{I}$. Together with Section 2, this result states that a countable structure $\mathcal{M}$ has the 1PXYEP if and only if $\mathcal{M}$ is XY-homogeneous. Similarly, by Section 4 a countable structure $\mathcal{M}$ is XZ-homogeneous if and only if it has the 1PXYEP and the 1P$\overline{\text{XY}}$EP, where (Z,Y) are as in $\mathscr{S}$ (Equation 1 on page 1).

By considering properties of partial maps and endomorphisms of structures, our next result places restrictions on certain types of homomorphism-homogeneity. We look at structures known as cores; a structure $\mathcal{M}$ is a core if every endomorphism of $\mathcal{M}$ is an embedding [4]. Widely studied examples of cores include the countable dense linear order without endpoints $(\mathbb{Q},<)$, the complete graph on countably many vertices $K^{\aleph_{0}}$, the $K^{n}$-free homogeneous graphs for $n\geq 3$ [32] and the Henson digraphs $M_{T}$ [12]. This straightforward result includes a restatement of Lemma 1.1 of [25].

Lemma \thelemma.

Let $\mathcal{M}$ be a countable $\sigma$-structure.

1. (1)

$\mathcal{M}$* is MI and MA-homogeneous (HI and HA-homogeneous) if and only if $\mathcal{M}$ is IA-homogeneous and every finite partial monomorphism (homomorphism) of $\mathcal{M}$ is an isomorphism.* 2. (2)

If $\mathcal{M}$ is HM or HB-homogeneous, then every finite partial homomorphism of $\mathcal{M}$ is also a monomorphism. 3. (3)

Let $\mathcal{M}$ be a core. If there exists a finite partial monomorphism of $\mathcal{M}$ that is not an isomorphism, then $\mathcal{M}$ is not MH-homogeneous.

Proof.

(1) is contained in Lemma 1.1 of [25]; notice that we cannot extend a map that is not a partial isomorphism of $\mathcal{M}$ to an isomorphism of the entire structure $\mathcal{M}$. The converse direction is clear. To show (2), note that if $h$ is a finite partial homomorphism of $\mathcal{M}$ that is not injective, then we cannot possibly extend this to an injective map and so $\mathcal{M}$ does not have the HMEP. For (3), let $h$ be a finite partial monomorphism of a core $\mathcal{M}$ that is not an isomorphism. As any endomorphism of $\mathcal{M}$ is an embedding, we cannot extend $h$. ∎

Remark*.*

Note that (1) and (2) also follow from Theorem 1.1 and Theorem 1.2.

Following the approach of [25] in classifying homomorphism-homogeneous posets, the idea of this section is to look at properties of graphs and digraphs to determine “maximal” homomorphism-homogeneity classes with respect to the containment order on $\mathfrak{H}$. We formally define what we mean by “maximal”.

Definition \thedefinition.

Let $\mathcal{M}$ be a first-order structure. A homomorphism-homogeneity class XY $\in\mathbb{H}$ is maximal for $\mathcal{M}$ if $\mathcal{M}$ is XY-homogeneous and $\mathcal{M}$ is not PQ-homogenenous, where PQ $\subseteq$ XY in $\mathbb{H}$. If this happens, we say that XY is a maximal homomorphism-homogeneity class (shortened to mhh-class) for $\mathcal{M}$.

Remark*.*

While this definition describes a minimal element in the poset $\mathbb{H}$, it is so named because of the strengths of different notions of homomorphism-homogeneity. For instance, HA-homogeneity is a stronger condition than IA-homogeneity, but HA $\subseteq$ IA in $\mathbb{H}$. This reflects the inverse correspondence between the relative strength of notions of homomorphism-homogeneity in $\mathfrak{H}$ and containment of classes in $\mathbb{H}$ (see the discussion on page 1).

For example, if $\mathcal{M}$ is MB-homogeneous but not MA or HB-homogeneous, then MB is a mhh-class for $\mathcal{M}$. A structure $\mathcal{M}$ may have more than one mhh-class. The set of mhh-classes for $\mathcal{M}$ completely determines the extent of homomorphism-homogeneity satisfied by $\mathcal{M}$; we therefore denote this set by $\mathbb{H}(\mathcal{M})$. As an example $\mathbb{H}((\mathbb{Q},<))=\{$HA$\}$; this example arose from the classification of homomorphism-homogeneous posets in [25].

If $\mathcal{M}$ is a countable $\sigma$-structure where there exists a finite partial monomorphism of $\mathcal{M}$ that is not an isomorphism, and a finite partial homomorphism of $\mathcal{M}$ that is not an monomorphism, then Section 5 implies that the “best possible” mhh-classes for $\mathcal{M}$ are IA, MB and HE. As an aside, these classes have important roles to play in the theory of generic endomorphisms [24].

Before we investigate some examples in the context of this article, we note the following direct consequence of Section 5 (3) with respect to Section 5.

Corollary \thecorollary.

Let $\mathcal{M}$ be a countable homogeneous core. If there exists a finite partial monomorphism of $\mathcal{M}$ that is not an isomorphism, then $\mathbb{H}(\mathcal{M})=\{$IA$\}$. ∎

Remark*.*

It has been shown that every $K_{n}$-free graph (for $n\geq 3$) [32], every Henson digraph $M_{T}$ (for any set of tournaments on more than $3$ vertices) and the myopic local order $S(3)$ [12] are cores. By Section 5, it follows that the mhh-class for each of these structures is IA.

In the rest of this section, we look at a selection of countable homogeneous graphs and digraphs encountered throughout the literature in order to determine sets of mhh-classes for these structures. By restricting ourselves to classes XY $\in\mathfrak{I}$, we can recall Section 5 and the remark that follows it; to show that $\mathcal{M}$ is XY-homogeneous it suffices to show that $\mathcal{M}$ has the 1PXYEP, and to show that $\mathcal{M}$ is XZ-homogeneous it suffices to show that it has the 1PXYEP and the 1P$\overline{\text{XY}}$EP.

5.1 Graphs

In this article, a graph $\Gamma$ is a set of vertices $V\Gamma$ together with a set of edges $E\Gamma$, where this edge set interprets a irreflexive and symmetric binary relation $E$. For $n\in\mathbb{N}\cup\{\aleph_{0}\}$, recall that a complete graph $K^{n}$ on $n$ vertices is a graph on vertex set $VK^{n}$ (with $|VK^{n}|=n$) with edges given by $u\sim v\in K^{n}$ if and only if $u\neq v\in VK^{n}$. The complement of the graph $K^{n}$ is called the null graph on $n$ vertices or an independent set on $n$ vertices, and is denoted by $\bar{K}^{n}$. Recall that the complement of a graph $\Gamma=(V,E)$ is the graph $\bar{\Gamma}=(V,[V]^{2}\smallsetminus E)$; that is, the graph $\Gamma$ on vertex set $V$ where all the edges in $\bar{\Gamma}$ are non-edges of $\Gamma$ and vice versa. A subset $X$ of $V$ is an independent set if $x\nsim x^{\prime}$ for all $x,x^{\prime}\in X$. For more on the basics of graph theory, see [13].

Example \theexample.

It is well-known (see [23]) that the complete graph on countably many vertices $K^{\aleph_{0}}$ is homogeneous. Suppose that $h:A\xrightarrow{}B$ is a homomorphism between two finite substructures of $K^{\aleph_{0}}$. Then as $h$ preserves edges, it cannot send two distinct vertices $x_{1},x_{2}\in VA$ to a single point $v\in VB$; hence $h$ is injective. As there are no non-edges to preserve, it must preserve non-edges and so $h$ is an embedding. It follows from Section 5 (1) that $K^{\aleph_{0}}$ is HA-homogeneous and so $\mathbb{H}(K^{\aleph_{0}})=\{$HA$\}$.

Its complement $\bar{K}^{\aleph_{0}}$, the infinite null graph, is also homogeneous and as every finite partial monomorphism of $\bar{K}^{\aleph_{0}}$ preserves non-edges, it is MA-homogeneous by Section 5 (1). We note that there exist non-injective finite partial homomorphisms of $\bar{K}^{\aleph_{0}}$ and hence it is not HM or HB-homogeneous by Section 5 (3). So if $h:A\xrightarrow{}B$ is any finite partial homomorphism, we can define a bijective map $g:\bar{K}^{\aleph_{0}}\smallsetminus A\xrightarrow{}\bar{K}^{\aleph_{0}}\smallsetminus B$ and note that the map $\alpha:\bar{K}^{\aleph_{0}}\xrightarrow{}\bar{K}^{\aleph_{0}}$ that acts like $h$ on $A$ and $g$ everywhere else is an epimorphism of $\bar{K}^{\aleph_{0}}$; so $\bar{K}^{\aleph_{0}}$ is HE. Hence $\mathbb{H}(\bar{K}^{\aleph_{0}})=\{$MA, HE$\}$.

Example \theexample.

Let $R$ be the random graph (see [7], for instance). Note that there exist finite partial monomorphisms of $R$ that are not isomorphisms and finite partial homomorphisms of $R$ that are not monomorphisms; hence $R$ is not MI or HM-homogeneous by Section 5. It was shown in [12] that $R$ is MB-homogeneous and in [9] that $R$ is HH-homogeneous; here, we show that $R$ is HE-homogeneous. To do this, we rely on the Alice’s restaurant property characteristic of $R$ (see [7]), which says:

(ARP) For any finite, disjoint subsets $U,V\subseteq V\Gamma$, there exists $x\in V\Gamma$ such that $x\sim u$ for all $u\in U$ and $x\nsim v$ for all $v\in V$.

Let $A\subseteq B\in$ Age$(R)$ with $B\smallsetminus A=\{b\}$ and suppose that $\bar{f}:A\xrightarrow{}R$ is an antihomomorphism such that $A\bar{f}$ is finite. Using ARP, we can find a vertex $v\in VR$ such that $v$ is independent of everything in $A\bar{f}$. Let $\bar{g}:B\xrightarrow{}R$ be the multifunction defined by such that $b\bar{g}=v$ and $\bar{g}|_{A}=\bar{f}$; this is an antihomomorphism as all non-edges from $A$ to $b$ are preserved. Therefore, $R$ has the 1P$\overline{\text{HH}}$EP and so $R$ is HE-homogeneous by Section 5 and Section 4. We conclude that $\mathbb{H}(R)=\{$IA, MB, HE$\}$.

Remark*.*

It was shown in [24, Theorem 5.3] that $R$ has a generic endomorphism. As $R$ is HE-homogeneous, it follows from Theorem 2.1 of the same source that this generic endomorphism must be in Epi$(R)$.

Example \theexample.

Let $H$ be the complement of the homogeneous $K_{n}$-free graph for $n\geq 3$. It was previously shown that $H$ is MM but not MB-homogeneous [12]. A result of [14] shows that $H$ has the 1PHHEP, and so is both MM and HH-homogeneous by these two results.

We now show that $H$ does not have the 1P$\overline{\text{HH}}$EP and hence cannot be HE-homogeneous. Let $A=\{a\}$ be a single vertex, and let $B$ be the independent pair of vertices $\{a,b\}$. Note that $A\subseteq B\in$ Age$(H)$. Let $\bar{f}:A\xrightarrow{}H$ be an antihomomorphism sending $A=\{a\}$ to an independent set $A\bar{f}$ of $n-1$ vertices in $H$; such a substructure exists by definition of $H$. Then as antihomomorphisms preserve non-edges and cannot send two points in a domain to a single point in the codomain, a potential image point for $b$ in $H$ must be a vertex $x$ independent of $A\bar{f}$; this cannot happen as $H$ would then induce an independent $n$-set. So $H$ does not have the 1P$\overline{\text{HH}}$EP. Therefore, $H$ is not HE-homogeneous and we see that $\mathbb{H}(H)=\{$IA, MM, HH$\}$.

The next result, detailing the rest of the disconnected, countably infinite homogeneous graphs, extend results of [9] and [35].

Proposition \theproposition.

Let $\Gamma=\bigsqcup_{i\in I}K^{n}_{i}$ be a disjoint union of $|I|$ many complete graphs, each of which have size $n\in\mathbb{N}\cup\{\aleph_{0}\}$.

1. (1)

If $|I|=m$ for some $m\in\mathbb{N}$ and $n=\aleph_{0}$, then $\mathbb{H}(\Gamma)=\{$IA, MM, HH$\,\}$. 2. (2)

If $|I|=\aleph_{0}$ and $n\in\mathbb{N}$, then $\mathbb{H}(\Gamma)=\{$IA, HE$\,\}$. 3. (3)

If $|I|=n=\aleph_{0}$, then $\mathbb{H}(\Gamma)=\{$IA, MB, HE$\,\}$.

Proof.

1. (1)

It is shown in [12] that the finite disjoint union of infinite complete graphs is MM but not MB-homogeneous. Furthermore, [9, Proposition 1.1] asserts that $\Gamma$ is HH-homogeneous. It remains to prove that $\Gamma$ is not HE-homogeneous; here, it is enough to show that $\Gamma$ does not have the 1P$\overline{\text{HH}}$EP by Section 5 and Section 4. As $\Gamma$ in this case does not embed an independent $n+1$-set, the proof of this is similar to Subsection 5.1. 2. (2)

In this case, $\Gamma$ is HH-homogeneous by [9, Proposition 1.1], but not MM-homogeneous by Proposition 2.5 of the same paper. We show that $\Gamma$ has the 1P$\overline{\text{HH}}$EP in this case; proving that $\Gamma$ is HE-homogeneous. Suppose that $A\subseteq B\in$ Age$(\Gamma)$ with $B\smallsetminus A=\{b\}$ and that $\bar{f}:A\to\Gamma$ is an antihomomorphism such that $A\bar{f}$ is finite. As $A\bar{f}$ is finite, it follows that $A\bar{f}\subseteq\bigsqcup_{j\in J}K^{n}_{j}$ for some finite set $J\subseteq I$. Select a $v\in K^{n}_{k}$, where $k\in I\smallsetminus J$; so $v$ is independent of every element of $A\bar{f}$. Define $\bar{g}:B\to\Gamma$ to act like $\bar{f}$ on $A$ with $b\bar{g}=v$. Then $\bar{g}$ is an antihomomorphism and so $\Gamma$ has the 1P$\overline{\text{HH}}$EP. 3. (3)

The proof that $\Gamma$ is MB-homogeneous can be found in [12, Proposition 3.4]; the proof that it is HE-homogeneous is similar to part (2).

∎

The only countable homogeneous graphs that have not yet been considered are the complements of the disconnected graphs described in Subsection 5.1. The following result deals with these cases. Recall from [35, Theorem 3, Theorem 5] that a countable graph is MH-homogeneous if and only if it is HH-homogeneous; if this graph is also connected, then it is HH-homogeneous if and only if it is MM-homogeneous.

Proposition \theproposition.

Let $\bar{\Gamma}=\overline{\bigsqcup_{i\in I}K^{n}_{i}}$ be the complement of a disjoint union of $|I|$ many complete graphs, each of which have size $n\in\mathbb{N}\cup\{\aleph_{0}\}$.

1. (1)

If $|I|=m$ for some $m\in\mathbb{N}$ and $n=\aleph_{0}$, then $\mathbb{H}(\bar{\Gamma})=\{$IA$\}$. 2. (2)

If $|I|=\aleph_{0}$ and $n\in\mathbb{N}$, then $\mathbb{H}(\bar{\Gamma})=\{$IA, MM, HH$\}$. 3. (3)

If $|I|=n=\aleph_{0}$, then $\mathbb{H}(\bar{\Gamma})=\{$IA, MB, HE$\}$.

Proof.

1. (1)

Notice that $\bar{\Gamma}$ does not contain an infinite complete graph; therefore it is not MM-homogeneous by [9, Proposition 2.5]. Since $\bar{G}$ is connected, it is not MH-homogeneous by the results of [35] mentioned above. 2. (2)

In this case, any finite set $U\subseteq\bar{\Gamma}$ of vertices has a vertex $x$ such that $x$ is adjacent to every element of $U$. (This is property $(\triangle)$ of [12].) Therefore, $\bar{G}$ is MM and HH-homogeneous by [9, Proposition 2.1]. If $\bar{\Gamma}$ were MB-homogeneous, then $\Gamma$ would be MB-homogeneous by [12, Proposition 3.1]; contradicting Subsection 5.1. So $\Gamma$ is not MB-homogeneous. It remains to show that $\bar{\Gamma}$ is not HE-homogeneous; the proof of this is similar to Subsection 5.1 (1). 3. (3)

This is MB-homogeneous by the remark following [12, Proposition 3.4]. It can be shown that this is HE-homogeneous using a similar argument to part (2) of Subsection 5.1.

∎

Remark*.*

Notice that part (2) of both Propositions 5.1 and 5.1 show that the complement of a HE-homogeneous graph is not necessarily HE-homogeneous.

These results mean that we have determined the mhh-classes for all countable homogeneous graphs in Lachlan and Woodrow’s classification [23]; the results are summarised in Table 2.

5.2 Digraphs

Following the results of Subsection 5.1 and similar work on posets [25], the next natural direction would be to determine mhh-classes for the countable homogeneous digraphs. Unless stated otherwise, in this article a digraph $\Delta=(V\Delta,A\Delta)$ is a set of vertices $V\Delta$ together with a set $A\Delta\subseteq V\Delta^{2}$ of ordered pairs, called arcs of the digraph. For two vertices $x,y$ of $\Delta$, we write $x\to y$ if $(x,y)\in A\Delta$, and $x\parallel y$ if neither $(x,y)$ nor $(y,x)$ are in $A\Delta$. All the digraphs in this article are loopless; so for all $x\in\Delta$, it follows that $(x,x)\notin\Delta$. Additionally, we stipulate that they do not contain $2$-cycles – there is a 2-cycle between $x$ and $y$ if and only if $x\to y$ and $y\to x$ – with the notable exception of Subsection 5.2.

We provide a few introductory observations here, and leave the further development of the subject as an open question. Our first example deals with the countable homogeneous tournaments, classified in [22].

Example \theexample.

Recall that a tournament is defined to be an oriented, loopless complete graph. By a similar argument to the complete graph in Subsection 5.1, every finite partial homomorphism of a tournament is an embedding. It follows from Section 5 (1) that every countable homogeneous tournament is HA-homogeneous. Therefore, the three countable homogeneous tournaments as classified by Lachlan [22], namely $(\mathbb{Q},<)$, the random tournament $\mathbb{T}$, and the local order $S(2)$, are all HA-homogeneous. So HA is the unique mhh-class for these three examples.

Example \theexample.

Let $D$ be the generic digraph without $2$-cycles; for a detailed definition, see [1]. This is the unique countable homogeneous structure whose age contains all finite digraphs without $2$-cycles. This structure is also MB-homogeneous [12, Example 4.4].

However, $D$ is not HH-homogeneous. First of all, as there exist finite partial homomorphisms of $D$ that are not monomorphisms (such as an independent $2$-set being mapped to a single point), $D$ is not HM-homogeneous by Section 5 (2). To show that $D$ is not HH-homogeneous, it is enough to prove that every endomorphism of $D$ is a monomorphism. Consider $\gamma\in$ End$(D)$, and suppose there exists $v\parallel w\in VD$ such that $v\gamma=w\gamma$. As $D$ is universal and homogeneous, there exists an oriented graph $A=\{v,w,x\}$ such that $x\xrightarrow{}v$ and $w\xrightarrow{}x$ (see Figure 11).

The image of $A$ under $\gamma$ is a 2-cycle and this is a contradiction as $D$ does not embed $2$-cycles. It follows that every endomorphism of $D$ is a monomorphism and so $D$ is not HH-homogeneous. We conclude that the mhh-classes of $D$ are IA and MB.

Corollary \thecorollary.

Let $\Delta$ be a countable connected homogeneous digraph that embeds the digraph $A$ as in Figure 11. Then Mon$(\Delta)=$ End$(\Delta)$ and $\Delta$ is not HH-homogeneous. ∎

Remark*.*

It was shown in [12] that the $I_{n}$-free digraph for $n\geq 3$ is MM but not MB-homogeneous. It follows from Subsection 5.2 that the mhh-classes for these digraphs are IA and MM.

In the disconnected case, the situation is slightly different. Here is an example of a HE-homogeneous digraph.

Example \theexample.

Let $\mathbb{T}$ be the countable, homogeneous random tournament (see [22]). Using homogeneity of $\mathbb{T}$, we can show that $\mathbb{T}$ satisfies the following property $(\leftrightarrows)$:

(Property $(\leftrightarrows)$) For all finite, disjoint subsets $U,V$ of $V\mathbb{T}$ there exists $x\in V\mathbb{T}$ such that $x\to u$ for all $u$ in $U$ and $v\to x$ for all $v\in V$.

Now, let $\mathcal{T}=\bigsqcup_{i\in\mathbb{N}}\mathbb{T}_{i}$ be the infinite disjoint union of isomorphic copies $\mathbb{T}_{i}$ of the countable random tournament. Our aim is to show that $\mathcal{T}$ is HE-homogeneous; so it is enough to show that $\mathcal{T}$ has both the 1PHHEP and 1P$\overline{\text{HH}}$EP by Section 5 and Section 4. Suppose then that $A\subseteq B\in$ Age$(\mathcal{T})$ is such that $B\smallsetminus A=\{b\}$. As $A\in$ Age$(\mathcal{T})$, we can write $A=\bigsqcup_{j\in J}T_{j}$, where each $T_{j}$ is a finite tournament and $J$ is finite. Let $f:A\to\mathcal{T}$ be a homomorphism. There are two cases to consider; either $b$ is independent of every tournament in $A$, or it is not. If $b$ is independent of every tournament in $A$, then choose any vertex $v\in V\mathcal{T}\smallsetminus Af$; the function $g:B\to\mathcal{T}$ acting like $f$ on $A$ and sending $b$ to $v$ is a homomorphism. If $b$ is not independent of $A$, then there is only one tournament $T_{j}\subseteq A$ that is related to $b$. Partition $T_{j}$ into two sets

[TABLE]

As $T_{j}$ is a tournament, it follows that $f|_{T_{j}}:T_{j}\to T_{j}f$ is bijective and so $T_{j}^{\rightarrow}(b)f$ and $T_{j}^{\leftarrow}(b)f$ partition $T_{j}f$. As $f:A\to\mathcal{T}$ is a homomorphism, $T_{j}f$ is a finite subtournament of $\mathbb{T}_{i}$ for some $i\in\mathbb{N}$; so both $T_{j}^{\rightarrow}(b)f$ and $T_{j}^{\leftarrow}(b)f$ are finite. We can use property $(\leftrightarrows)$ of $\mathbb{T}_{i}$ to find a vertex $w$ such that $cf\to w$ for all $cf\in T_{j}^{\rightarrow}(b)f$ and $w\to df$ for all $df\in T_{j}^{\leftarrow}(b)f$. Now, define a function $g:B\to\mathcal{T}$ that acts like $f$ on $A$ and sends $b$ to $w$; here, $g$ preserves all relations and so is a homomorphism. Therefore, $\mathcal{T}$ has the 1PHHEP.

The proof that $\mathcal{T}$ has the 1P$\overline{\text{HH}}$EP follows from a similar argument to Subsection 5.1 (2). Hence $\mathcal{T}$ is HE-homogeneous by Section 5 and Section 4. Adapting this proof for the 1PMMEP and 1P$\overline{\text{MM}}$EP, we can show that $\mathcal{T}$ is MB-homogeneous; so $\mathbb{H}(\mathcal{T})=\{$IA, MB, HE$\}$.

Changing our definition of digraph to include $2$-cycles also increases the flexibility of the structure.

Example \theexample.

Let $D^{*}$ be the generic digraph with $2$-cycles; similar to $D$, it is the unique countable homogeneous digraph whose age contains all finite digraphs with $2$-cycles. Recall (from [31, ch4] or [11, ch2]) that $D^{*}$ has a characteristic extension property known as the directed Alice’s restaurant property (DARP), which says:

(DARP) For any finite and pairwise disjoint sets of vertices $U,V,W,X$ of $D^{*}$, there exists a vertex $z$ of $D^{*}$ such that: there is an arc from $z$ to every element of $U$, an arc to $z$ from every element of $V$, a 2-cycle between $z$ and every element of $W$, and $z$ is independent of every vertex in $X$. (See Figure 12 for a diagram of an example.)

It was mentioned in [12] that $D^{*}$ is MB-homogeneous. Using the DARP, we show that $D^{*}$ is HE-homogeneous. Let $A\subseteq B\in$ Age$(D^{*})$ with $B\smallsetminus A=\{b\}$ and suppose that $f:A\xrightarrow{}D^{*}$ is a homomorphism. As $Af$ is finite, we can use DARP to find a vertex $v\in VD^{*}$ such that there is a 2-cycle between $v$ and every element in $Af$. Let $g:B\xrightarrow{}D^{*}$ be the map such that $bg=v$ and $g|_{A}=f$; this is a homomorphism as all arcs from $A$ to $b$ are preserved. Therefore $D^{*}$ has the 1PHHEP. The proof to show that $D^{*}$ has the 1P$\overline{\text{HH}}$EP is similar; we use DARP to instead find a vertex $w\in VD^{*}$ that is independent of the finite set $Af$. The resulting multifunction $\bar{g}$ is an antihomomorphism as it preserves all non-relations. Therefore, $D^{*}$ is HE-homogeneous by Section 5 and Section 4. So $\mathbb{H}(D^{*})=\{$IA, MB, HE$\}$.

Remark*.*

Note the difference between the mhh-classes of $D$, the generic digraph without 2-cycles, and $D^{*}$, the generic digraph with 2-cycles.

The work in this section leads to a natural open question.

Question \thequestion.

Investigate countable homomorphism-homogeneous digraphs (both with and without $2$-cycles) in more detail. In particular, determine the mhh-classes for those countable homogeneous digraphs in Cherlin’s classification [10].

Acknowledgements: The author would like to thank Christian and Maja Pech for pointing out an oversight in the introduction.