Hereditary classes of ordered binary structures

TL;DR

This paper investigates hereditary classes of ordered binary structures, establishing a dichotomy in their profile growth based on the existence of finite monomorphic decompositions, with implications for class complexity and structure embeddings.

Contribution

It introduces the notion of monomorphic decomposition to analyze hereditary classes, proving a finite basis for structures without such decompositions and characterizing their profile growth.

Findings

Structures without finite monomorphic decomposition have a finite basis.

Profiles of certain classes grow at least as fast as the Fibonacci sequence.

A dichotomy exists: polynomial-bounded profiles versus Fibonacci lower bounds.

Abstract

Balogh, Bollob\'{a}s and Morris (2006) have described a threshold phenomenon in the behavior of the profile of hereditary classes of ordered graphs. In this paper, we give an other look at their result based on the notion of monomorphic decomposition of a relational structure introduced in \cite{P-T-2013}. We prove that the class $\mathfrak S$ of ordered binary structures which do not have a finite monomorphic decomposition has a finite basis (a subset $\mathfrak A$ such that every member of $\mathfrak S$ embeds some member of $\mathfrak A$). In the case of ordered reflexive directed graphs, the basis has 1242 members and the profile of their ages grows at least as the Fibonacci function. From this result, we deduce that the following dichotomy property holds for every hereditary class $\mathfrak C$ of finite ordered binary structures of a given finite type. Either there is an integer $\ell$ such that every member of $\mathfrak C$ has a monomorphic decomposition into at most $\ell$ blocks and in this case the profile of $\mathfrak C$ is bounded by a polynomial of degree $\ell-1$ (and in fact is a polynomial), or $\mathfrak C$ contains the age of a structure which does not have a finite monomorphic decomposition, in which case the profile of $\mathfrak C$ is bounded below by the Fibonacci function.

Equations

$${\rho }_i({\psi }_1({\overrightarrow{\alpha }}_1),...,{\psi }_{m_i}({\overrightarrow{\alpha }}_{m_i}))={\rho }_i({\psi }_1(h({\overrightarrow{\alpha }}_1)),...,{\psi }_{m_i}(h({\overrightarrow{\alpha }}_{m_i})))$$

$$\rho (\psi (\overrightarrow{\alpha }),\psi (\overrightarrow{\beta }))=\rho (\psi (h(\overrightarrow{\alpha })),\psi (h(\overrightarrow{\beta })))$$

$$\begin{array}{r}\text{For every local isomorphism}f\text{of}(L,\le ),\text{the map}(f,Id)\text{on}L\times K\\ \text{extended}\text{by the identity on}F\text{induces a local isomorphism of}\mathcal{R}.\end{array}$$

$$d(x,y)=d(x^{\prime },y)$$

$$d(x,z)=d(z,y)\text{for every}z\in ]x,y[$$

$$d(x,z)=d(y,z)\text{ for every}z\notin [x,y].$$

$$x{\le }^{(1)}y\iff F(x)\le F(y)\text{and}{d}^{(1)}(x,y)=d(F(x),F(y))$$

$${\mathcal{R}}^{(1)}{\upharpoonright }_{\{(n,0),(n,1)\}}\text{ and }{\mathcal{R}}^{(1)}{\upharpoonright }_{\{(m,0),(n,1)\}}\text{ are not isomorphic}$$

$$d^{(1)}((0,0),(0,1))\ne d^{(1)}((0,1),(1,0)).$$

$$x{\le }^{(2)}y\iff F(x)\le F(y)\text{and}{d}^{(2)}(x,y)=d(F(x),F(y))$$

$$\forall n<m\in \mathbb{N},\text{leavevmode}{\mathcal{R}}^{(2)}{\upharpoonright }_{\{a,(n,0),(n,1)\}}\text{ and }{\mathcal{R}}^{(2)}{\upharpoonright }_{\{a,(n,1),(m,0)\}}\text{ are not isomorphic.}$$

$$x{\le }^{(3)}y\iff F^{\prime \prime }(x)\le F^{\prime \prime }(y)\text{and}{d}^{(3)}(x,y)=d(F^{\prime \prime }(x),F^{\prime \prime }(y))$$

$$d^{(3)}((n,0),(m,0))\ne d^{(3)}((n,0),(m,1)),$$

$$d^{(3)}((n,0),(m,0))\ne d^{(3)}((n,1),(m,1)),$$

$$d^{(3)}((n,0),(m,1))\ne d^{(3)}((n,1),(m,1)).$$

Full text

*The author was supported by CMEP-Tassili grant

AMS Subject Classification: 05C30, 06F99, 05A05, 03C13.

Keywords: profile, monomorphic decomposition, ordered graphs, ordered binary relational structures, ordered set, well quasi-ordering.

1. Introduction and presentation of the results

The framework of this paper is the theory of relations. It is about a counting function, the profile. The profile of a class $\mathfrak{C}$ of finite relational structures (also called the speed by other authors) is the integer function $\varphi_{\mathfrak{C}}$ which counts for each non negative integer $n$ the number $\varphi_{\mathfrak{C}}(n)$ of members of $\mathfrak{C}$ defined on $n$ elements, isomorphic structures being identified. For the last fifteen years, the behavior of this function has been discussed in many papers, particularly when $\mathfrak{C}$ is hereditary (that is contains every substructure of any member of $\mathfrak{C}$) and is made of graphs (directed or not), of tournaments, of ordered sets, of ordered graphs and of ordered hypergraphs. As observed by P. J. Cameron [7], numerous results obtained about classes of permutations obtained during the same period fall under the frame of the profile of hereditary classes of relational structures, namely bichains, that is structures made of two linear orders on the same set. The results show that the profile cannot be arbitrary: there are jumps in its possible growth rates. Typically, its growth is polynomial or faster than every polynomial ([28] for ages, see [30] for a survey) and for several classes of structures, it is either at least exponential (e.g. for tournaments [3, 5], ordered graphs and hypergraphs [1, 2, 17] and permutations [16]) or at least with the growth of the partition function (e.g. for graphs [4]). For more, see the survey of Klazar [18] and for permutations the survey of Vatter [34].

This paper is motivated by Balogh, Bollobás and Morris results about the profile of hereditary classes of ordered graphs. They show in [2] that if $\mathfrak{C}$ is a hereditary class of finite ordered graphs then its profile $\varphi_{\mathfrak{C}}$ is either polynomial or is ranked by the Fibonacci functions.

Their theorem states:

Theorem 1.1**.**

If $\mathfrak{C}$ is a hereditary class of finite ordered graphs, then one of the following assertions holds.

1. (a)

$\varphi_{\mathfrak{C}}(n)$* is bounded above and there exist $M,N\in{\mathbb{N}}$ such that $\varphi_{\mathfrak{C}}(n)=M$ for every $n\geq N$.* 2. (b)

$\varphi_{\mathfrak{C}}(n)$* is a polynomial in $n$. There exist $k\in{\mathbb{N}}$ and integers $a_{0},\dots,a_{k}$ such that, $\varphi_{\mathfrak{C}}(n)=\underset{i=0}{\overset{k}{\sum}}a_{i}\binom{n}{i}$ for all sufficiently large $n$, and $\varphi_{\mathfrak{C}}(n)\geq n$ for every $n\in{\mathbb{N}}$.* 3. (c)

$F_{n,k}\leq\varphi_{\mathfrak{C}}(n)\leq p(n)F_{n,k}$* for every $n\in{\mathbb{N}}$, for some $2\leq k\in{\mathbb{N}}$ and some polynomial $p$, so in particular $\varphi_{\mathfrak{C}}(n)$ is exponential.* 4. (d)

$\varphi_{\mathfrak{C}}(n)\geq 2^{n-1}$* for every $n\in{\mathbb{N}}$.*

Here, $F_{n,k}$ denote the $n^{\text{th}}$ generalized Fibonacci number of order $k$, defined by $F_{n,k}=0$ if $n<0$, $F_{0,k}=1$ and $F_{n-1,k}=F_{n-2,k}+F_{n-3,k}+\dots+F_{n-k,k}$ for every $n\geq 1$. For $k=2$, $F_{n,2}$ is the Fibonacci number $F_{n}$.

Their result extends Kaiser and Klazar result for classes of permutations (see [16]).

In this paper, we give an other look at Balogh, Bollobás and Morris result using notions of the theory of relations and notably the notion of monomorphic decomposition of a relational structure (described in Pouzet and Thiéry 2013 [32]).

Our technique allows to characterize the hereditary classes of ordered directed graphs and more generally ordered binary structures which have a polynomially bounded profile. It gives the jump of the profile between polynomials and the ordinary Fibonacci function $F_{-,2}$ but does not gives the hierarchy given in $(c)$ and $(d)$ of Theorem 1.1.

We recall that a monomorphic decomposition of a relational structure $\mathcal{R}$ defined on a set $V$ is a partition $(V_{i})_{i\in I}$ of $V$ such that the induced structures $\mathcal{R}_{\restriction A}$ and $\mathcal{R}_{\restriction A^{\prime}}$ on two finite subsets $A,A^{\prime}$ of $V$ are isomorphic provided that the sets $A\cap V_{i}$ and $A^{\prime}\cap V_{i}$ have the same cardinality for each $i\in I$. We also recall that the age of a relational structure $\mathcal{R}$ is the set $\operatorname{Age}(\mathcal{R})$ of finite induced substructures of $\mathcal{R}$ considered up to isomorphism. We will call *profile of * $\mathcal{R}$, denoted by $\varphi_{\mathcal{R}}$, the profile of its age $\operatorname{Age}(\mathcal{R})$.

We prove first a dichotomy result about this notion:

Theorem 1.2**.**

Let $\mathfrak{C}$ be a hereditary class of finite relational structures with a fixed finite signature. Then

1. (1)

Either $\mathfrak{C}$ is a finite union of ages of relational structures, each admitting a finite monomorphic decomposition. 2. (2)

Or $\mathfrak{C}$ contains the age $\mathfrak{D}$ of a structure $\mathcal{R}$ that does not have a finite monomorphic decomposition, this age being minimal with this property.

From this follows easily a characterization of classes satisfying the first item of Theorem 1.2.

Corollary 1.3**.**

A class $\mathfrak{C}$ is a finite union of ages of relational structures that have a finite monomorphic decomposition if and only if there is an integer $\ell$ such that every member of $\mathfrak{C}$ has a monomorphic decomposition into at most $\ell$ blocks.

Clearly, the profile $\varphi_{\mathcal{R}}$ of a structure $\mathcal{R}$ admitting a monomorphic decomposition is bounded above by a polynomial (in fact, $\varphi_{\mathcal{R}}(n)\leq\binom{n+\ell-1}{\ell-1}$, where $\ell$ is the number of blocks of the monomorphic decomposition). Hence, if $\mathfrak{C}$ satisfies (1) of Theorem 1.2, the profile of $\mathfrak{C}$ is bounded by a polynomial. It was shown in [32] that $\varphi_{\mathcal{R}}$, hence $\varphi_{\mathfrak{C}}$, is a quasi-polynomial (that is a polynomial $a_{\ell-1}(n)n^{\ell-1}$ $+\cdots$ $+a_{1}(n)n$ $+a_{0}(n)$ whose coefficients $a_{\ell-1}(n)$, $\cdots$, $a_{0}(n)$ are periodic functions). In the case of ordered structures, this profile is in fact polynomial (see [25]).

The profile of a class verifying (2) of Theorem 1.2 is bounded below by the profile of the age $\mathfrak{D}$. For arbitrary relational structures, this profile can be bounded above by a polynomial. But, in the case of ordered structures, it is necessarily at least exponential [25] and, as we will see in Proposition 1.7, in the case of ordered binary structures, the profile of $\mathfrak{D}$ is bounded below by the Fibonacci function.

Consequently:

Theorem 1.4**.**

If $\mathfrak{C}$ is a hereditary class of ordered binary structures then

1. (1)

Either the profile is bounded above by a polynomial, and in this case there is an integer $\ell$ such that every member of $\mathfrak{C}$ has a monomorphic decomposition in at most $\ell$ blocks. 2. (2)

Or the profile is bounded below by the Fibonacci function.

Ordered structures that have a finite monomorphic decomposition have a particularly simple form. If $\mathcal{R}:=(V,\leq,\rho_{1},\dots,\rho_{k})$ is such a structure, the chain $(V,\leq)$ decomposes into finitely many intervals $V_{i}$ such that the union of any local isomorphisms $f_{i}$ of $(V_{i},\leq_{\restriction V_{i}})$ is a local isomorphism of $\mathcal{R}$ (see Theorem 3.3 below). If $\mathcal{R}$ is made of binary relations, these relations are quite close to the given order. For example, if $\mathcal{R}$ is a bichain, that is $\mathcal{R}:=(V,\leq,\leq^{\prime})$, where $\leq^{\prime}$ is a linear order, then $\leq^{\prime}$ coincides with $\leq$ or its opposite on each $V_{i}$. Note that any ordered binary structure can be viewed as superposition of graphs (symmetric and irreflexive) and unary relations on the same ordered set and this superposition has the same local isomorphisms as $\mathcal{R}$ hence the same profile. Indeed, if $\mathcal{R}:=(V,\leq,\rho_{1},\dots,\rho_{k})$ is an ordered binary structure, replace each $\rho_{i}$ by the graphs $\rho_{i}^{+}$, $\rho_{i}^{-}$ and the unary relation $u(\rho_{i})$ defined by setting $\rho_{i}^{+}:=\{\{x,y\}\in[V]^{2}:x<y\text{ and }(x,y)\in\rho_{i}\}$, $\rho_{i}^{-}:=\{\{x,y\}\in[V]^{2}:x<y\text{ and }(y,x)\in\rho_{i}\}$ and $u(\rho_{i}):=\{x\in V:(x,x)\in\rho_{i}\}$. For example, if $\mathcal{R}$ has a finite monomorphic decomposition $(V_{i})_{i:=1,\dots,\ell}$ then the graphs are full or empty on each $V_{i}$.

The following result indicates that for ordered binary structures, the study of their profile reduces (roughly) to the case of single ordered binary relations, and in fact to unary ordered structures or to ordered graphs.

Proposition 1.5**.**

Let $k\in\mathbb{N}$. An ordered binary structure $\mathcal{R}:=(V,\leq,\rho_{1},\dots,\rho_{k})$ has a finite monomorphic decomposition if and only if every structure $\mathcal{R}_{i}:=(V,\leq,\rho_{i})$, $(1\leq i\leq k)$, has such a decomposition.

The case of an ordered unary relation is handled by a result of P. Jullien [14]: the profile is either polynomial or bounded below by the exponential function $n\hookrightarrow 2^{n}$. The case of ordered graphs was handled by Balogh, Bollobás and Morris [2].

Finite ordered structures, of the form $\mathcal{R}:=(V,\leq,u_{1},\dots,u_{k})$, consisting of a linear order $\leq$ and unary relations $u_{i}$, can be represented by words over a finite alphabet $A$ (namely the set $\{0,1\}^{k}$). Embedding between structures correspond to the subword ordering. A famous result of Higman [13] asserts that the set $A^{\ast}$ of words over a finite alphabet $A$ is well-quasi-ordered for this ordering. Hence, hereditary classes of words are finite unions of ideals. According to [14] (see Chapter 6, page 103), each ideal decomposes into a finite product of elementary ideals, sets of the form $\{\square,a\}$, where $\square$ is the empty word and $a\in A$ and of starred ages, sets of the form $B^{\ast}$ where $B^{\ast}$ is the set of words over $B\subseteq A$ (cf. [15] for an extension to an ordered alphabet). The profile of $B^{\ast}$ satisfies $\varphi_{B^{\ast}}(n)=|B|^{n}$. Thus the profile of a hereditary class of words is either polynomial or at least exponential.

In this paper, we give more information about the binary ordered structures yielding an exponential profile. The case of ordered structures not necessarily binary is handled in a forthcoming paper [25].

We say that ordered structures of the form $\mathcal{R}:=(V,\leq,\rho_{1},\dots,\rho_{k})$ have type $k$. As a consequence of Ramsey’s theorem, we obtain:

Theorem 1.6**.**

The collection $\mathfrak{S}_{k}$ of ordered binary structures of type $k$ that do not have a finite monomorphic decomposition has a finite basis, that is contains a finite set $\mathfrak{A}_{k}$ such that every member of $\mathfrak{S}_{k}$ embeds some member of $\mathfrak{A}_{k}$. If $k=1$, the subset $\check{\mathfrak{A}}_{1}$ of $\mathfrak{A}_{1}$ made of ordered reflexive graphs has one thousand two hundred and forty two members and none embeds into an other one.

Members $\mathcal{R}$ of $\mathfrak{A}_{k}$ have the following common features. First, their domain $V$ is either ${\mathbb{N}}\times\{0,1\}$, or $({\mathbb{N}}\times\{0,1\})\cup\{a\}$ for some fixed element $a$. Next, if $\mathcal{R}$ is such a structure and $u$ is any one-to-one order-preserving map on $({\mathbb{N}},\leq)$, then the map $(u,Id)$ on ${\mathbb{N}}\times\{0,1\}$ defined by $(u,Id)(x,i)=(u(x),i)$ for $i\in\{0,1\}$ and that fixes $a$ if $a\in V$ preserves $\mathcal{R}$. They are almost multichainable in the sense of [30].

Ordered reflexive directed graphs $\mathcal{G}$ belonging to $\check{\mathfrak{A}}_{1}$ depend on three parameters $p,l,k$. There are integers such that $1\leq p\leq 10$, $1\leq\ell\leq 6$ and the value of $k$ depends upon $p,l$ (see Section 6).

We show that the profile of every element of $\check{\mathfrak{A}}_{1}$ is at least exponential:

Proposition 1.7**.**

The profile of a member of $\check{\mathfrak{A}}_{1}$ is either given by one of the five following functions: $\varphi_{1}(n):=2^{n}-1$, $\varphi_{2}(n):=2^{n}-n$, $\varphi_{3}(n):=2^{n-1}$, $\varphi_{4}(n):=2^{n-1}+1$ and the Fibonacci sequence, or is bounded below by one of them.

This paper is organized as follows. Section 2 contains the definitions and basic notions. Section 3 contains a presentation of the notion of monomorphic decomposition. We give the proof of Theorem 1.2 and Corollary 1.3 in Subsection 3.2 and the proof of Proposition 1.5 in Subsection 3.3. In section 5 we give the detailed proof of the first part of Theorem 1.6 with a description of members of $\mathfrak{A}_{k}$. The proof of the second part is given in Section 7, with a study of profiles of members of $\check{\mathfrak{A}}_{1}$ from which Proposition 1.7 follows.

The results presented in this paper are included in Chapter 8 of our doctoral thesis [22]. They are mentionned in an abstract posted on ArXiv [24]. Our results have been presented at the $9$th International Colloquium on Graph Theory and combinatorics (ICGT 2014) held in Grenoble (France), June 30-July 4, 2014, and at the conference-school on Discrete Mathematics and Computer Science (DIMACOS’2015) held in Sidi Bel Abbès (Algeria), November 15-19, 2015. We are pleased to thank the organizers of these conferences for their help.

2. Basic concepts

Our terminology follows Fraïssé [10]. For properties of profiles we refer to the survey of Pouzet [30].

2.1. Relational structures, embedabbility, hereditary classes and ages

Let $m$ be a non-negative integer. A $m$-ary relation with domain $V$ is a subset $\rho$ of $V^{m}$. When needed, we identify $\rho$ with its characteristic function$\chi_{\rho}$ sending every element of $\rho$ on $1$, hence, $\rho$ becomes a map from $V^{m}$ to $\{0,1\}$. A relational structure is a pair $\mathcal{R}:=(V,(\rho_{i})_{i\in I})$ made of a set $V$, the base or domain of $\mathcal{R}$, also denoted by $V(\mathcal{R})$ and a family $(\rho_{i})_{i\in I}$ of $n_{i}$-ary relations $\rho_{i}$ on $V$. The family $\mu:=(n_{i})_{i\in I}$ is the signature of $\mathcal{R}$. If $I$ is finite, we say that the signature $\mu$ is finite. When all the relations $\rho_{i},\leavevmode\nobreak\ i\in I$, are binary, we have a binary relational structure, (binary structure for short). If one specified relation is a linear order, we have an ordered relational structure. A structure which has the form $\mathcal{R}:=(V,\leq,\rho_{1},\dots,\rho_{k})$, where $k$ is a non negative integer, $\leq$ is a linear order on the set $V$ and each $\rho_{i}$ is a binary relation on $V$, is an ordered binary structure of type $k$. Basic examples of ordered binary structures are chains ($k=0$), bichains ($k=1$ and $\rho_{1}$ is a linear order on $V$) and ordered directed graphs ($k=1$); if in this last case $\rho_{1}$ is an irreflexive and symmetric binary relation on $V$, we just say that the structure is an ordered graph. Let $\mathcal{R}$ be a relational structure. The substructure induced by $\mathcal{R}$ on a subset $A$ of $V$, simply called the *restriction * of $\mathcal{R}$ to $A$, is the relational structure $\mathcal{R}\restriction_{A}:=(A,(\rho_{i}\restriction_{A})_{i\in I})$. The notion of isomorphism between relational structures is defined in the natural way. A local isomorphism of $\mathcal{R}$ is any isomorphism between two restrictions of $\mathcal{R}$. A relational structure $\mathcal{R}$ is embeddable into a relational structure $\mathcal{R}^{\prime}$, in notation $\mathcal{R}\leq\mathcal{R}^{\prime}$, if $\mathcal{R}$ is isomorphic to some restriction of $\mathcal{R}^{\prime}$. Embeddability is a quasi-order on the class of structures having a given signature. We denote by $\Omega_{\mu}$ the class of finite relational structures of signature $\mu$; it is quasi-ordered by embeddability. Most of the time, we consider its members up to isomorphism. The age of a relational structure $\mathcal{R}$ is the set $\operatorname{Age}(\mathcal{R})$ of restrictions of $\mathcal{R}$ to finite subsets of its domain, these restrictions being considered up to isomorphism. A class $\mathfrak{C}$ of structures is hereditary if it contains every relational structure which can be embedded in some member of $\mathfrak{C}$ (i.e., if $\mathcal{R}\in\mathfrak{C}$ and $\mathcal{S}\leq\mathcal{R}$ then $\mathcal{S}\in\mathfrak{C}$). In order theoretic terms, a class of finite structures is hereditary iff this is an initial segment of $\Omega_{\mu}$. If $\mathfrak{B}$ is any subset of $\Omega_{\mu}$ the set $Forb(\mathfrak{B}):=\{\mathcal{R}\in\Omega_{\mu}:\mathcal{B}\not\leq\mathcal{R}\;\text{for all}\;\mathcal{B}\in\mathfrak{B}\}$ is a hereditary class. A bound of a hereditary class $\mathfrak{C}$ of finite structures is any element of $\Omega_{\mu}\setminus\mathfrak{C}$ which is minimal w.r.t. embeddability. Each hereditary class $\mathfrak{C}$ of $\Omega_{\mu}$ is determined by its bounds (in fact $\mathfrak{C}=Forb(\mathfrak{B})$ where $\mathfrak{B}$ is the set of bounds of $\mathfrak{C}$). Any age is an ideal of $\Omega_{\mu}$, that is a non empty initial segment of $\Omega_{\mu}$ which is up-directed. One of the most basic result of the theory of relations asserts that the converse (almost) holds.

Lemma 2.1**.**

(Fraïssé, 1954) If $\mu$ is finite, every ideal of $\Omega_{\mu}$ is the age of a countable structure.

Note 2.1**.**

In the sequel, all relational structures we consider will have a finite signature.

2.2. Invariant structures

We borrow the following notion and results to [8]; see [5] for an illustration.

Let $C:=(L,\leq)$ be a chain. For each integer $n$, let $[C]^{n}$ be the set of $n$-tuples $\overrightarrow{a}:=(a_{1},...,a_{n})\in L^{n}$ such that $a_{1}<...<a_{n}$. This set will be identified with the set $[L]^{n}$ of the $n$-element subsets of $L$.

For every local automorphism $h$ of $C$ with domain $D$, set $h(\overrightarrow{a}):=(h(a_{1}),...,h(a_{n}))$ for every $\overrightarrow{a}\in[D]^{n}$. Let $\mathfrak{L}:=\left\langle C,{\mathcal{R}},\Phi\right\rangle$ be a triple made of a chain $C$ on $L$, a relational structure ${\mathcal{R}}:=(V,(\rho_{i})_{i\in I})$ and a set $\Phi$ of maps, each one being a map $\psi$ from $[C]^{a(\psi)}$ into $V$, where $a(\psi)$ is an integer, the arity of $\psi$.

We say that $\mathfrak{L}$ is invariant if:

[TABLE]

for every $i\in I$ and every local automorphism $h$ of $C$ whose domain contains $\overrightarrow{\alpha}_{1},...,\overrightarrow{\alpha}_{m_{i}},$ where $m_{i}$ is the arity of $\rho_{i}$, $\psi_{1},...,\psi_{m_{i}}\in\Phi$, $\overrightarrow{\alpha}_{j}\in[C]^{a(\psi_{j})}$ for $j=1,...,m_{i}.$

Condition (2.1) expresses the fact that each $\rho_{i}$ is invariant under the transformation of the $m_{i}$-tuples of $V$, that are induced on $V$, by the local automorphisms of $C$. For example, if ${\rho}$ is a binary relation and $\Phi=\{\psi\}$ then

[TABLE]

means that $\rho(\psi(\overrightarrow{\alpha}),\psi(\overrightarrow{\beta}))$ depends only upon the relative positions of $\overrightarrow{\alpha}$ and $\overrightarrow{\beta}$ on the chain $C$.

If $\mathfrak{L}:=\left\langle C,{\mathcal{R}},\Phi\right\rangle$ and $L^{\prime}$ is a subset of $L$, set $\Phi_{\upharpoonleft_{L^{\prime}}}:=\{\psi_{\upharpoonleft_{[L^{\prime}]^{a(\psi)}}}:\psi\in\Phi\}$ and $\mathfrak{L}_{\upharpoonleft_{L^{\prime}}}:=\left\langle C_{\upharpoonleft_{L^{\prime}}},{\mathcal{R}},\Phi_{\upharpoonleft_{L^{\prime}}}\right\rangle$ the restriction of $\mathfrak{L}$ to $L^{\prime}.$

The following result (see [8]) is a consequence of Ramsey’s theorem:

Theorem 2.2**.**

Let $\mathfrak{L}:=\left\langle C,{\mathcal{R}},\Phi\right\rangle$ be a triple such that the domain $L$ of $C$ is infinite, ${\mathcal{R}}$ consists of finitely many relations and $\Phi$ is finite. Then there is an infinite subset $L^{\prime}$ of $L$ such that $\mathfrak{L}_{\upharpoonleft_{L^{\prime}}}$ is invariant.

2.3. Almost multichainable relational structure

A relational structure $\mathcal{R}$ of signature $\mu$ is almost multichainable if its domain $V$ decomposes into $F\cup(L\times K)$ where $F$ and $K$ are two finite sets and there is a linear order $\leq$ on $L$ such that:

[TABLE]

If $F$ is empty and $|K|=1$, Condition (2.2) reduces to say that there is a linear order $\leq$ on $V$ such that every local isomorphism of $(V,\leq)$ is a local isomorphism of $\mathcal{R}$. Relational structures with this property are said chainable, a notion invented by Fraïssé [10] (the relationship with monomorphy is given in Section 3.1).

Almost multichainable structures were introduced in [28], see [30]. They fall under the frame of invariant structures. Indeed, let $\mathcal{R}$ be a relational structure; suppose that its domain $V$ decomposes into $V=F\cup(L\times K)$ where $F$ and $K$ are two finite sets and that $\leq$ is a linear order on $L$. Denote by $\psi_{k}$ the map from $L$ into $V$ defined by $\psi_{k}(a):=(a,k)$ if $k\in K$ and $\psi_{k}(a):=k$ if $k\in F$. Set $\Phi:=\{\psi_{k}:k\in K\cup F\}$ and $C:=(L,\leq)$. Then the structure $\left\langle C,{\mathcal{R}},\Phi\right\rangle$ is said invariant iff Condition 2.2 holds. Hence, Theorem 2.2 allows to extract from any $\mathcal{R}$ an almost multichainable structure.

2.4. Well-quasi-ordering

We recall that a poset $P$ is well-quasi-ordered (w.q.o.) if $P$ contains no infinite antichain and no infinite descending chain. We recall the following result  [29, 3.9 p. 329]. This is a special instance of a property of posets which is similar to Nash-William’s lemma on minimal bad sequences [21]).

Lemma 2.3**.**

If a hereditary class $\mathfrak{C}$ is not w.q.o., it contains an age $\mathfrak{D}$ which is w.q.o. and has infinitely many bounds.

We recall the following notion and result (see Chapter 13 p. 354, of [10]). A class $\mathfrak{C}$ of finite structures is very beautiful if for every integer $k$, the collection $\mathfrak{C}(k)$ of structures $(S,U_{1},\dots,U_{k})$, where $S\in\mathfrak{C}$ and $U_{1},\dots,U_{k}$ are unary relations with the same domain as $S$, has no infinite antichain w.r.t. embeddability. A crucial property is the following:

Theorem 2.4**.**

[26]**. A very beautiful age has only finitely many bounds.

In the case of binary structures, Theorem 2.4 has a simple proof.

A straightforward consequence of Higman’s theorem on words (see [13]) is that the age of an almost multichainable structure is very beautiful. As a consequence:

Lemma 2.5** ([30, Theorem 4.20]).**

The age of an almost multichainable structure has only finitely many bounds.

The finiteness of the bounds is a deep result. Frasnay [11] proved by means of a clever finite combinatorial analysis that chainable relational structures have finitely many bounds. The argument using their very beautiful character is shorter (but less precise: it gives no estimate on the size of bounds).

3. Monomorphic decomposition of a relational structures

We present in this section the notion of monomorphic decomposition of a relational structure. This notion was introduced in [32]. The introduction of an equivalence relation makes the presentation simpler and the proofs easier. This equivalence relation appeared in [24] and for hypergraphs in [31]. Its study is developped in full in [22] (cf. the third part of the thesis).

3.1. Monomorphic decomposition: basic properties

Let $\mathcal{R}:=(V,(\rho_{i})_{i\in I})$ be a relational structure. A subset $B$ of $V$ is a monomorphic part of $\mathcal{R}$ if for every non-negative integer $k$ and every pair $A,\leavevmode\nobreak\ A^{\prime}$ of $k$-element subsets of $V(\mathcal{R})$, the induced structures on $A$ and $A^{\prime}$ are isomorphic whenever $A\setminus B=A^{\prime}\setminus B$. A monomorphic decomposition of $\mathcal{R}$ is a partition $\mathcal{P}$ of $V$ into monomorphic parts. Equivalently, it is a partition $\mathcal{P}$ of $V(\mathcal{R})$ into blocks such that for every integer $n$, the induced structures on two $n$-element subsets $A$ and $A^{\prime}$ of $V$ are isomorphic whenever $|A\cap B|=|A^{\prime}\cap B|$ for every $B\in\mathcal{P}$. Each monomorphic part is included into a maximal one. This monomorphic part is unique and called a monomorphic component. The monomorphic components of $\mathcal{R}$ form a monomorphic decomposition of $\mathcal{R}$ of which every monomorphic decomposition of $\mathcal{R}$ is a refinement (Proposition 2.12, page 14 of [32]).

Let $x$ and $y$ be two elements of $V$. Let $F$ be a finite subset of $V\setminus\{x,y\}$. We say that $x$ and $y$ are $F$-equivalent and we set $x\simeq_{F,\mathcal{R}}y$ if the restrictions of $\mathcal{R}$ to $\{x\}\cup F$ and $\{y\}\cup F$ are isomorphic. Let $k$ be a non-negative integer, we set $x\simeq_{k,\mathcal{R}}y$ if $x\simeq_{F,\mathcal{R}}y$ for every $k$-element subset $F$ of $V\setminus\{x,y\}$. We set $x\simeq_{\leq k,\mathcal{R}}y$ if $x\simeq_{k^{\prime},\mathcal{R}}y$ for every $k^{\prime}\leq k$. Finally, we say that they are equivalent and we set $x\simeq_{\mathcal{R}}y$ if $x\simeq_{k,\mathcal{R}}y$ for every integer $k$.

The fundamental property, whose proof is easy (see [31], [24], [22]) is this:

Lemma 3.1**.**

The relations $\simeq_{k,\mathcal{R}},\leavevmode\nobreak\ \simeq_{\leq k,\mathcal{R}}$ and $\simeq_{\mathcal{R}}$ are equivalence relations. Furthermore, the equivalence classes of $\simeq_{\mathcal{R}}$ are the monomorphic components of $\mathcal{R}$.

Using this equivalence relation, the proof of the following result, which improves Proposition 2.4 of [32], is straigthforward.

Lemma 3.2**.**

A relational structure $\mathcal{R}$ admits a finite monomorphic decomposition if and only if there is some integer $\ell$ such that every member of its age $\operatorname{Age}(\mathcal{R})$ admits a monomorphic decomposition into at most $\ell$ classes.

Let $\mathcal{R}$ be a relational structure; if for some non-negative integer $n$ all the restrictions to the $n$-element subsets of its domain are isomorphic, we say that $\mathcal{R}$ is $n$-monomorphic. We say that $\mathcal{R}$ is $(\leq n)$-monomorphic if it is $m$-monomorphic for every $m\leq n$ and that $R$ is monomorphic if it is $n$-monomorphic for every integer $n$. Since two finite chains with the same cardinality are isomorphic, chains are monomorphic. More generally, if there is a linear order $\leq$ on the domain $V$ of a relational structure $\mathcal{R}$ such that the local isomorphisms of $C:=(V,\leq)$ are local isomorphisms of $\mathcal{R}$, then $\mathcal{R}$ is monomorphic. Structures with this propery are called chainable. Every infinite monomorphic relational structure is chainable (Fraïssé, 1954, [10]). Every monomorphic relational structure of finite cardinality large enough (depending on the signature) is chainable (Frasnay 1965 [11]). It is an easy exercice to show that every binary relational structure on at least $4$ elements which is $(\leq 3)$-monomorphic is chainable. The extension to larger arities is a deep result of Frasnay 1965 ([11]). He shown the existence of an integer $p$ such that every $p$-monomorphic relational structure $R$ whose maximum of the signature is at most $m$ and domain infinite or sufficiently large is chainable (note that any $p$-monomorphic relational structure on $2p-1$-element is $(\leq p)$-monomorphic [27]). The least integer $p$ in the sentence above is the monomorphy treshold, $p(m)$. Its value vas given by Frasnay [12] in 1990: $p(1)=1$, $p(2)=3$, $p(m)=2m-2$ for $m\geq 3$. For a detailed exposition of this result, see [10] Chapter 13, notably p. 378.

Relational structures with a finite monomorphis decomposition have a form close to the chainable ones, as shown by the following result proved in [32], cf. Theorem 1.8 p. 10.

Theorem 3.3**.**

A relational structure $\mathcal{R}:=(V,(\rho_{i})_{i\in I})$ admits a finite monomorphic decomposition if and only if there exists a linear order $\leq$ on $V$ and a finite partition $(V_{x})_{x\in X}$ of $V$ into intervals of $(V,\leq)$ such that every local isomorphism of $(V,\leq)$ which preserves each interval is a local isomorphism of $\mathcal{R}$.

We may notice that if $\mathcal{R}$ is ordered then the given order $\leq$ has the property stated in Theorem 3.3.

If a structure $\mathcal{R}$ admits a finite monomorphic decomposition, then it has some restriction with the same age which is almost multichainable. As a consequence of Lemma 2.5 we have:

Corollary 3.4**.**

The age $\operatorname{Age}(\mathcal{R})$ of a relational structure $\mathcal{R}$ with a finite monomorphic decomposition is very beautiful and has finitely many bounds.

We may give an almost self-contained proof. The w.q.o. character of $\operatorname{Age}(\mathcal{R})$ is Dickson’s Lemma. Indeed, if $V_{1},\dots,V_{\ell}$ is a partition of $V$ into monomorphic blocks, associate to every finite subset $A$ of $V$ the sequence $\vartheta(A):=(|A\cap V_{i}|)_{i:=1,\dots,\ell}$. If $A$ and $A^{\prime}$ are two finite subsets and $\vartheta(A)\leq\vartheta(A^{\prime})$ in the direct product ${\mathbb{N}}^{\ell}$ then $\mathcal{R}_{\restriction A}$ is embeddable into $\mathcal{R}_{\restriction A^{\prime}}$. Since ${\mathbb{N}}^{\ell}$ is w.q.o. by Dickson’s Lemma, $\operatorname{Age}(\mathcal{R})$ is w.q.o. Adding $k$ unary predicates to $\mathcal{R}$ amounts to replace each $a_{i}:=|A\cap V_{i}|$ by a word of length $a_{i}$ over an alphabet on $2^{k}$ letters. The w.q.o. character follows from Higman’s Theorem on words. The fact that $\mathcal{R}$ has finitely many bounds follows from Theorem 2.4.

3.2. Proof of Theorem 1.2 and Corollary 1.3

3.2.1. Proof of Theorem 1.2.

Suppose that (1) does not hold. We consider two cases.

$a)$ $\mathfrak{C}$ is w.q.o. by embeddability.

In this case, $\mathfrak{C}$ is a finite union of ideals $\mathfrak{C}_{i}$ (this is a special case of a general result about posets due to Erdös-Tarski [9]). According to Lemma 2.1, each ideal $\mathfrak{C}_{i}$ is the age of a structure $\mathcal{R}_{i}$. Since $\mathfrak{C}$ does not satisfy (1), there is some $\mathcal{R}_{i}$ that cannot have a finite monomorphic decomposition. Since $\mathfrak{C}$ is w.q.o., the set of hereditary subclasses is well founded (Higman [13]). Since $\mathfrak{C}$ contains the age of a structure with no finite monomorphic decomposition, it contains a minimal age with this property.

$b)$ $\mathfrak{C}$ is not well-quasi ordered by embeddability.

In this case, it contains an infinite antichain. According to Lemma 2.3, it contains an age $\mathfrak{D}$ which is w.q.o. and has infinitely many bounds. According to Lemma 3.4, no relation having this age can have a finite monomorphic decomposition. Since $\mathfrak{D}$ is w.q.o., it contains an age minimal with this property. $\Box$

3.2.2. Proof of Corollary 1.3.

Suppose that $\mathfrak{C}$ is a finite union of ages $\operatorname{Age}(\mathcal{R}_{i})$ of relational structures $\mathcal{R}_{i}$ and each $\mathcal{R}_{i}$ has a monomorphic decomposition into $\ell_{i}$ components. Then, for $\ell:=Max\{\ell_{i}:i\}$, each member of $\mathfrak{C}$ has a monomorphic decomposition into at most $\ell$ components.

Now, suppose that there is an integer $\ell$ such that every member of $\mathfrak{C}$ has a monomorphic decomposition into at most $\ell$ blocks. In this case $\mathfrak{C}$ cannot satisfy (2) of Theorem 1.2. Otherwise, $\mathfrak{C}$ contains an age $\mathfrak{D}$ such that no relational structure $\mathcal{R}$ having this age has a finite monomorphic decomposition. According to Lemma 3.2, there is no bound on the size of the monomorphic decompositions of the member of $\mathfrak{D}$, and so the members of $\mathfrak{C}$. This fact contradicts our hypothesis. Then, $\mathfrak{C}$ satisfy (1) of Theorem 1.2. $\Box$

3.3. Proof of Proposition 1.5.

We prove a little more, namely:

The equivalence relation $\simeq_{\mathcal{R}}$ associated to the ordered structure $\mathcal{R}:=(V,\leq,(\rho_{i})_{i\in I})$ is the intersection of the equivalence relations $\simeq_{\mathcal{R}_{i}}$ associated to $\mathcal{R}_{i}:=(V,\leq,\rho_{i})$ for $i\in I$.

Clearly $\simeq_{\mathcal{R}}$ is included into $\simeq_{\mathcal{R}_{i}}$ for each $i\in I$. Conversely, let $x,y\in V$ such that $x\simeq_{\mathcal{R}_{i}}y$ for every $i\in I$. Let $F$ be a finite subset of $V\setminus\{x,y\}$. Since $x\simeq_{\mathcal{R}_{i}}y$ there is a local isomorphism $f_{i}$ of $\mathcal{R}_{i}$ which carries $F\cup\{x\}$ onto $F\cup\{y\}$. Since $f_{i}$ is a local isomorphism of $C:=(V,\leq)$ which carries $F\cup\{x\}$ onto $F\cup\{y\}$, $f_{i}$ is independent of $i$. Hence, this is a local isomorphism of $\mathcal{R}$ proving that $x\simeq_{\mathcal{R}}y$. $\Box$

4. The case of ordered binary structures

In this section, and the next, we consider ordered binary structures. A crucial property of ordered structures is that if two finite substructures are isomorphic, there is just one isomorphism from one to the other. A repeated use of this property allows us describe the form of equivalence classes when these structures are binary. We start by describing the case of one class. It is immediate to show this:

Lemma 4.1**.**

An ordered binary structure $\mathcal{R}:=(V,\leq,(\rho_{i})_{i\in I})$ is monomorphic iff it is $(\leq 2)$-monomorphic iff each relation $\rho_{i}$ which is reflexive is either a chain which coincide with $\leq$ or its dual, a reflexive clique or an antichain and every relation $\rho_{i}$ which is irreflexive is either an acyclic tournament which coincide with the strict order $<$ or its dual, a clique or an independent set.

In order to describe the equivalence classes in general, we introduce some notation. Let $\mathcal{R}:=(V,\leq,(\rho_{i})_{i\in I})$ be an ordered binary structure. Identifying a relation to its characteristic function, we set $d_{i}(x,y):=(\rho_{i}(x,y),\rho_{i}(y,x))$ for all $x,y\in V$ and $i\in I$ and $d(x,y):=(d_{i}(x,y))_{i\in I}$. We may note that the value of $d(x,y)$ determines the value of $d(y,x)$. In fact, set $\overline{u}:=(\beta,\alpha)$ for every $u:=(\alpha,\beta)\in\{0,1\}\times\{0,1\}$ and $\overline{(u_{i})_{i\in I}}:=(\overline{u}_{i})_{i\in I}$ for every sequence of members of $\{0,1\}\times\{0,1\}$. Doing so, we have $d(y,x)=\overline{d(x,y)}$.

Set $C:=(V,\leq)$. Let $x,y\in V$. If $x<y$ in $V$, we set $[x,y]:=\{z\in V:x\leq z\leq y\}$ and $]x,y[:=\{z\in V:x<z<y\}$. If the order between $x$ and $y$ is not given, we denote by $I_{\leq}(x,y)$ the least interval of $C$ containing $x$ and $y$. We recall that a subset $A$ of $V$ is an interval of $\mathcal{R}$ if it is an interval of $C$ and

[TABLE]

for every $x,x^{\prime}\in A$ and $y\not\in A$.

The following fact is obvious:

Fact 1**.**

Two elements $x,y$ of $V$ such that $x<y$ are $1$-equivalent if and only if

[TABLE]

and

[TABLE]

Lemma 4.2**.**

Let $\mathcal{R}:=(V,\leq,(\rho_{i})_{i\in I})$ be an ordered binary structure. Two elements $x,y$ of $V$ such that $x<y$ are ($\leq 2$)-equivalent if and only if $[x,y]$ is a $1$-monomorphic interval of $\mathcal{R}$ and the restrictions of $\mathcal{R}$ to any two $2$-element subsets of $[x,y]$ distinct of $\{x,y\}$ are isomorphic.

**Proof. ** Suppose that $x$ and $y$ are ($\leq 2$)-equivalent. We claim that $[x,y]$ is a $1$-monomorphic interval of $\mathcal{R}$. If $[x,y]=\{x,y\}$ then since $x$ and $y$ are [math]-equivalent, $\mathcal{R}_{\restriction\{x\}}$ and $\mathcal{R}_{\restriction\{y\}}$ are isomorphic and thus $[x,y]$ is $1$-monomorphic. Since $x$ and $y$ are $1$-equivalent, the fact that $[x,y]$ is an interval of $\mathcal{R}$ follows from $(\ref{equ:second equality})$ of Fact 1. If $[x,y]\neq\{x,y\}$, let $z\in]x,y[$ and $a\notin[x,y]$ (if any). Since $x$ and $y$ are $1$-equivalent, $\mathcal{R}_{\restriction\{x,z\}}$ and $\mathcal{R}_{\restriction\{z,y\}}$ are isomorphic (and the only isomorphism sends $x$ to $z$ and $z$ to $y$), in particular $\mathcal{R}_{\restriction\{x\}}$ and $\mathcal{R}_{\restriction\{z\}}$ are isomorphic, hence $[x,y]$ is $1$-monomorphic. Since $x$ and $y$ are $2$-equivalent, the restrictions of $\mathcal{R}$ to $\{x,a,z\}$ and $\{y,a,z\}$ are isomorphic; the only isomorphism sends $a$ onto $a$, $x$ to $z$ and $z$ to $y$, hence $d(a,z)=d(a,x)=d(a,y)$, proving that $[x,y]$ is an interval of $\mathcal{R}$. We look at the $2$-element subsets of $[x,y]$ which are distinct from $\{x,y\}$. If $[x,y]=\{x,y\}$ there are none and there is nothing to prove. Otherwise, $[x,y]\not=\{x,y\}$. Let $z\in]x,y[$. Since $x$ and $y$ are $1$-equivalent, $\mathcal{R}_{\restriction\{x,z\}}$ and $\mathcal{R}_{\restriction\{z,y\}}$ are isomorphic. If there is no other element than $z$ this yields the conclusion of the lemma. If there is an other element $z^{\prime}$ we may suppose $z<z^{\prime}$. Since $x$ and $y$ are $2$-equivalent, there is an isomorphism of $\mathcal{R}_{\restriction\{x,z,z^{\prime}\}}$ onto $\mathcal{R}_{\restriction\{z,z^{\prime},y\}}$ and in fact a unique one. It sends $x$ to $z$, $z$ to $z^{\prime}$ and $z^{\prime}$ to $y$. It follows that the five restrictions $\mathcal{R}_{\restriction\{z,z^{\prime}\}}$, $\mathcal{R}_{\restriction\{x,z\}}$, $\mathcal{R}_{\restriction\{x,z^{\prime}\}}$, $\mathcal{R}_{\restriction\{y,z\}}$, $\mathcal{R}_{\restriction\{y,z^{\prime}\}}$ are isomorphic. This property yields immediately the conclusion of the lemma if there is no other element. If there are others, then this property also says that all the restrictions of $\mathcal{R}$ to all pairs $\{x,z\}$ for which $z\not=y$ are isomorphic; since the restrictions of $\mathcal{R}$ to pairs $\{z,z^{\prime}\}$ are isomorphic to the restrictions of $\mathcal{R}$ to such pairs $\{x,z\}$, the restrictions of $\mathcal{R}$ to all pairs distinct of $\{x,y\}$ are isomorphic, as claimed.

Conversely, suppose that $[x,y]$ is a $1$-monomorphic interval of $\mathcal{R}$ and the restrictions of $\mathcal{R}$ to any two $2$-element subsets of $[x,y]$ distinct of $\{x,y\}$ are isomorphic. Let $F$ be a finite subset of $V\setminus\{x,y\}$ and $\varphi$ be the unique order-isomorphism from $F\cup\{x\}$ onto $F\cup\{y\}$. Let $z,z^{\prime}\in F$. If $z\in F\setminus[x,y]$ then $\varphi(z)=z$. If $z^{\prime}\in[x,y]$ then $\varphi(z^{\prime})\in[x,y]$ and since $[x,y]$ is a $1$-monomorphic interval of $\mathcal{R}$, the map $\varphi$ induces an isomorphism of $\mathcal{R}_{\restriction\{z,z^{\prime}\}}$ onto $\mathcal{R}_{\restriction\{\varphi(z),\varphi(z^{\prime})\}}$. If $z,z^{\prime}\in[x,y]$ then $\varphi(z),\varphi(z^{\prime})\in[x,y]$. From the $1$-monomorphy of $[x,y]$ and the fact that the $2$-element subsets of $[x,y]$ distinct of $\{x,y\}$ are isomorphic, $\varphi$ induces an isomorphism of $\mathcal{R}_{\restriction\{z,z^{\prime}\}}$ onto $\mathcal{R}_{\restriction\{\varphi(z),\varphi(z^{\prime})\}}$. It follows that $\varphi$ is an isomorphism of $\mathcal{R}_{\restriction F\cup\{x\}}$ onto $\mathcal{R}_{\restriction F\cup\{y\}}$ hence $x\simeq_{F,\mathcal{R}}y$. In particular $x$ and $y$ are $(\leq 2)$-equivalent.

Corollary 4.3**.**

Let $\mathcal{R}:=(V,\leq,(\rho_{i})_{i\in I})$ be an ordered binary structure and $A$ be a subset of $V$. If $A$ is an interval of $(V,\leq)$ included into some $(\leq 1)$-equivalence class then it is included into a $(\leq 2)$-equivalence class.

**Proof. ** We apply Lemma 4.2. Let $x,y\in A$ with $x<y$. Since $A$ is an interval of $(V,\leq)$ and all the elements of $A$ are $(\leq 1)$-equivalent, $[x,y]$ is a $1$-monomorphic interval of $\mathcal{R}$ and the restrictions of $\mathcal{R}$ to any two $2$-element subsets of $[x,y]$ distinct of $\{x,y\}$ are isomorphic, hence by Lemma 4.2, $x$ and $y$ are $(\leq 2)$-equivalent.

Remark 4.4**.**

The fact that two elements $x$ and $y$ are $(\leq 2)$-equivalent does not imply that the interval $[x,y]$ is $2$-monomorphic. Indeed, let $\mathcal{R}:=(V,\leq,\rho)$ where $\rho$ coincides with $\leq$ except on one ordered pair $(x,y)$ with $x<y$. In such an example, the set $V$ decomposes into four equivalence classes, namely $\{x,y\}$ and the three open intervals of $(V,\leq)$: $]\leftarrow x[$, $]x,y[$ and $]x\rightarrow[$ determined by $x$ and $y$.

The form of the equivalence classes for an arbitrary ordered binary structure is given in Theorem 4.3 below. Before, we extract the following result from Lemma 4.2.

Theorem 4.5**.**

On an ordered binary structure $\mathcal{R}:=(V,\leq,(\rho_{i})_{i\in I})$, the equivalence relations $\simeq_{\leq 2,\mathcal{R}}$ and $\simeq_{\mathcal{R}}$ coincide.

**Proof. ** Suppose by contradiction that there are two elements $x,y\in V$ with $x<y$, $x\simeq_{\leq 2,\mathcal{R}}y$ and $x\not\simeq_{F,\mathcal{R}}y$ for some finite $F\subseteq V\setminus\{x,y\}$. We may choose $F$ minimal w.r.t inclusion. We claim that $F\subseteq[x,y]$. Indeed, set $F^{\prime}:=[x,y]\cap F$ and $V^{\prime}:=F\cup\{x,y\}$. If $F^{\prime}\neq F$ then, by minimality of $F$, $\mathcal{R}_{\restriction{F^{\prime}\cup\{x\}}}$ and $\mathcal{R}_{\restriction{F^{\prime}\cup\{y\}}}$ are isomorphic. Since $[x,y]$ is an interval of $\mathcal{R}$, $[x,y]\cap V^{\prime}$ is an interval of $\mathcal{R}\restriction_{V^{\prime}}$, hence any isomorphism of $\mathcal{R}_{\restriction_{F^{\prime}\cup\{x\}}}$ onto $\mathcal{R}_{\restriction_{F^{\prime}\cup\{y\}}}$ extended by the identity on $F\setminus[x,y]$ is an isomorphism of $\mathcal{R}_{\restriction_{F\cup\{x\}}}$ onto $\mathcal{R}_{\restriction_{F\cup\{y\}}}$. But then, $x\simeq_{F,\mathcal{R}}y$. This contradicts our hypothesis and proves our claim. According to Lemma 4.2, the restrictions of $\mathcal{R}$ to $F\cup\{x\}$ and $F\cup\{y\}$ are isomorphic.

Theorem 4.6**.**

An equivalence class $A$ of an ordered binary structure $\mathcal{R}:=(V,\leq,(\rho_{i})_{i\in I})$ is either an interval of $(V,\leq)$ or consists of two distinct elements $x$, $y$ with $x<y$ such that the interval $]x,y[$ forms an other equivalence class.

**Proof. ** Let $A$ be an equivalence class. Lemma 4.2 ensures that for every $x<y$ in $A$ the open interval $]x,y[$ is included into a $(\leq 2)$-equivalence class. From this fact and Theorem 4.5, it follows that if $|A|\geq 3$, $A$ is an interval of $(V,\leq)$. Thus, if $A$ is not an interval of $(V,\leq)$ then $A$ is made of two elements $x$, $y$ with $x<y$ and there is some $z\not\in A$ such that $x<z<y$. By Lemma 4.2, the open interval $]x,y[$ is included into a $(\leq 2)$-equivalence class distinct from $A$. To conclude, we neeed to prove that $]x,y[$ is equal to this equivalence class. This amounts to prove the following claim:

Claim 1**.**

Let $x<x^{\prime}<y<y^{\prime}$ with $x\simeq_{\leq 2,\mathcal{R}}y$ and $x^{\prime}\simeq_{\leq 2,\mathcal{R}}y^{\prime}$, then $x\simeq_{\leq 2,\mathcal{R}}y^{\prime}$.

**Proof. ** $a)$ We prove that the conclusion holds if there is an fifth element $z$ in the interval $[x,y^{\prime}]$. For example, suppose $z\in]x,y[$. In this case, from Lemma 4.2, we have $x^{\prime}\simeq_{\leq 2,\mathcal{R}}z$. Now, if $x^{\prime}<z<y$, we have $z\simeq_{\leq 2,\mathcal{R}}y$ from Lemma 4.2; if not, then $x<z<x^{\prime}$; since $z\simeq_{\leq 2,\mathcal{R}}y^{\prime}$, we have $x^{\prime}\simeq_{\leq 2,\mathcal{R}}y$ from Lemma 4.2. In both cases we have $x^{\prime}\simeq_{\leq 2,\equiv\mathcal{R}}y$. If $x^{\prime}<z<y$ the conclusion is similar.

$b)$ From $a)$ we may suppose that $]x,y^{\prime}[=\{x^{\prime},y\}$. In this case, it suffices to prove that $[x,y^{\prime}]$ is a $(\leq 2)$-monomorphic $\mathcal{R}$-interval. That is the restrictions of $\mathcal{R}$ on the six ordered pairs $x,x^{\prime}$, $x,y$, $x,y^{\prime}$, $x^{\prime},y$, $x^{\prime},y^{\prime}$, $y,y^{\prime}$ are isomorphic. From the fact that $x\simeq_{\leq 2,\mathcal{R}}y$ the restrictions of $\mathcal{R}$ to $\{x,x^{\prime},y^{\prime}\}$ and to $\{x^{\prime},y,y^{\prime}\}$ are isomorphic Similarly, from the fact that $x^{\prime}\simeq_{\leq 2,\mathcal{R}}y^{\prime}$ the restrictions of $\mathcal{R}$ to $\{x,x^{\prime},y\}$ and to $\{x,y,y^{\prime}\}$ are isomorphic. The first isomorphism yields that the restrictions of $\mathcal{R}$ on $x,x^{\prime}$ and $x^{\prime},y$ are isomorphic, the same for $x,y^{\prime}$ and $x^{\prime},y^{\prime}$ and also for $x^{\prime}y^{\prime}$ and $y,y^{\prime}$. Hence, at least the restrictions of $\mathcal{R}$ to three of these pairs, namely $x,y^{\prime}$, $x^{\prime},y^{\prime}$, $y,y^{\prime}$ are isomorphic. The second isomorphism yields that the restrictions of $\mathcal{R}$ on $x,x^{\prime}$ and $x,y$ are isomorphic, the same for $x,y$ and $x,y^{\prime}$ and also for $x^{\prime}y$ and $y,y^{\prime}$. Also, three of these pairs, namely $x,x^{\prime}$, $x,y$, $x,y^{\prime}$ are isomorphic. The pair $x,y^{\prime}$ belonging to these two sets, the restrictions of $\mathcal{R}$ to these five pairs are all isomorphic; since the restrictions of $\mathcal{R}$ to the remaining pairs $y,y^{\prime}$ and to $x^{\prime},y$ are isomorphic, the restrictions of $\mathcal{R}$ to all these pairs are isomorphic as claimed.  

 

With this, the proof of the theorem is complete.

Remark 4.7**.**

Here is an example for which none of the $2$-element equivalence classes is an interval. Let $V:={\mathbb{N}}\times\{0,1,2\}$. Order $V$ by $(n,i)\leq(n^{\prime},i^{\prime})$ if either $n<n^{\prime}$ or $n=n^{\prime}$ and $i\leq i^{\prime}$; the order type of $(V,\leq)$ is the lexicographical product $3\cdot\omega$. Let $\rho$ be the lexicographical product $C_{3}\cdot\omega$, that is $(n,i)\rho(n^{\prime},i^{\prime})$ if either $n<n^{\prime}$ or $n=n^{\prime}$ and $i^{\prime}=i+1$ (modulo $3$). Let $\mathcal{R}:=(V,\leq,\rho)$. Then, the equivalence classes are the pairs $\{(n,0),(n,2)\}$ and the singletons $\{(n,1)\}$ for $n\in{\mathbb{N}}$.

From Lemma 4.1 and Theorem 4.6, Theorem 3.3 restricted to binary ordered structures follows. Indeed, let $\mathcal{R}:=(V,\leq,(\rho_{i})_{i\in I})$ be such a structure. Then, each equivalence class of $\simeq_{\mathcal{R}}$ which is not an interval of $(V,\leq)$ has just two elements; replacing each of these classes by two blocks made of these two elements will give a partition of $V$ into monomorphic parts. On each part, says $A$, every local isomorphism of $(A,\leq_{\restriction A})$, extended by the identity outside, is a local isomorphism of $\mathcal{R}$, hence every local isomorphism of $(V,\leq)$ which preserves each interval of this new partition is a local isomorphism of $\mathcal{R}$. If $\simeq_{\mathcal{R}}$ has only finitely many classes, the new partition of $V$ has only finitely many blocks and the conclusion of Theorem 3.3 holds.

If the number of relations is finite, we have the following separation lemma;

Lemma 4.8**.**

If an ordered binary structure $\mathcal{R}:=(V,\leq,\rho_{1},\dots,\rho_{k})$ of type $k$ has infinitely many equivalence classes then one of these two cases occurs:

1. (1)

There is an infinite subset $A\subseteq V$ such that any two distinct elements of $A$ are [math]-equivalent but not $1$-equivalent. 2. (2)

There are two disjoint infinite subsets $A_{1},A_{2}$ of $V$ such that any two distinct elements of $A_{i},i\in\{1,2\}$ are $1$-equivalent but not $2$-equivalent and for every $x,y\in A_{i}$, with $i\in\{1,2\}$ and $x<y$, we have $[x,y]\cap A_{j}\neq\emptyset,$ for $i\neq j$.

**Proof. **

Case 1**.**

$\mathcal{R}$* has infinitely many classes of $1$-equivalence.*

Since $\mathcal{R}$ is made of finitely many relations, $V$ consists of only finitely many classes of [math]-equivalence. Since $V$ is made of infinitely many classes of $1$-equivalence, one class of [math]-equivalence, say $X_{0}$, contains infinitely many classes of $1$-equivalence. Pick one element from every class of $1$-equivalence belonging to $X_{0}$ to form a subset $A$ of $V$. The set $A$ satisfies the Assertion (1) of Lemma 4.8.

Case 2**.**

$\mathcal{R}$* has finitely many classes of $1$-equivalence.*

According to Theorem 4.5, every $(\leq 2)$-equivalence class is an equivalence class, hence $\mathcal{R}$ has infinitely many $(\leq 2)$-equivalence classes. Since $V$ is made of finitely many classes of $1$-equivalence, some $1$-equivalence class, say $X_{1}$, contains infinitely many $(\leq 2)$-equivalence classes. Pick an element from each class of $(\leq 2)$- equivalence class included into $X_{1}$. Let $A$ be the resulting set. Let $a,b\in A$ with $a<b$. Then the interval $[a,b]$ cannot be contained in $X_{1}$, otherwise by Item (1) of Lemma 4.3, $a$ and $b$ would be $(\leq 2)$-equivalent. Hence, there exists $c\in[a,b]$ which belongs to an other class of $1$-equivalence $X^{\prime}\neq X_{1}$. We can extract from $A$ a sequence $(a_{i})_{i\geq 0}$ which is monotonic w.r.t. $\leq$. With no loss of generality, we may suppose this sequence increasing. According to the above remark, for every $i\geq 0$, there exists $c_{i}\in[a_{i},a_{i+1}]$ with $c_{i}$ belonging to a class of $1$-equivalence which is different from $X_{1}$. Since the number of $1$-equivalence classes is finite, we can then find an infinite subsequence $(c_{i}^{\prime})_{i\geq 0}$ of $(c_{i})_{i\geq 0}$ whose elements are in the same class of $1$-equivalence. Let then $(a_{i}^{\prime})_{i\geq 0}$ be a subsequence of $(a_{i})_{i\geq 0}$ such that $c_{i}^{\prime}\in[a_{i}^{\prime},a_{i+1}^{\prime}]$. Set $A_{1}=\{a_{i}^{\prime},i\in\mathbb{N}\}$ and $A_{2}=\{c_{i}^{\prime},i\in\mathbb{N}\}$. The sets $A_{1}$ and $A_{2}$ satisfy Assertion (2) of Lemma 4.8.

5. Proof of the first part of Theorem 1.6

We give a proof of the first part of Theorem 1.6. We prove that the collection $\mathfrak{S}_{k}$ of ordered binary structures of type $k$ which do not have a finite monomorphic decomposition, has a finite basis $\mathfrak{A}_{k}$. The proof of the second part is given in section 6.

The proof of Theorem 1.6 goes as follow. Let $\mathcal{R}:=(V,\leq,\rho_{1},\dots,\rho_{k})$ be an ordered binary structure which has infinitely many equivalence classes. According to Lemma 4.8, we have two cases.

5.1. Case 1.

$\mathcal{R}$ satisfies Assertion $(1)$ of Lemma 4.8.

In this case, let $f:{\mathbb{N}}\longrightarrow V$ be a $1$-to-$1$ map such that $f({\mathbb{N}})=A$ where $A$ is the set given by Assertion $(1)$ of Lemma 4.8. Since $f(n)$ and $f(m)$ are not $1$-equivalent for every $n<m$, we may find some element $g(n,m)$ witnessing this fact, meaning that the restrictions of $\mathcal{R}$ to $\{f(n),g(n,m)\}$ and $\{f(m),g(n,m)\}$ are not isomorphic.

Let $\Phi:=\{f,g\}$ and $\mathfrak{L}:=\left\langle\omega,\mathcal{R},\Phi\right\rangle$, where $\omega$ is the chain made of ${\mathbb{N}}$ and the natural order.

Ramsey’s theorem in the version of Theorem 2.2, allows us to find an infinite subset $X\subseteq{\mathbb{N}}$ such that $\mathfrak{L}\restriction_{X}$ is invariant. By relabeling $X$ with non-negative integers, we may suppose $X={\mathbb{N}}$ and hence that $\mathfrak{L}$ is invariant.

Claim 2**.**

The maps $f$ and $g$ satisfy the following properties:

1. (1)

$f(n)\leq f(m)\Leftrightarrow f(n^{\prime})\leq f(m^{\prime}),\leavevmode\nobreak\ \forall n<m,\leavevmode\nobreak\ n^{\prime}<m^{\prime}$. 2. (2)

$d(f(n),f(m))=d(f(n^{\prime}),f(m^{\prime})),\leavevmode\nobreak\ \forall n<m,\leavevmode\nobreak\ n^{\prime}<m^{\prime}$. 3. (3)

$g(n,m)\leq g(k,l)\Leftrightarrow g(n^{\prime},m^{\prime})\leq g(k^{\prime},l^{\prime}),\leavevmode\nobreak\ \forall n<m\leq k<l,\leavevmode\nobreak\ n^{\prime}<m^{\prime}\leq k^{\prime}<l^{\prime}$. 4. (4)

$d(g(n,m),g(k,l))=d(g(n^{\prime},m^{\prime}),g(k^{\prime},l^{\prime})),\leavevmode\nobreak\ \forall n<m\leq k<l,\leavevmode\nobreak\ n^{\prime}<m^{\prime}\leq k^{\prime}<l^{\prime}$. 5. (5)

$g(n,m)\in[f(n),f(m)]\Leftrightarrow g(n^{\prime},m^{\prime})\in[f(n^{\prime}),f(m^{\prime})],\leavevmode\nobreak\ \forall n<m,\leavevmode\nobreak\ n^{\prime}<m^{\prime}$. 6. (6)

$g(n,m)\leq f(k)$* for some integers $n<m<k\Leftrightarrow g(n,m)\leq f(l)$ for every $l>m$.* 7. (7)

$d(f(n),g(n,m))=d(f(k),g(k,l)),\leavevmode\nobreak\ \forall n<m,\leavevmode\nobreak\ k<l$. 8. (8)

$d(g(n,m),f(k))=d(g(p,q),f(l)),\leavevmode\nobreak\ \forall n<m<k,\leavevmode\nobreak\ p<q<l$. 9. (9)

If the restrictions $\mathcal{R}_{\restriction_{\{f(n),g(n,m)\}}}$ and $\mathcal{R}_{\restriction_{\{f(k),g(n,m)\}}}$ are isomorphic for some integers $n<m<k$ then $\mathcal{R}_{\restriction_{\{f(n^{\prime}),g(n^{\prime},m^{\prime})\}}}$ and $\mathcal{R}_{\restriction_{\{f(k^{\prime}),g(n^{\prime},m^{\prime})\}}}$ are isomorphic for every $n^{\prime}<m^{\prime}<k^{\prime}$. 10. (10)

$g(n,m)$* and $f(k)$ are different for every distinct integers $n<m$ and $k$.* 11. (11)

$g(n,m)\neq g(n^{\prime},m^{\prime})$* for every $n<m<n^{\prime}<m^{\prime}$.*

Proof. The nine first items follow from invariance. To prove Item (10), suppose that there are integers $n,m,k$ with $n<m$ such that $g(n,m)=f(k)$. By construction of the functions $f$ and $g$, the sets $\{f(n),g(n,m)\}$ and $\{f(m),g(n,m)\}$ have two elements each, hence $k$ cannot be equal to $n$ or to $m$. According to Item (1) and (2), the restrictions of $\mathcal{R}$ to $\{f(n),f(k)\}$ and $\{f(m),f(k)\}$ are isomorphic; thus, if $g(n,m)=f(k)$ we get that the restrictions of $\mathcal{R}$ to $\{f(n),g(n,m)\}$ and to $\{f(m),g(n,m))\}$ are isomorphic, contradicting the choice of $g(n,m)$. For the proof of Item (11), suppose that there are integers $n<m<n^{\prime}<m^{\prime}$ such that $g(n,m)=g(n^{\prime},m^{\prime})$. The transformation fixing $n$, $m$ and sending $n^{\prime}$ onto $m^{\prime}$ is a local isomorphism of the chain, hence the restrictions of $\mathcal{R}$ to $\{f(n^{\prime}),g(n,m)\}$ and $\{f(m^{\prime}),g(n,m)\}$ are isomorphic. Replacing $g(n,m)$ by $g(n^{\prime},m^{\prime})$, we get that the restrictions to $\{f(n^{\prime}),g(n^{\prime},m^{\prime})\}$ and $\{f(m^{\prime}),g(n^{\prime},m^{\prime})\}$ are isomorphic which is a contradiction with the choice of $g(n^{\prime},m^{\prime})$. $\diamond$

We define a map $F:{\mathbb{N}}\times\{0,1\}\longrightarrow V(\mathcal{R})$ and an ordered binary structure of type $k$, $\mathcal{R}^{(1)}:=(V^{(1)},\leq^{(1)},\rho_{1}^{(1)},\dots,\rho_{k}^{(1)})$ with vertex set $V^{(1)}:={\mathbb{N}}\times\{0,1\}$ such that $F$ is an embedding from $\mathcal{R}^{(1)}$ into $\mathcal{R}$.

We define first $F$. We set $F(n,1):=g(2n,2n+1)$ for $n\in{\mathbb{N}}$. From Item (9) of Claim 2, we have two cases:

Case $(a)$. $\mathcal{R}\restriction_{\{f(n),g(n,m)\}}$ and $\mathcal{R}\restriction_{\{f(k),g(n,m)\}}$ are isomorphic for some integers $n<m<k$; from invariance, this property holds for all $n<m<k$.

Case $(b)$. Case $(a)$ does not hold.

In Case $(a)$, we set $F(n,0):=f(2n+1)$ for every $n\in{\mathbb{N}}$. In Case $b$, we set $F(n,0):=f(2n)$ for every $n\in{\mathbb{N}}$.

The map $F$ is $1$-to-$1$ (by construction $f$ is $1$-to-$1$, hence the restriction of $F$ to ${\mathbb{N}}\times\{0\}$ is $1$-to-$1$; the restriction of $F$ to ${\mathbb{N}}\times\{1\}$ is also $1$-to-$1$ by $(11)$ of Claim 2; the images of ${\mathbb{N}}\times\{0\}$ and ${\mathbb{N}}\times\{1\}$ are disjoint by $(10)$ of Claim 2.

Since $F$ is $1$-to-$1$, we take for $\mathcal{R}^{(1)}$ the inverse image of $\mathcal{R}$.

This amounts to

[TABLE]

for every $x,y\in{\mathbb{N}}\times\{0,1\}$, where $d^{(1)}(x,y):=(d_{i}^{(1)}(x,y))_{i=1,\dots,k}$ and $d_{i}^{(1)}(x,y):=(\rho_{i}^{(1)}(x,y),\rho_{i}^{(1)}(y,x))$ for every $1\leq i\leq k$.

Claim 3**.**

For every $n<m\in{\mathbb{N}}$, $(n,0)$ and $(m,0)$ are not $1$-equivalent, hence $\mathcal{R}^{(1)}$ has infinitely many equivalence classes.

Proof.

It suffices to prove that:

[TABLE]

By definition, $\mathcal{R}_{\restriction\{f(2n),g(2n,2n+1)\}}$ and $\mathcal{R}_{\restriction\{f(2n+1),g(2n,2n+1)\}}$ are not isomorphic. In Case $(a)$, $\mathcal{R}\restriction_{\{f(2n),g(2n,2n+1)\}}$ and $\mathcal{R}\restriction_{\{f(2m+1),g(2n,2n+1)\}}$ are isomorphic, hence $\mathcal{R}\restriction_{\{f(2n+1),g(2n,2n+1)\}}$ and $\mathcal{R}\restriction_{\{f(2m+1),g(2n,2n+1)\}}$ are not isomorphic; since $F(n,0):=f(2n+1),F(m,0)=f(2m+1)$ and $F(n,1)=g(2n,2n+1)$ that means that $\mathcal{R}^{(1)}\restriction_{\{(n,0),(n,1)\}}$ and $\mathcal{R}^{(1)}\restriction_{\{(m,0),(n,1)\}}$ are not isomorphic, proving that (5.1) holds. In Case $(b)$, $\mathcal{R}_{\restriction\{f(2n),g(2n,2n+1)\}}$ and $\mathcal{R}_{\restriction\{f(2m+1),g(2n,2n+1)\}}$ are not isomorphic. By invariance, $\mathcal{R}_{\restriction\{f(2m),g(2n,2n+1)\}}$ and $\mathcal{R}_{\restriction\{f(2m+1),g(2n,2n+1)\}}$ are isomorphic, hence

$\mathcal{R}\restriction_{\{f(2n),g(2n,2n+1)\}}$ and $\mathcal{R}\restriction_{\{f(2m),g(2n,2n+1)\}}$ are not isomorphic; since $F(n,0):=f(2n),F(m,0)=f(2m)$ and $F(n,1)=g(2n,2n+1)$ that means that $\mathcal{R}^{(1)}\restriction_{\{(n,0),(n,1)\}}$ and $\mathcal{R}^{(1)}\restriction_{\{(m,0),(n,1)\}}$ are not isomorphic, proving that (5.1) holds. $\diamond$

Claim 4**.**

The set $\mathfrak{A}_{k}^{1}$ of ordered binary structures $\mathcal{R}^{(1)}$ obtained by this process is finite.

Proof. According to Claim 2, $\mathcal{R}^{(1)}$ is entirely defined by its values on the pairs $((i,j),(i^{\prime},j^{\prime}))$ of vertices taken among the four vertices $(0,0)$, $(0,1)$, $(1,0)$ and $(1,1)$; the values on the other pairs will be deduced by taking local isomorphisms of $C:=({\mathbb{N}},\leq^{(1)})$ and using Claim 2. $\diamond$

Let us give a hint about the form of the structures which arise (the full description in the case of a single binary relation is given in Section 6).

Due to its invariance, the map $d^{(1)}$ is determined by its values on the five ordered pairs $((0,0),(1,0))$, $((0,1),(1,1))$, $((0,0),(0,1))$, $((0,0),(1,1))$, $((0,1),(1,0))$. The only requirement for those pairs, due to Condition (5.1), is that

[TABLE]

On each ordered pair, $d^{(1)}$ can take $4^{k}$ values. On five pairs, this gives $4^{5k}$ possibilities, but $4^{4k}$ are forbidden (those for which $d^{(1)}$ takes the same values on the pairs $((0,0),(0,1))$ and $((0,1),(1,0))$). This gives $4^{4k}.(4^{k}-1)$ possibilities. In fact, some of the resulting structures embed into some others. Thus the number of non equimorphic structures is a bit less.

5.2. Case 2.

$\mathcal{R}$ satisfies the Assertion $(2)$ of Lemma 4.8.

Since $A_{1}$ is infinite, there is an countable sequence $(x_{n})_{n\in{\mathbb{N}}}$ of elements of $A_{1}$ which is either increasing or decreasing. Let $(y_{n})_{n\in{\mathbb{N}}}$ such that $y_{n}\in I_{\leq}(x_{n},x_{n}+1)\cap A_{2}$. Set $A^{\prime}_{0}:=\{x_{n}:n\in{\mathbb{N}}\}$, $A^{\prime}_{1}:=\{y_{n}:n\in{\mathbb{N}}\}$.

Then $(A^{\prime}_{1}\cup A^{\prime}_{2},\leq_{\restriction_{A^{\prime}_{1}\cup A^{\prime}_{2}}})$ is ordered as $\omega$ or $\omega^{*}$. Each of the sets $A^{\prime}_{1}$ and $A^{\prime}_{2}$ is contained into a class of $1$-equivalence. Conditions (4.2) and (4.3) of Fact 1 are satisfied for every $x,y\in A_{i}$ and $z\in A_{j}$, $j\neq i$. We have then two situations:

First situation. There exists $x_{0}\in V\setminus(A^{\prime}_{1}\cup A^{\prime}_{2})$ which witnesses the fact that $A^{\prime}_{1}$ and $A^{\prime}_{2}$ are contained in two different classes of $1$-equivalence. As $A^{\prime}_{1}\cup A^{\prime}_{2}$ is ordered as $\omega$ or $\omega^{\star}$, we may suppose that we have either $x_{0}\leq a$ or $x_{0}\geq a$, for every $a\in A^{\prime}_{1}\cup A^{\prime}_{2}$, because otherwise, we can find some cofinite subset of $A^{\prime}_{1}\cup A^{\prime}_{2}$ for which we have this condition.

We may then find maps $f^{\prime},g^{\prime},g^{\prime\prime}:{\mathbb{N}}\longrightarrow V$ such that, $f^{\prime}({\mathbb{N}})=A^{\prime}_{1}$, $g^{\prime}({\mathbb{N}})=A^{\prime}_{2}$, $g^{\prime\prime}({\mathbb{N}}):=\{x_{0}\}$, $g^{\prime}(n)\in I_{\leq}(f^{\prime}(n),f^{\prime}(n+1)),\leavevmode\nobreak\ \forall n\in{\mathbb{N}}$ and the restrictions of $\mathcal{R}$ to $\{f^{\prime}(n),g^{\prime}(n),g^{\prime\prime}(0)\}$ and $\{f^{\prime}(m),g^{\prime}(n),g^{\prime\prime}(0)\}$ are not isomorphic for every $n,m\in{\mathbb{N}}$ (in fact the restrictions of $\mathcal{R}$ to $\{f^{\prime}(n),g^{\prime\prime}(0)\}$ and $\{g^{\prime}(m),g^{\prime\prime}(0)\}$ are not isomorphic for every $n,m\in{\mathbb{N}}$.

As in Case (5.1), we define a map $F^{\prime}:\{a\}\cup({\mathbb{N}}\times\{0,1\})\longrightarrow V(\mathcal{R})$, with $a\notin{\mathbb{N}}\times\{0,1\}$, such that $F^{\prime}(a):=g^{\prime\prime}(0)$, $F^{\prime}(n,0):=f^{\prime}(n)$ and $F^{\prime}(n,1):=g^{\prime}(n)$.

The map $F^{\prime}$ being $1$-to-$1$, we may define an ordered binary structure of type $k$, $\mathcal{R}^{(2)}:=(V^{(2)},\leq^{(2)},\rho_{1}^{(2)},\dots,\rho_{k}^{(2)})$ with vertex set $V^{(2)}:=\{a\}\cup({\mathbb{N}}\times\{0,1\})$ such that $F^{\prime}$ is an embedding from $\mathcal{R}^{(2)}$ into $\mathcal{R}$. This amounts to

[TABLE]

for every $x,y\in{\mathbb{N}}\times\{0,1\}$, where $d^{(2)}(x,y):=(d_{i}^{(2)}(x,y))_{i=1,\dots,k}$ and $d_{i}^{(2)}(x,y):=(\rho_{i}^{(2)}(x,y),\rho_{i}^{(2)}(y,x))$ for every $1\leq i\leq k$.

By construction, $\mathcal{R}^{(2)}$ satisfies Condition ((5.3)) below, hence has infinitely many equivalence classes.

[TABLE]

The ordered binary structure $\mathcal{R}^{(2)}$ satisfies Observation 1 stated below which follows directly from the fact that the elements of $A_{i}$ for $i\in\{1,2\}$ are $1$-equivalent.

Observation 1**.**

1. (1)

$a\leq^{(2)}(n,i)\Leftrightarrow a\leq^{(2)}(m,i),\leavevmode\nobreak\ \forall n<m,\leavevmode\nobreak\ i\in\{0,1\}$. 2. (2)

$a\leq^{(2)}(n,0)\Leftrightarrow a\leq^{(2)}(n,1),\leavevmode\nobreak\ \forall n\in{\mathbb{N}}$. 3. (3)

$d^{(2)}(a,(n,i))=d^{(2)}(a,(m,i)),\leavevmode\nobreak\ \forall n<m,\leavevmode\nobreak\ i\in\{0,1\}$. 4. (4)

$d^{(2)}((n,i),(m,i))=d^{(2)}((n^{\prime},i),(m^{\prime},i)),\forall n<m,\leavevmode\nobreak\ n^{\prime}<m^{\prime},\leavevmode\nobreak\ i\in\{0,1\}$. 5. (5)

$d^{(2)}((n,0),(n,1))=d^{(2)}((n,1),(m,0)),\leavevmode\nobreak\ \forall n<m$. 6. (6)

$d^{(2)}((n,0),(n,1))=d^{(2)}((m,0),(m,1)),\leavevmode\nobreak\ \forall n<m$. 7. (7)

$d^{(2)}((n,0),(m,1))=d^{(2)}((m,1),(m+1,0)),\leavevmode\nobreak\ \forall n<m$.

With this we obtain a finite subset $\mathfrak{B}_{k}^{1}$ of ordered binary structures with the same vertex set ${\mathbb{N}}\times\{0,1\}\cup\{a\}$.

Second situation. There is no vertex $x_{0}$ as above. Then, since the elements of $A^{\prime}_{i}$, $i\in\{1,2\}$, are $1$-equivalent, we can deduce from Fact 1 that two vertices $x,y$ of $A^{\prime}_{i}$ are separated by two vertices $z,z^{\prime}$ such that $z\in I_{\leq}(x,y)\cap A^{\prime}_{j}$ with $j\neq i$ and $z^{\prime}\notin I_{\leq}(x,y)$. In this case and by Lemma 4.8, the relation between two elements of $A^{\prime}_{i}$, for at least one $i=1,2$, is different from the relation between two elements $x,y$, with $x\in A^{\prime}_{1}$ and $y\in A^{\prime}_{2}$. We can then define two maps $f_{1},g_{1}:{\mathbb{N}}\longrightarrow V(\mathcal{R})$ such that, $f_{1}({\mathbb{N}})=A^{\prime}_{1}$, $g_{1}({\mathbb{N}})=A^{\prime}_{2}$, $g_{1}(n)\in I_{\leq}(f_{1}(n),f_{1}(n+1)),\leavevmode\nobreak\ \forall n\in{\mathbb{N}}$.

Set $F^{\prime\prime}:{\mathbb{N}}\times\{0,1\}\longrightarrow V(\mathcal{R})$ such that $F^{\prime\prime}(n,0):=f(n)$ and $F^{\prime\prime}(n,1):=g(n)$.

We can define an ordered binary structure $\mathcal{R}^{(3)}:=(V^{(3)},\leq^{(3)},\rho_{1}^{(3)},\dots,\rho_{k}^{(3)})$ with vertex set $V^{(3)}={\mathbb{N}}\times\{0,1\}$ such that

[TABLE]

for every $x,y\in{\mathbb{N}}\times\{0,1\}$, where $d^{(3)}(x,y):=(d_{i}^{(3)}(x,y))_{i=1,\dots,k}$ and $d_{i}^{(3)}(x,y):=(\rho_{i}^{(3)}(x,y),\rho_{i}^{(3)}(y,x))$ for every $1\leq i\leq k$. As we said before, by construction of $\mathcal{R}^{(3)}$, the order $\leq^{(3)}$ is isomorphic to $\omega$ or $\omega^{*}$ with $(n,1)\in I_{\leq^{(3)}}((n,0),(n+1,0))$ and for every $n<m$, we have either

[TABLE]

or

[TABLE]

or

[TABLE]

Then, with the fact that $A^{\prime}_{1}$ and $A^{\prime}_{2}$ are, each one, included into a class of $1$-equivalence, $\mathcal{R}^{(3)}$ satisfies the following observation.

Observation 2**.**

1. (1)

$d^{(3)}((n,0),(n,1))=d^{(3)}((m,0),(m,1)),\leavevmode\nobreak\ \forall n<m$. 2. (2)

$d^{(3)}((n,0),(n,1))=d^{(3)}((n,1),(m,0)),\leavevmode\nobreak\ \forall n<m$. 3. (3)

$d^{(3)}((n,0),(m,1))=d^{(3)}((m,1),(m+1,0)),\leavevmode\nobreak\ \forall n<m$. 4. (4)

$d^{(3)}((n,i),(m,i))=d^{(3)}((n^{\prime},i),(m^{\prime},i)),\leavevmode\nobreak\ \forall n<m,\leavevmode\nobreak\ n^{\prime}<m^{\prime},\leavevmode\nobreak\ \leavevmode\nobreak\ i\in\{0,1\}$.

It is then clear that $\mathcal{R}^{(3)}$ has infinitely many equivalence classes. By construction, the set $\mathfrak{B}_{k}^{2}$ of ordered binary structure obtained in this case is finite.

First, we conclude that $\mathfrak{A}_{k}^{1}\cup\mathfrak{B}_{k}^{1}\cup\mathfrak{B}_{k}^{2}$ is finite. Hence, the set $\mathfrak{A}_{k}$ of minimal ordered binary structures (w.r.t embeddability) of $\mathfrak{A}_{k}^{1}\cup\mathfrak{B}_{k}^{1}\cup\mathfrak{B}_{k}^{2}$ is finite. Next, by construction, $\mathfrak{A}_{k}$ is a basis. With that, the proof of the first part of Theorem 1.6 is complete.

6. Description of the ordered directed graphs

We give in this section the proof of the second part of Theorem 1.6.

Let $\mathfrak{A}_{1}$ be the set of ordered binary structures $\mathcal{R}:=(V,\leq,\rho)$ defined in the previous section and let $\check{\mathfrak{A}}_{1}$ be the subset made of the ordered reflexive directed graphs $\mathcal{G}:=(V,\leq,\rho)$ of the set $\mathfrak{A}_{1}$. The members of $\check{\mathfrak{A}}_{1}$ are almost multichains on $F\cup(L\times K)$ such that $L:={\mathbb{N}}$, $|K|=2$ and $|F|\leq 1$. We prove that the set $\check{\mathfrak{A}}_{1}$ contains one thousand two hundred and forty two members, such that $|\check{\mathfrak{A}}_{1}\cap\mathfrak{A}_{1}^{1}|=1122$, $|\check{\mathfrak{A}}_{1}\cap\mathfrak{B}_{1}^{1}|=48$ and $|\check{\mathfrak{A}}_{1}\cap\mathfrak{B}_{1}^{2}|=72$.

According to the nature of these graphs due, in part to the nature of the order $\leq$, we classify these graphs into several classes which we describe below.

Denote by $\mathcal{G}_{\ell,k}^{(p)}:=(V_{\ell,k}^{(p)},\leq_{\ell,k}^{(p)},\rho_{\ell,k}^{(p)})$ the ordered directed graphs of $\check{\mathfrak{A}}_{1}$, where $p,\ell$ and $k$ are non-negative integers such that $1\leq p\leq 10$, $1\leq\ell\leq 6$. The set of vertices $V_{\ell,k}^{(p)}$ is either ${\mathbb{N}}\times\{0,1\}$ (if $\mathcal{G}_{\ell,k}^{(p)}$ is in $\mathfrak{A}_{1}^{1}\cup\mathfrak{B}_{1}^{2}$) or ${\mathbb{N}}\times\{0,1\}\cup\{a\}$ (if $\mathcal{G}_{\ell,k}^{(p)}$ is in $\mathfrak{B}_{1}^{1}$).

The ordered graphs with the same value of $p$ are said of class $p$, their restrictions to $A:={\mathbb{N}}\times\{0\}$ are identical and their restrictions to $B:={\mathbb{N}}\times\{1\}$ also. If they have the same value of $\ell$, then the linear orders $(V,\leq)$ have the same order-type, $\ell$ takes values from $1$ to $6$ if the linear order is isomorphic to respectively $\omega$, $\omega^{*}$, $\omega+\omega$, $\omega^{*}+\omega$, $\omega+\omega^{*}$, $\omega^{*}+\omega^{*}$. We do not consider the cases where the linear order $\leq$ is isomorphic to $\underline{2}^{*}.\omega$, or to $\underline{2}.\omega^{*}$ because all the ordered directed graphs which are obtained in this case are isomorphic to some ones for which the order $\leq$ is isomorphic to $\omega$ or $\omega^{\star}$. The integer $k$ enumerates the graphs for all values of $p$ and $\ell$. Different classes have not necessarily the same cardinalities.

For $p=1$ if $\ell=1,2$ we have $1\leq k\leq 18$ and if $3\leq\ell\leq 6$ we have $1\leq k\leq 15$. For $p=2,3,4$, if $\ell=1,2$ we have $1\leq k\leq 21$ and if $3\leq\ell\leq 6$ we have $1\leq k\leq 15$.

For $5\leq p\leq 10$, if $\ell=1,2$ we have $1\leq k\leq 22$ and if $3\leq\ell\leq 6$ we have $1\leq k\leq 24$. The total is one thousand two hundred and forty two as claimed.

In each class $p$, when $\ell=1$, the linear order $\leq_{\ell,k}^{(p)}$ is isomorphic to $\omega$. In this case we have, $(0,0)<_{\ell,k}^{(p)}(0,1)<_{\ell,k}^{(p)}(1,0)$ when the vertex set is ${\mathbb{N}}\times\{0,1\}$, and $a<_{\ell,k}^{(p)}(0,0)<_{\ell,k}^{(p)}(0,1)<_{\ell,k}^{(p)}(1,0)$ when this set is ${\mathbb{N}}\times\{0,1\}\cup\{a\}$. The order is reversed when $\ell=2$. If $\ell\geq 3$, the vertex set is ${\mathbb{N}}\times\{0,1\}$. All the ordered directed graphs given for $\ell\geq 3$ belong to $\mathfrak{A}_{1}^{1}$ and for $p\geq 5$ they all belong to $\mathfrak{A}_{1}^{1}\cup\mathfrak{B}_{1}^{2}$.

We will give a graphical representations for some classes. All these representations are done on the following six vertices $(0,0)$, $(1,0)$, $(2,0)$, $(0,1)$, $(1,1)$, $(2,1)$ for the graphs of $\mathfrak{A}_{1}^{1}\cup\mathfrak{B}_{1}^{2}$ and on the following seven vertices $a$, $(0,0)$, $(1,0)$, $(2,0)$, $(0,1)$, $(1,1)$, $(2,1)$ for those of $\mathfrak{B}_{1}^{1}$ (the loops are not shown). These representations are given for $\ell=1$ (the linear order $\leq_{\ell,k}^{(p)}$ is isomorphic to $\omega$).

Recall that a graph $G$ which is isomorphic to its dual is said self-dual, and if $G$ is a directed graph, the symmetrized of $G$ is the graph $G^{\prime}$ obtained from $G$ by adding every edge $u:=(x,y)$ such that $u^{-1}:=(y,x)$ is an edge of $G$. Thus $G^{\prime}$ may be considered as an undirected graph.

Let $\mathcal{G}:=(V,\leq,\rho)$ be an ordered reflexive directed graph. The subset $E$ of $V^{2}$ such that $(x,y)\in E$ if and only if $\rho(x,y)=1$ is the edge set of $\mathcal{G}$ and $G:=(V,E)$ is the directed graph associated to the ordered directed graph $\mathcal{G}$. For $x,y\in V$, set $d(x,y):=(\rho(x,y),\rho(y,x))$.

We are now ready to describe our ordered reflexive directed graphs of $\check{\mathfrak{A}}_{1}$ given in Theorem 1.6. According to our notations, we have just to describe the associated directed graphs $G_{\ell,k}^{(p)}=(V_{\ell,k}^{(p)},E_{\ell,k}^{(p)})$.

For $n\in{\mathbb{N}}$, set $a_{n}:=((n,0),(n,1))$.

Class $p=1$: The restrictions of $G_{\ell,k}^{(1)}$ to sets $A:={\mathbb{N}}\times\{0\}$ and $B:={\mathbb{N}}\times\{1\}$ are both antichains.

I) If $\ell=1,2$ then $1\leq k\leq 18$.

The graphs $\mathcal{G}_{\ell,k}^{(1)}$ for $1\leq k\leq 9$ are in $\mathfrak{A}_{1}^{1}$, they are in $\mathfrak{B}_{1}^{2}$ for $10\leq k\leq 12$ and in $\mathfrak{B}_{1}^{1}$ for $13\leq k\leq 18$.

$\bullet$ For $1\leq k\leq 12$. A pair $(x,x^{\prime})$ of vertices, where $x=(n,i),x^{\prime}=(n^{\prime},i^{\prime})$, is

• $\centerdot$

an edge of $G_{\ell,1}^{(1)}$ if $n=n^{\prime}$ and $i<i^{\prime}$;

• $\centerdot$

an edge of $G_{\ell,2}^{(1)}$ if $(x^{\prime},x)$ is an edge of $G_{\ell,1}^{(1)}$. Thus $G_{\ell,2}^{(1)}$ is the dual of $G_{\ell,1}^{(1)}$;

• $\centerdot$

an edge of $G_{\ell,3}^{(1)}$ if $n=n^{\prime}$ and $i\neq i^{\prime}$. The graph $G_{\ell,3}^{(1)}$ is self-dual;

• $\centerdot$

an edge of $G_{\ell,4}^{(1)}$ if $n\leq n^{\prime}$ and $i<i^{\prime}$;

• $\centerdot$

an edge of $G_{\ell,5}^{(1)}$ if $(x^{\prime},x)$ is an edge of $G_{\ell,4}^{(1)}$. Thus $G_{\ell,5}^{(1)}$ is the dual of $G_{\ell,4}^{(1)}$;

• $\centerdot$

an edge of $G_{\ell,6}^{(1)}$ if it is either an edge of $G_{\ell,4}^{(1)}$ or an edge of $G_{\ell,5}^{(1)}$. Thus $G_{\ell,6}^{(1)}$ is the symmetrized of $G_{\ell,4}^{(1)}$ (and of $G_{\ell,5}^{(1)}$), it is self-dual;

• $\centerdot$

an edge of $G_{\ell,7}^{(1)}$ if $i<i^{\prime}$. The graph $G_{\ell,7}^{(1)}$ is equimorphic to its dual.

• $\centerdot$

an edge of $G_{\ell,8}^{(1)}$ if either ($n\leq n^{\prime}$ and $i<i^{\prime}$) or ($n>n^{\prime}$ and $i<i^{\prime}$) or ($n<n^{\prime}$ and $i>i^{\prime}$).

• $\centerdot$

an edge of $G_{\ell,9}^{(1)}$ if $(x^{\prime},x)$ is an edge of $G_{\ell,8}^{(1)}$. Thus $G_{\ell,9}^{(1)}$ is the dual of $G_{\ell,8}^{(1)}$;

• $\centerdot$

an edge of $G_{\ell,10}^{(1)}$ if either ($n\leq n^{\prime}$ and $i<i^{\prime}$) or ($n<n^{\prime}$ and $i>i^{\prime}$);

• $\centerdot$

an edge of $G_{\ell,11}^{(1)}$ if $(x^{\prime},x)$ is an edge of $G_{\ell,10}^{(1)}$. Thus $G_{\ell,11}^{(1)}$ is the dual of $G_{\ell,10}^{(1)}$;

• $\centerdot$

an edge of $G_{\ell,12}^{(1)}$ if $i\neq i^{\prime}$. The graph $G_{\ell,12}^{(1)}$ is the symmetrized of $G_{\ell,10}^{(1)}$ (and of $G_{\ell,11}^{(1)}$).

Denote by $\underline{2}$ the poset made of $2:=\{0,1\}$ ordered so that $0<1$. The poset $\underline{2}^{*}$ is its dual. Denote by $K_{2}$ the reflexive clique on two vertices and by $\Delta_{{\mathbb{N}}}$ the antichain with ${\mathbb{N}}$ as vertex set. With this notation, $G_{\ell,1}^{(1)}$ is isomorphic to $\underline{2}.\Delta_{{\mathbb{N}}}$, the lexicographic product of $\underline{2}$ by $\Delta_{{\mathbb{N}}}$ (that is the antichain $\Delta_{{\mathbb{N}}}$ where every vertex is replaced by the chain $\underline{2}$). The graph $G_{\ell,2}^{(1)}$ is isomorphic to $\underline{2}^{*}.\Delta_{{\mathbb{N}}}$, the graph $G_{\ell,3}^{(1)}$ is isomorphic to $K_{2}.\Delta_{{\mathbb{N}}}$, the graph $G_{\ell,6}^{(1)}$ is the half complete bipartite graph of Schmerl- Trotter [33] and the graph $G_{\ell,7}^{(1)}$ is isomorphic to the ordinal sum $\Delta_{{\mathbb{N}}}+\Delta_{{\mathbb{N}}}$.

$\bullet$ For $13\leq k\leq 18$, the vertex set of the graph is $\{a\}\cup A\cup B$.

A pair $(x,x^{\prime})$ of vertices is

• $\centerdot$

an edge of $G_{\ell,13}^{(1)}$ if $x=a,x^{\prime}=(n,1)$;

• $\centerdot$

an edge of $G_{\ell,14}^{(1)}$ if $(x^{\prime},x)$ is an edge of $G_{\ell,13}^{(1)}$. Thus $G_{\ell,14}^{(1)}$ is the dual of $G_{\ell,13}^{(1)}$;

• $\centerdot$

an edge of $G_{\ell,15}^{(1)}$ if it is either an edge of $G_{\ell,13}^{(1)}$ or an edge of $G_{\ell,14}^{(1)}$. Thus $G_{\ell,15}^{(1)}$ is the symmetrized of $G_{\ell,13}^{(1)}$ (and also of $G_{\ell,14}^{(1)}$);

• $\centerdot$

an edge of $G_{\ell,16}^{(1)}$ if either $x=a$ and $x^{\prime}=(n,0)$ or $x=(n,1)$ and $x^{\prime}=a$; this graph is self-dual;

• $\centerdot$

an edge of $G_{\ell,17}^{(1)}$ if it is either an edge of $G_{\ell,16}^{(1)}$ or an edge of $G_{\ell,13}^{(1)}$;

• $\centerdot$

an edge of $G_{\ell,18}^{(1)}$ if $(x^{\prime},x)$ is an edge of $G_{\ell,17}^{(1)}$. Thus $G_{\ell,18}^{(1)}$ is the dual of $G_{\ell,17}^{(1)}$;

The graphical representations of these graphs are given in Figure 6.

II) If $3\leq\ell\leq 6$, we have the same examples for each value of $\ell$ and their number is $15$, according to the linear order $\leq_{\ell,k}^{(1)}$, the elements of $A$ are placed before those of $B$. In these cases, $\mathcal{G}_{\ell,k}^{(1)}\in\mathfrak{A}_{1}^{1}$ for every $1\leq k\leq 15$.

$\centerdot$ $G_{\ell,k}^{(1)}=G_{1,k}^{(1)}$ for every $1\leq k\leq 6$.

$\centerdot$ $G_{\ell,k}^{(1)}=G_{1,k+1}^{(1)}$ for every $7\leq k\leq 10$.

$\centerdot$ $G_{\ell,11}^{(1)}$ is obtained from $G_{1,7}^{(1)}$ by adding all edges $((n,1),(m,0))$ for $n\geq m$.

$\centerdot$ $G_{\ell,12}^{(1)}$ is obtained from $G_{1,10}^{(1)}$ by adding all edges $((n,1),(m,0))$ for $n\geq m$, the graph $G_{\ell,12}^{(1)}$ is the dual of $G_{\ell,11}^{(1)}$

$\centerdot$ $G_{\ell,13}^{(1)}$ is undirected. Its edge set is $\{\{(n,0),(n^{\prime},1)\};n\neq n^{\prime}\}$.

$\centerdot$ $G_{\ell,14}^{(1)}$ is obtained from $G_{\ell,13}^{(1)}$ by adding all edges $a_{n}$ for $n\in{\mathbb{N}}$.

$\centerdot$ $G_{\ell,15}^{(1)}$ is obtained from $G_{\ell,13}^{(1)}$ by adding all edges $a_{n}^{-1}$ for $n\in{\mathbb{N}}$.