some information about the regular D-classes and number of J-classes of End(B).
Lemma 7.42. There exist2ℵ0 non-isomorphic group H -classes of End(B) and hence 2ℵ0 distinct regular D-classes where the group H-classes in dif- ferent D-classes are non-isomorphic.
Proof. In the proof of Corollary 3.27 we saw that there exist 2ℵ0 non-isomorphic countable groups. Corollary 7.39 tells us that End(B) has a group H -class isomorphic to each of these and hence there exist 2ℵ0 non-isomorphic group
H -classes of End(B). However, we also know that if two group H-classes are contained in the same D-class, then they must be isomorphic. Hence these 2ℵ0 non-isomorphic maximal group H -classes must be contained in 2ℵ0 distinct regular D-classes of End(B) and the result follows.
Theorem 7.43. There exist 2ℵ0 distinct regular D-classes of End(B) for which any two group H -classes are isomorphic.
Proof. In Theorem 2.10 we saw that iff andgare two elements ofE(End(B)) then fDg if and only if the induced bipartite graphs himfi and himgi are isomorphic. By the details of Theorem 7.38, if Γ is a countable graph, then there exist 2ℵ0 sets Σ ⊆
N\ {0,1} such that HfΣ ∼= Aut(Γ) and such that
himfΣi himfΨi for any ψ ⊆ N\ {0,1} with Σ 6= Ψ. Therefore these idempotents are contained in 2ℵ0 distinct regular D-classes of End(B) but the groupH -classesHfΣ are all isomorphic. Since any other groupH -class contained in one of these D-classes must then also be isomorphic to Aut(Γ) the result follows.
So far all of the results obtained have been analogous to a result proved for the random graph, R. The next example illustrates that End(B) has
D-classes which are ‘smaller’ than any of those of End(R), thus exhibiting a difference in the semigroup theoretic structure of these two semigroups.
Example 7.44. Letf ∈E(End(B)) and suppose that imf is isomorphic to Λ = ({u, v},({(u, v),(v, u)},{(u, u),(v, v)})).
Then Df contains exactly countably many group H-classes.
Proof. First we note that since Λ is algebraically closed an application of Theorem 7.19 guarantees the existence of an idempotent f ∈ End(B) with imf ∼= Λ. By Theorem 2.10, we know that an idempotent g ∈ End(B) lies in Df if and only if img ∼= Λ. Since VB is countable and since B is
existentially closed, the number of finite subsets U ⊆VΓ such that hUi ∼= Λ is ℵ0. Furthermore, we can show that there exists exactly one idempotent g such that img = hUi for each fixed U = {x, y} as follows. First note that we can assume without loss of generality that x ∈ V0 and y ∈ V1, where
VB = V0 ∪V1 is the bipartition of B. Thus if such an idempotent g existed then g|U = 1U and so by Lemma 7.2 we can conclude that V0g = x and
V1g =y. Clearly, this lone map is an idempotent homomorphism of B and is then the unique map such that img = hUi. Thus there exists exactly ℵ0 idempotents g ∈Df and hence exactly ℵ0 group H-classes in Df.
Theorem 7.45. Let g ∈E(End(B)) and suppose that img is strongly alge- braically closed. Then Dg contains 2ℵ0 group H-classes.
Proof. Since img is algebraically closed we can apply Lemma 7.19 to show that there are 2ℵ0 distinct idempotents with image isomorphic to img. By Theorem 2.10 these idempotents are all D-related and therefore lie in Dg.
However since no groupH-class can contain more than one idempotent they lie in distinct H-classes and the result follows.
Theorem 7.46. There exist 2ℵ0 distinct J-classes of End(B).
Proof. By Lemma 2.11, we know that two maps g, h ∈ E(End(B)) are J- related, if and only ifhimhican be embedded in to himgiandvice versa. By Lemma 3.31, there exists a set P of 2ℵ0 subsets of
N\ {0,1} such that if Σ,
Ψ∈P then Σ +k6⊆Ψ and Ψ +k 6⊆Σ for allk ∈N. Hence by Lemma 7.30 ΛΣ cannot be embedded into ΛΨ for any Σ, Ψ∈P. Furthermore by Lemma 7.27 it follows that Λ‡Σ cannot be embedded into Λ‡Ψ for any Σ, Ψ ∈ P. Now by Lemma 7.32, Λ†Σ is an algebraically closed bipartite graph for all Σ ∈ P. Thus for all Σ ∈ P there exists an idempotent fΣ ∈ End(B) such that imfΣ ∼= Λ
‡
Σ. By our previous observations, fΣ and fΨ cannot be J- related for sets Σ 6= Ψ in P. Since P had size 2ℵ0 it now follows that these idempotents must be contained in 2ℵ0 distinctJ-classes of End(B).
Chapter 8
The Total Order
Q
In this chapter we consider the well known total order Q. We will find that it is a great deal more complicated to determine the maximal subgroups of End(Q). We will first show that we can characterise the exact subsets of Q which are the image of some idempotent from E(End(Q)). We call these subsets retracts of Q. We will then show that if Ω is a total order and there exists an embedding f : Ω → Q such that imf is a retract of
Q, then there exist 2ℵ0 maximal subgroups of End(Q) which are isomorphic
to Aut(Ω). We will also show that there exist regular D-classes of End(Q) which contain countably many group H -classes as well as regular D-classes which contain 2ℵ0 group H-classes.
8.1
Defining Properties
It is well known that the class of all finite linear orders has the hereditary, joint embedding and amalgamation properties and thus that this class has a unique Fra¨ıss´e limit. We will first show that this Fra¨ıss´e limit has particular properties.
Recall that a total order Ω = (VΩ,≤Ω) is called dense if for all u, v ∈VΩ with u <Ω v there exists x ∈ VΩ such that u <Ω x <Ω v. Additionally, Ω is said to be without endpoints if for allu∈VΩ there existsy, z ∈VΩ such that
y <Ω u <Ω z.
Notice that if Ω = (VΩ,≤Ω) is a dense total order, then for any u, v ∈VΩ withu <Ω vthere must exist infinitely many elementsxsuch thatu <Ω x <Ω
v. Similarly if the total order Ω is without endpoints, then there must exist infinitely many elements y and infinitely many elements z such that y <Ω u
and v <Ωz. Clearly, total orders which are dense or without endpoints must then be infinite. Furthermore, the following theorem is well known.
Theorem 8.1. Let Ω = (VΩ,≤) be a countable dense total order without endpoints. Then every countable total order can be embedded into Ω.
Proof. First, let VΩ ={qj :j ∈N}. Now let Λ = (VΛ,≤Λ) be any countable total order. Enumerate the elements in VΛ as {vi : i ∈ N}, replacing the
natural numbers by a finite set when necessary. Now inductively define a sequence of functions as follows. Let f0 :{v0} →VΩ be defined byv0f0 =q0. Then clearly f0 is an embedding of hv0i into Ω. Now suppose that forn∈N we have defined fn :{v0, . . . , vn} →Ω which is an embedding of hv0, . . . , vni
into Ω. Let
N− ={vi :vi <Λvn+1,0≤i≤n}, and let
N+ ={vi :vn+1 <Λvi,0≤i≤n}.
Suppose that N−, N+ 6=∅. Then since Ω is dense there exists qj ∈ VΩ such that (N−)fn < qj <(N+)fn. On the other hand if N− =∅ or N+ =∅, then since Ω is without endpoints there exists qj ∈VΩ such that qj < (N+)fn or
(N−)fn < qj respectively. In any case define fn+1 :{v0, . . . , vn+1} →Ω by,
vkfn+1 =
(
vkfn if k = 0, . . . , n
qj if k =n+ 1.
Then clearly fn+1 is an injective map since by assumption fn was and since
qj 6= vkfn for k = 0, . . . , n. Furthermore, vn+1 <Λ vk for some k = 0, . . . , n
if and only if qj =vn+1fn+1 < vkfn+1 and similarly vk <Λ vn+1 if and only if
vkfn+1 < vn+1fn+1 =qj. Thus fn+1 is an embedding of hv0, . . . , vn+1iinto Ω. Now let
f = [
n∈N
fn.
Then as the union of embeddings fn such thatfn+1 is an extension of fn for
all n∈N, f is an embedding of Λ into Ω.
By Theorem 8.1, the age of any countable dense total order without endpoints is exactly the class of all finite total orders. It thus follows that any countable total order which is dense and without endpoints is isomorphic to the unique homogeneous Fra¨ıss´e limit of the class of finite total orders. Of course (Q,≤), the set of rational numbers with the natural ordering, is a countable total order which is both dense and without endpoints. Thus
(Q,≤)is the Fra¨ıss´e limit of the class of finite total orders. For convenience, we will often abuse notation in this chapter and write Q to mean (Q,≤).
As usual (Q,≤) can be thought of as a substructure of the relational structure (R∪ {−∞,∞},≤), the set of affinely extended real numbers with the natural ordering. This allows for the definition of an interval in Q with real or infinite endpoints as follows. Forp, q ∈R∪{−∞,∞}define theclosed interval in Qwith closed endpoints p≤q by
[p, q] ={x∈Q:p≤x≤q}.
The open interval in Q with open endpointsp and q will be defined by (p, q) ={x∈Q:p < x < q}.
We similarly define the right closed interval in Q with left open endpoint p
and right closed endpoint q by
(p, q] ={x∈Q:p < x≤q}
and the left closed interval in Q with left closed endpoint p and right open endpoint q by,
[p, q) = {x∈Q:p≤x < q}.
Note that if p ∈ (R\Q)∪ {−∞} then the intervals in Q given by (p, q) and [p, q) are equal as are the intervals [p, q] and (p, q]. Similarly if q ∈
(R\Q)∪ {∞} then (p, q) = (p, q] and [p, q] = [p, q). Also worth mentioning is that if p∈R∪ {−∞,∞}, then (p, p] = [p, p) = (p, p) =∅. Furthermore if
p∈Q then [p, p] =pand if p6∈Q then [p, p] =∅.
The term non-closed interval will be used to mean an interval which cannot be written in the form [p, q] where p, q ∈ R∪ {−∞,∞}. Thus the empty set is not a non-closed interval since [p, p] = ∅ for all p ∈ R\Q. As we will see in the next subsection, non-closed intervals play a key part in the structure of the images of idempotents from End(Q).