Corollary 4.20. There are 2ℵ0 distinct regular D-classes of End(D) such that no two group H -classes from distinct D-classes are isomorphic.
Proof. By Corollary 4.14 there exist 2ℵ0 non-isomorphic group H-classes of End(D). Since each of these must lie in a distinct regularD-class the result follows.
Corollary 4.21. There are 2ℵ0 distinct regular D-classes of End(D) whose group H-classes are all isomorphic.
Proof. By the proof of Theorem 4.11, if Γ is a (symmetric) directed graph then there exist 2ℵ0 idempotents f
Σ ∈ EEnd(D) such that HfΣ ∼= Aut(Γ) but such that himfΣi 6∼= himfΨi for Σ 6= Ψ. Hence by Theorem 2.10, each
fΣ is contained in a distinct regular D-class but the HfΣ are all isomorphic to Aut(Γ). Since every group H -class contained in one of these D-classes must also be isomorphic to Aut(Γ) the result is complete.
Corollary 4.22. Each regularD-class ofEnd(D)contains2ℵ0 distinct group
H -classes.
Proof. If aD-class is regular, it contains at least one groupH -class. Letf ∈ E(End(D)) be the identity of the groupH-class. Since imf is algebraically closed Theorem 4.10 guarantees the existence 2ℵ0 distinct idempotents whose images induce subgraphs which are isomorphic to himfi. By Theorem 2.10 these idempotents all lie in the same D-class, but since no H -class can contain more than one idempotent they lie in distinct group H -classes.
Corollary 4.23. There are 2ℵ0 distinct J-classes of End(D) .
Proof. In the proof of 3.32 we saw that there exist 2ℵ0 algebraically closed (symmetric) directed graphs which are mutually non-embeddable. By The- orems 4.10 and 2.11, there thus exist 2ℵ0 idempotents in End(D) which are not J-related and hence the result follows.
Chapter 5
The Random Tournament
In this chapter we will briefly discuss the random tournament and see why, in the context of this thesis, its endomorphism monoid is a somewhat less interesting structure.
5.1
Defining Properties and Constructions
Recall that a tournament is a directed graph in which for every pair of distinct vertices there exists exactly one edge between them (in one direction or the other). It is not hard to show that the class of finite tournaments has the hereditary, joint embedding and amalgamation properties. Therefore, the class of finite tournaments has a unique homogeneous Fra¨ıss´e limit which we will call the random tournament, T. We can show that T has the following properties.We will say that a tournament Γ is existentially closed in the class of tournaments if for all finite subsets U1, U2 ∈ VΓ there exists a vertex x ∈
VΓ\(U1∪U2) such that there exists an edge from xto every vertex inU1 and from every vertex inU2 tox. For the remainder of this chapter a tournament which is said to be existentially closed should be assumed to be existentially closed in the class of tournaments.
Clearly any existentially closed tournament must be infinite, for if Γ is a finite tournament then VΓ is a finite set for which there exists no vertex x with an edge from x to every member of VΓ.
Theorem 5.1. Let Γ be an existentially closed tournament. Then Γ is ho- mogeneous and every finite tournament can be embedded into Γ.
For a proof see for example [Hod97] or alternatively, the construction described in Definition 5.3 will make this clear. Theorem 5.1 tells us that the age of any existentially closed tournament Γ is exactly the class of all finite tournaments. Since the class of finite tournaments has a unique homogeneous Fra¨ıss´e limit it follows that if Γ is an existentially closed tournament then Γ∼=T. As one might expect, we can probabilistically carry out a construction of an existentially closed tournament as follows.
Theorem 5.2. Let Λ be a countable tournament constructed as follows. Let
VΛ be a countably infinite set, and for any two distinct vertices u, v ∈ VΛ chose either (u, v) or (v, u) to be in the edge set (each with probability 12) independently from any other pair of distinct vertices. Then with probability 1, Λ is existentially closed.
Proof. Let U1 and U2 be finite subsets of VΛ. Suppose that |U1| = m and
|U2|=n for m, n∈N. We will say that a vertex x∈VΛ\(U1∪U2) is joined correctly to U1 and U2 if there exists an edge from x to every vertex of U1 and an edge from every vertex of U2 tox. The probability that a vertex xis not joined correctly is
1− 1
2m+n
and is independent from the probability that any other distinct vertex y is not joined correctly. Now since VΛ is infinite, the probability that no vertex of VΛ\(U1∪U2) is joined correctly to U1 and U2 is,
lim k→∞ 1− 1 2m+n k = 0.
Thus the probability that existential closure is not satisfied for the sets U1 and U2 is 0. Since there are only countably many choices for the setsU1 and
U2 it follows that the probability that Λ is not existentially closed is 0 and hence it is existentially closed with probability 1.
As in the other settings, there exists a standard explicit construction of the random tournament from any given tournament.
Definition 5.3. Starting with any countable tournament Γ we can create a new tournament J(Γ) by the addition of vertices and edges. Since Γ is countable we can enumerate the finite subsets of VΓ as {Ui}i∈N where the natural numbers can be replaced by a finite set if Γ is finite. Now for each finite set Ui add a vertex vi and edges from vi to every vertex in Ui and
tournament we need to have an edge between each pair{vi, vj}. The direction
of these edges turns out to be irrelevant. So we let
VJ(Γ)=VΓ∪ {vi :i∈N},
and
EJ(Γ)=EΓ∪ {(vi, u),(w, vi) :u∈Ui, w ∈VΓ\Ui}
∪ {(vi, vj) :i, j ∈N, i < j}.
If Γ is a finite graph then |VJ(Γ)|= 2|VΓ|+|VΓ|and henceJ(Γ) is also a finite tournament. If Γ is in fact countably infinite, then since the set of all finite sets of VΓ is also countably infinite, J(Γ) is countably infinite itself.
Now inductively define a sequence of tournaments by setting Γ0 = Γ and Γn+1 = J(Γn) for all n ∈ N\ {0}. Let Γ∞ be the limit of this process so that, Γ∞ = [ n∈N Γn = [ n∈N VΓn, [ n∈N EΓn ! .
Since Γ∞ is a countable union of tournaments Γn such that Γn−1 is con- tained in Γnfor alln ∈N\{0}, it should be easy to see that Γ∞is a countable tournament itself.
Example 5.4. [Construction of Γ1 given Γ.]
Γ0 = Γ ◦ // ◦ Γ1 ◦ // //WW // / // // / GGGG## G G G G G G G G G G G G h h ◦ W W // // / // // / G G { { wwwwww ww wwwwww ww v v • oo o o o o • oo o o • oo •
Since the construction of Γ∞is dependent on the enumeration of the finite setsU ⊆VΓ, it may seem plausible that taking a different enumeration would give us a different (non-isomorphic) graph. However the following theorem proves that this is not true.
Theorem 5.5. Let Γ be a countable tournament. Then Γ∞ is existentially closed and thus Γ∞ ∼=T.
Proof. Suppose that U1 and U2 are finite and disjoint subsets of VΓ∞. Then U1, U2 ⊆ Γk for some k ∈ N. By construction of Γk+1 there exists a vertex
v ∈ VΓk+1 \VΓk such that there is an edge from v to every vertex in U1 and
from every vertex in VΓ\U1 to v. In particular this means that there is an edge from v to every vertex in U1 and from every vertex in U2 to v in Γk+1.
Since Γk+1is contained as an induced substructure of Γ∞it follows that there is an edge from v to every vertex inU1 and from every vertex in U2 to v in Γ∞. Thus since U1 and U2 are arbitrary, Γ∞ is existentially closed.
The construction of Γ∞ from a countable tournament Γ should make it clear that any finite tournament can be embedded into T.