1 “BULLYING” O ACOSO ESCOLAR
2. LA RELACIÓN DE ACOSO ESCOLAR COMO PROCESO
The notion of self-duality is a central notion for the original Wadge hierarchy. We recall that we call a set A∈ P(R)self-dual if and only ifA≡∗
WR\A, which in fact is equivalent toA≤∗WR\A. The distinction between self-dual and non-self-dual sets is important for the Wadge hierarchy in particular when we try to define operations on Wadge degrees, since often their properties differ depending on whether they are applied to self-dual or non-self-dual Wadge degrees. Many examples for this can be found in the book drafts [And01] and [And00].
On first glance one could conjecture that the usefulness of this distinction between self-dual and non-self-dual Wadge degrees is intimately connected to the fact that the original Wadge hierarchy has antichains of length 2 and that these are all of the form{[A]∗W,[R\A]∗W}for someA ∈ P(R). As a consequence there would be no such distinction for the hierarchy of norms, which underADis a linear order. However, this is not true. Benedikt L¨owe observed in unpublished notes [L¨ow10] based on ideas from Duparc’s article [Dup03] that there is a notion of self-duality for the hierarchy of norms sharing many properties with the notion of self-duality in the case of the original Wadge hierarchy. He makes the following definition.
Definition 3.4.1 We call a regular norm ϕself-dualiff PlayerIIhas a winning strategy in the game
G<W(ϕ, ϕ).
We will use this notion throughout Chapter 4, in which we will define several operations on regular norms which have certain desirable properties only when restricted to self-dual regular norms. Unifying
the concept of self-duality for the original Wadge hierarchy and the hierarchy of regular norms, we now define the following general notion of self-duality.
Definition 3.4.2 Let(Q,≤)be a quasi-order. We call aQ-normϕ L-self-dualiff PlayerIIhas a winning strategy in the gameG6≥L(ϕ, ϕ). We call aQ-normϕ W-self-dualiff PlayerIIhas a winning strategy in the gameG6≥W(ϕ, ϕ). We call aQ-normϕself-dualiff it is bothL-self-dual andW-self-dual.
This notions extends to degrees as follows.
Lemma 3.4.3. Let(Q,≤)be a quasi-order. Letϕand ψbeQ-norms such thatϕ ≡L ψ. Thenϕis L-self-dual if and only ifψis L-self-dual. The same is true forLreplaced withW.
Proof. We only show the claim for the Lipschitz case. The Wadge case is completely analogous. We assume thatϕ≡L ψandϕis L-self-dual. Then we have three Lipschitz functionsf, g, h:R→Rsuch
that for allx∈R
ϕ(f(x))6≤ϕ(x), ϕ(x)≤ψ(g(x)), ψ(x)≤ϕ(h(x)).
Then we consider the Lipschitz functiong◦f ◦h: R→ Rand claim that for allx∈Rwe have that ψ(x)6≥ψ(g◦f ◦h(x)). To see this we assume for a contradiction that it is not the case, i.e., we take an x∈Rsuch thatψ(g◦f ◦h(x))≤ψ(x). But then we get by the properties of the functionf, g, hthat
ϕ(f◦h(x))≤ψ(g◦f◦h(x))≤ψ(x)≤ϕ(h(x))
and so we have thatϕ(f(h(x)))≤ϕ(h(x)), contradicting our choice off.
In light of this result we call a(Q,≤)-Wadge degreecself-dual iff everyϕ∈ cis W-self-dual and we call a(Q,≤)-Lipschitz degreec self-dual iff every everyϕ ∈ c is L-self-dual. Next we see that these general notion of self-duality coincides with the ones for the original Wadge hierarchy and for the hierarchy of norms. For the original Wadge hierarchy we note the following sequence of equivalences
Ais self-dual ⇔ A≤∗WR\A
⇔ there is continuousf :R→Rsuch that for allx∈R: x∈A⇔x /∈A
⇔ there is continuousf :R→Rsuch that for allx∈R: χA(x)6≥TVχA(f(x))
⇔ PlayerIIwins the gameG6≥TVW (χA, χA),
which shows that the new notion of W-self-duality in this case coincides with the old notion. For the hierarchy of norms we note that(Θ,≤)is a linear order and so for any twoα, β∈Θwe have thatα < β if and only ifα6≥βand so vacuously we have for anyϕ∈Nthat
PlayerIIwinsG<W(ϕ, ϕ) ⇔ PlayerIIwinsG6≥W(ϕ, ϕ).
Next we will state and prove the natural generalization of the Steel-Van Wesep theorem to this context. This proof is original work.
Theorem 3.4.4(Steel-Van Wesep). Let(Q,≤)be a vsBQO. Then aQ-normϕis L-self-dual if and only if it is W-self-dual.
Proof. Clearly for anyQ-normϕ, if PlayerIIhas a winning strategy in the gameG6≥L(ϕ, ϕ), then this is also a winning strategy in the gameGW6≥(ϕ, ϕ). Hence ifϕisL-self-dual, thenϕis alsoW-self-dual.
We show the other direction by contraposition. For this we assume that there is aQ-normϕsuch that ϕisW-self-dual, but notL-self-dual. Then we show that(Q,≤)is no vsBQO.
We fix a winning strategyσ2:ω<ω\ {∅} →ω∪ {p}for PlayerIIin the gameG2:= G6≥
W(ϕ, ϕ)and byADfurthermore a winning strategyσ0:ω<ω→ωfor PlayerIin the gameG0:= G6≥L(ϕ, ϕ). Finally we define a winning strategyσ1 :ωω\ {∅} →ωfor PlayerIIin the gameG1 := G=L(ϕ, ϕ)by setting for alls∈ωω\ {∅}
Now for anyx∈3ωwe define a game sequenceGx:=hhGx(n), σx(n)i |n∈ωi. To construct a bad
generalized(Q,≤)-array we first construct a sequencehMk |k∈ωiof natural numbers such that for all
x∈3ωwith the property thatx(M
k)∈ {1,2}for allk∈ωand thatx(n) = 0for alln /∈ {Mk |k∈ω}
the game sequenceGxadmits a global play.
We define this sequence recursively. First we setM0 := 0. Then we assume that we have already constructed the sequencehMk |k≤jifor somej ∈ω. Now for anys∈3Mj+1such that
∀i≤j(s(Mi)∈ {1,2}), (∗)
and ∀m≤Mj (m /∈ {Mk|k≤j} ⇒s(m) = 0), (∗∗)
we define a natural numberms as follows. Letxs := s_0(ω). Then by Proposition 2.2.4 the game
sequenceGxs admits a global playFs:ω →R. Now letzs:=Fs(Mj+ 1). Then for anyi≤Mj we
define a functiongi:ω≤ω→ω≤ωsuch thatgiR:R→Ris continuous by setting for anyy∈ω≤ω
gi(y) := filter(σs(i)(y)).
Then we setg := g0◦g1◦ · · · ◦gMj. As a composition of continuous functionsgR : R → Ris
also a continuous function and furthermore we have that Fs(0) = g(zs). The continuous function
gR : R → Ris induced bygω<ω : ω<ω → ω<ω and thus we letms be the leastm ∈ ωsuch that
g(zsm)⊇g(zs)Mj+1. Using this construction we finally defineMj+1by setting
Mj+1:=Mj+ max{ms|s∈3Mj+1andssatisfies(∗)and(∗∗)},
concluding the construction ofhMk |k∈ωi.
Now for anyX ∈ P(ω)we define a sequencezX∈3ωby setting for anyi∈ω
zX(i) := 1, if there isk∈ωs.t.i=Mkandk /∈X, 2, if there isk∈ωs.t.i=Mkandk∈X, 0, otherwise.
But then by construction of the sequence hMk | k ∈ ωiwe have that for every X ∈ P(ω) the
game sequenceGzX admits a unique global play, which we callF
X. Using this we define a generalized
(Q,≤)-arrayG: [ω]ω→Qby setting for anyX ∈[ω]ω
G(X) :=ϕ(FX(MminX+ 1)).
Now we show thatGis bad. For this we fixX ∈[ω]ωand note that since for allj ≥minX∗ we have
j ∈X if and only ifj ∈X∗, we get thatFX(Mmin
X∗+ 1) = FX ∗
(MminX∗+ 1)and therefore that
ϕ(FX(MminX∗+ 1)) =ϕ(FX ∗
(MminX∗+ 1)). Now we show that for allj ∈ωwith
MminX+ 1≤j≤MminX∗
we have thatϕ(FX(M
minX∗)) ≤ ϕ(FX(j))using induction onMminX∗−k. Ifj = minX∗, then
nothing is to show. Now we assume for somej withMminX + 1 < j ≤MminX∗ that the claim has
been established, i.e., that we have thatϕ(FX(M
minX∗))≤ϕ(FX(j)). Now we have to distinguish two
cases.
Case 1is that(j−1)∈ {Mk |k∈ω}, sayj−1 =Mi. Then sinceMminX < Mi < MminX∗we
have thati /∈Xand sozX(j−1) = 1, which implies thatFX(j−1) = filter(σ1(FX(j))). But sinceσ1
is a winning strategy for PlayerIIin the gameG=
L(ϕ, ϕ), this implies thatϕ(FX(j−1)) =ϕ(FX(j)) and so by induction hypothesisϕ(FX(Mmin
X∗))≤ϕ(FX(j−1)).
Case 2 is that (j −1) ∈ {M/ k | k ∈ ω}. Then zX(j − 1) = 0 and so FX(j − 1) =
filter(σ0(FX(j))). But sinceσ0is a winning strategy for PlayerIin the gameG6≥W(ϕ, ϕ), this implies thatϕ(FX(j−1))≥ϕ(FX(j))and so by induction hypothesisϕ(FX(MminX∗))≤ϕ(FX(j−1)).
This concludes the induction, in particular showing that
ϕ(FX(MminX∗))≤ϕ(FX(MminX+ 1)).
But now we note that since clearlyminX∗∈X∗we have that
But since σ2 is a winning strategy for Player II in the game G6≥(ϕ, ϕ) we thus get that ϕ(FX(M
minX∗+ 1))6≥ϕ(FX(MminX∗)). Sinceϕ(FX(MminX∗ + 1)) = ϕ(FX ∗
(MminX∗+ 1))
we thus in total have that
ϕ(FX∗(MminX∗+ 1))6≥ϕ(FX(MminX∗))
and
ϕ(FX(MminX∗))≤ϕ(FX(MminX+ 1)).
Now we assume towards a contradiction thatϕ(FX(Mmin
X+ 1))≤ϕ(FX ∗ (MminX∗+ 1)). Then we haveϕ(FX(Mmin X∗))≤ϕ(FX ∗
(MminX∗+ 1)), a contradiction. Hence we have that
ϕ(FX(MminX+ 1))6≤ϕ(FX
∗
(MminX∗+ 1)),
i.e., G(X) 6≤ G(X∗), which shows thatGis a bad generalized(Q,≤)-array and hence that Qis no vsBQO, concluding the proof.
The two corollaries for the original Wadge hierarchy and the hierarchy of norms re-establishing The- orem 2.3.5 and an analogous result for the hierarchy of regular norms are the following.
Corollary 3.4.5. For anyA∈ P(R)we have thatAis W-self-dual if and only ifAis L-self-dual. Thus A≤∗
WR\Aif and only ifA≤∗LR\A.
Corollary 3.4.6. For any regular normϕwe have thatϕis W-self-dual if and only ifϕis L-self-dual. Thus PlayerIIwins the gameG<W(ϕ, ϕ)if and only if PlayerIIwins the gameG<L(ϕ, ϕ).
We conclude this section with noting a very useful property of non-self-dualQ-norms, which we are going to use in Chapter 4.
Proposition 3.4.7. AssumeAD. Let(Q,≤)be an arbitrary vsBQO. Forψa non-self-dualQ-norm and ϕan arbitraryQ-norm we have thatϕ6≥Wψif and only if PlayerIIwins the gameG6≥W(ϕ, ψ). Also we have thatϕ6≥L ψif and only if PlayerIIwins the gameG6≥L(ϕ, ψ).
Proof. We only show the first part of the proposition. The second part is completely analogous. For the left-to-right direction we assume that PlayerII wins the gameG6≥W(ϕ, ψ), but ϕ ≥W ψ. Then there are continuous functionsf, g : R → Rsuch that for allx ∈ Rwe haveϕ(x) 6≥ ψ(f(x))and
ψ(x)≤ϕ(g(x)). We claim that thus for allx∈ Rwe haveϕ(x) 6≥ ϕ(g◦f(x)). To see this we take
towards a contradiction anx∈Rsuch thatϕ(x)≥ϕ(g◦f(x)). Then in particular
ϕ(x)≥ϕ(g(f(x)))≥ψ(f(x)),
a contradiction. But sinceg◦f is a continuous function it now follows that PlayerIIwins the game
G6≥W(ϕ, ϕ), contradicting the non-self-duality ofϕ.
For the right-to-left direction we assume that PlayerIIloses the gameG6≥W(ϕ, ψ). But then byAD
PlayerIhas a winning strategyσ : (ω∪ {p})<ω → ω inG6≥W(ϕ, ψ). Then we can define a winning strategyτ:ω<ω\ {∅} →ωfor PlayerIIin the gameG≤
L(ψ, ϕ)by setting for anys∈ω<ω\ {∅} τ(s) :=σ(slh(s)−1).
This is indeed winning forIIsince for anyx∈Rwe haveσ(x) =τ(x)and so
ψ(x)≤ϕ(σ(x)) =ϕ(τ(x)). Thus we haveψ≤Lϕand soψ≤Wϕ.
An important corollary of this proposition is that for any non-self-dual(Q,≤)-norm we have that
[ϕ]W= [ϕ]L.
Corollary 3.4.8. AssumeAD. Let(Q,≤)be an arbitrary vsBQO. Then for any non-self-dualQ-norm ϕand anyQ-normψwe have thatϕ≡Lψif and only ifϕ≡Wψ.
Proof. The left-to-right direction is obvious, since ϕ ≤L ψimplies ϕ ≤W ψ andψ ≤L ϕimplies ψ≤Lψ.
For the right-to-left direction we assume towards a contradiction thatϕ≡W ψ, butϕ6≡L ψ. Since ϕ ≡W ψ andϕis non-self-dual by assumption we get that also ψis non-self-dual. Without loss of generality we now assume thatϕ 6≥L ψ. Then we have by Proposition 3.4.7 that PlayerIIwins the gameG6≥L(ϕ, ψ), i.e., that there is a Lipschitz functionf : R→Rsuch that for allx∈Rwe have that ϕ(x)6≥ψ(f(x)). But since every Lipschitz function is also continuous we note that PlayerIIalso wins the gameG6≥W(ϕ, ψ), which implies thatϕ6≥Wψ, but this contradicts the assumption thatϕ≡Wψ.