We let $n\in{\rm Nature}$ and
$K_{0},\dots,K_{n},H_{0},\dots,H_{n}\in\mathcal{D}$ be as defined in Fact 8.4.
We may now prove that
$S\subseteq\mathcal{I}_{{Bool\/}(\mathcal{D})}\left[L,\rho\right]$. Let $s\in
S$. By hypothesis on $S$, we have $s\in f_{2n}(\tau_{2n}(L))$. Therefore,
since $\\{K_{i}\setminus H_{i}\mid i\leq n\\}$ is by definition a cover of
$L$, it is immediate from Lemma 8.3 that there exists $j\leq n$ such that
$s\leq\rho(K_{j}\setminus H_{j})$. Since $\\{K_{i}\setminus H_{i}\mid i\leq
n\\}$ is an optimal ${Bool\/}(\mathcal{D})$-cover of $L$ for $\rho$, this
implies that $s\in\mathcal{I}_{{Bool\/}(\mathcal{D})}\left[L,\rho\right]$
which concludes the main proof.

We turn to the proof of Lemma 8.3. The argument is an induction on $n\in{\rm
Nature}$. We start with the base case $n=0$.

Base case. Consider $L,K_{0},H_{0}\in\mathcal{D}$ such that $\\{K_{0}\setminus
H_{0}\\}$ is a cover of $L$ and let $s\in f_{0}(\tau_{0}(L))$. We have to show
that $s\leq\rho(K_{0}\setminus H_{0})$. By definition of $f_{0}$, we get
$r_{1},\dots,r_{k}\in\tau_{0}(L)$ such that $s\leq r_{1}+\cdots+r_{k}$.
Moreover, the definition of $\tau_{0}$ yields that for every $i\leq k$,
$r_{i}=\rho(w_{i})$ for some $w_{i}\in L$. Therefore, $r_{i}\leq\rho(L)$ for
every $i\leq k$ and since $R$ is idempotent for addition, $s\leq
r_{1}+\cdots+r_{k}\leq\rho(L)$. Finally, since $\\{K_{0}\setminus H_{0}\\}$ is
a cover of $L$, we have $L\subseteq K_{0}\setminus H_{0}$ and we get
$s\leq\rho(K_{0}\setminus H_{0})$, finishing the argument for the base case.

Inductive step. We now assume that $n\geq 1$. Let
$L,K_{0},\dots,K_{n},H_{0},\dots,H_{n}\in\mathcal{D}$ such that
$\\{K_{i}\setminus H_{i}\mid i\leq n\\}$ is a cover of $L$ and let $s\in
f_{2n}(\tau_{2n}(L))$. We have to exhibit $j\leq n$ such that
$s\leq\rho(K_{j}\setminus H_{j})$. Using the hypothesis that $s\in
f_{2n}(\tau_{2n}(L))$, we prove the following fact.

######  Fact 8.5.

There exists $(r,U)\in\tau_{2n}(L)$ such that $s\in f_{2n-1}(U)$.

###### Proof.

By definition of $f_{2n}$, the hypothesis that $s\in f_{2n}(\tau_{2n}(L))$
yields
$(r^{\prime}_{1},U^{\prime}_{1}),\dots,(r^{\prime}_{k},U^{\prime}_{k})\in\tau_{2n}(L)$
such that $r^{\prime}_{1}+\cdots+r^{\prime}_{k}\in f_{2n-1}(U^{\prime}_{i})$
for every $i\leq k$ and $s\leq r^{\prime}_{1}+\cdots+r^{\prime}_{k}$. We let
$(r,U)=(r^{\prime}_{1},U^{\prime}_{1})\in\tau_{2n}(L)$. We have
$r^{\prime}_{1}+\cdots+r^{\prime}_{k}\in f_{2n-1}(U)$ and $s\leq
r^{\prime}_{1}+\cdots+r^{\prime}_{k}$. Hence, since $f_{2n-1}(U)$ is closed
under downset (by definition), this implies $s\in f_{2n-1}(U)$. ∎

Recall that by definition, $\tau_{2n}$ is the rating map
$\zeta_{\mathcal{D}}^{\rho_{*}}[\tau_{2n-1}]$. Hence, $(r,U)\in\tau_{2n}(L)$
means that:

$U\in\mathcal{I}_{\mathcal{D}}\left[\rho_{*}^{-1}(r)\cap
L,\tau_{2n-1}\right],$

which yields, by Fact 4.6, that,

$U\in\mathcal{I}_{\mathcal{D}}\left[L,\tau_{2n-1}\right].$

Since $\\{K_{i}\setminus H_{i}\mid i\leq n\\}$ is a cover of $L$,
$L,K_{1},\dots,K_{n}\in\mathcal{D}$ and $\mathcal{D}$ is a lattice, it follows
that $\\{L\cap K_{i}\mid i\leq n\\}$ is a $\mathcal{D}$-cover of $L$.
Therefore, since $U\in\mathcal{I}_{\mathcal{D}}\left[L,\tau_{2n-1}\right]$, we
obtain some $\ell\leq n$ such that $U\subseteq\tau_{2n-1}(L\cap K_{\ell})$.

Furthermore, since $s\in f_{2n-1}(U)$, we may unravel the definition of
$f_{2n-1}$ which yields $(r_{1},U_{1}),\dots,(r_{k},U_{k})\in U$ such that
$r_{1}+\cdots+r_{k}\in f_{2(n-1)}(U_{m})$ for every $m\leq k$ and $s\leq
r_{1}+\cdots+r_{k}$. Note that by definition of $f_{2(n-1)}$, this also
implies that $s\in f_{2(n-1)}(U_{m})$ for every $m\leq k$. We now distinguish
two sub-cases.

Sub-case 1: Assume that for every $m\leq k$, we have,

$\rho_{*}^{-1}(r_{m})\cap(K_{\ell}\setminus H_{\ell})\neq\emptyset$

This means that for every $m\leq k$, we have $w_{m}\in K_{\ell}\setminus
H_{\ell}$ such that $\rho(w_{m})=r_{m}$. In particular,
$r_{m}=\rho(w_{m})\leq\rho(K_{\ell}\setminus H_{\ell})$ for every $m\leq k$.
Finally, since $R$ is idempotent for addition, we get
$r_{1}+\cdots+r_{k}\leq\rho(K_{\ell}\setminus H_{\ell})$. Since $s\leq
r_{1}+\cdots+r_{k}$, we get $s\leq\rho(K_{\ell}\setminus H_{\ell})$ and Lemma
8.3 holds for $j=\ell$ in this case.

Sub-case 2: Conversely, assume that there exists $m\leq k$ such that,

$\rho_{*}^{-1}(r_{m})\cap(K_{\ell}\setminus H_{\ell})=\emptyset$

This implies that $K_{\ell}\cap\rho_{*}^{-1}(r_{m})\subseteq K_{\ell}\cap
H_{\ell}$. Recall that $(r_{m},U_{m})\in U$ and that
$U\subseteq\tau_{2n-1}(L\cap K_{\ell})$. Since $\tau_{2n-1}$ is the rating map
$\zeta_{\rho_{*},\tau_{2(n-1)}}^{\mathcal{D}}[b]$ y definition, it follows
that,

$U_{m}\in\mathcal{I}_{\mathcal{D}}\left[\rho_{*}^{-1}(r_{m})\cap L\cap
K_{\ell},\tau_{2(n-1)}\right]$

Combined with the inclusion $K_{\ell}\cap\rho_{*}^{-1}(r_{m})\subseteq
K_{\ell}\cap H_{\ell}$ and Fact 4.6, this yields that,

$U_{m}\in\mathcal{I}_{\mathcal{D}}\left[L\cap K_{\ell}\cap
H_{\ell},\tau_{2(n-1)}\right]$

Since $\mathcal{D}$ is a lattice, it is clear that $\\{L\cap K_{\ell}\cap
H_{\ell}\\}$ is a $\mathcal{D}$-cover of $L\cap K_{\ell}\cap H_{\ell}$. Thus,
we obtain that $U_{m}\subseteq\tau_{2(n-1)}(L\cap K_{\ell}\cap H_{\ell})$.
Therefore, since $s\in f_{2(n-1)}(U_{m})$ by definition of $U_{m}$, we obtain
from Fact 8.1 that,

$s\in f_{2(n-1)}(\tau_{2(n-1)}(L\cap K_{\ell}\cap H_{\ell}))$

Finally, since $\\{K_{i}\setminus H_{i}\mid i\leq n\\}$ was a cover of $L$, it
is clear that $\\{K_{i}\setminus H_{i}\mid i\leq n\text{ and }i\neq\ell\\}$
(of size $n-1$) is a cover of $L\cap K_{\ell}\cap H_{\ell}$. Therefore, it
follows by induction on $n$ in Lemma 8.3 that there exists $j\leq n$ (with
$j\neq\ell$) such that $s\leq\rho(K_{j}\setminus H_{j})$, finishing the proof.
∎

###  8.2. Soundness proof for Theorem 5.3

We may now come back to our main objective: soundness in Theorem 5.3. We fix a
finite quotient-closed Boolean algebra $\mathcal{C}$ and a multiplicative
rating map $\rho:2^{A^{*}}\to R$. We show that for every
${BPol\/}(\mathcal{C})$-saturated subset
$S\subseteq({A^{*}}/{\sim_{\mathcal{C}}})\times R$ (for $\rho$), we have
$S\subseteq\mathcal{P}_{{BPol\/}(\mathcal{C})}^{\mathcal{C}}[\rho]$.

######  Remark 8.6.

Note that we do not use the hypothesis that $\rho$ is nice for this direction.
This is only needed for completeness.

By Theorem 3.2, ${Pol\/}(\mathcal{C})$ is a lattice. Therefore, we may
instantiate the definitions and results presented at the beginning of the
section for our nice multiplicative rating map $\rho:2^{A^{*}}\to R$ in the
special case when $\mathcal{D}={Pol\/}(\mathcal{C})$. We keep using the same
notation: we have the rating maps $\tau_{n}:2^{A^{*}}\to Q_{n}$ (as we prove
below, they are now quasi-multiplicative since
$\mathcal{D}={Pol\/}(\mathcal{C})$ and $\rho$ is multiplicative) and the maps
$f_{n}:Q_{n}\to 2^{R}$.

We complete Proposition 8.2 with another statement specific to this special
case. In fact, the proof of this second proposition is based on Theorem 7.1,
the characterization of ${Pol\/}(\mathcal{C})$-optimal imprints (we apply it
to the rating maps $\tau_{n}$). Together, these two results imply soundness in
Theorem 5.3.

######  Proposition 8.7.

Consider $S\subseteq({A^{*}}/{\sim_{\mathcal{C}}})\times R$ which is
${BPol\/}(\mathcal{C})$-saturated for $\rho$. Then, the following inclusion
holds for every $D\in{A^{*}}/{\sim_{\mathcal{C}}}$,

$S(D)\subseteq\bigcap_{n\in{\rm Nature}}f_{n}(\tau_{n}(D)).$

When put together, Proposition 8.2 and Proposition 8.7 imply soundness in
Theorem 5.3. Indeed, consider a ${BPol\/}(\mathcal{C})$-saturated set
$S\subseteq({A^{*}}/{\sim_{\mathcal{C}}})\times R$. We show that
$S\subseteq\mathcal{P}_{{BPol\/}(\mathcal{C})}^{\mathcal{C}}[\rho]$. This
amounts to proving that for every $D\in{A^{*}}/{\sim_{\mathcal{C}}}$, we have,

$S(D)\subseteq\mathcal{P}_{{BPol\/}(\mathcal{C})}^{\mathcal{C}}[\rho](D)=\mathcal{I}_{{BPol\/}(\mathcal{C})}\left[D,\rho\right].$

It is immediate from Proposition 8.7 that,

$S(D)\subseteq\bigcap_{n\in{\rm Nature}}f_{n}(\tau_{n}(D)).$

Moreover, since $D\in{A^{*}}/{\sim_{\mathcal{C}}}$, we have $D\in\mathcal{C}$,
which implies that $D\in{Pol\/}(\mathcal{C})$. Therefore, Proposition 8.2
yields that,

$\bigcap_{n\in{\rm
Nature}}f_{n}(\tau_{n}(D))\subseteq\mathcal{I}_{{BPol\/}(\mathcal{C})}\left[D,\rho\right].$

Altogether, we get the desired inclusion:
$S(D)\subseteq\mathcal{I}_{{BPol\/}(\mathcal{C})}\left[D,\rho\right]$. This
concludes the soundness proof.

It remains to prove Proposition 8.7. We start with a few additional results
about the rating maps $\tau_{n}$ that we are able to prove using our new
hypotheses (_i.e._ , that $\rho$ is multiplicative and
$\mathcal{D}={Pol\/}(\mathcal{C})$).

Preliminaries. Recall that $\rho$ is multiplicative. Hence, $R$ is a semiring
$(R,+,\cdot)$. Since $Q_{0}=2^{R}$ and $Q_{n}=2^{R\times Q_{n-1}}$ for all
$n\geq 1$, we may lift the multiplication of $R$ to all the rating sets
$Q_{n}$ in the natural way. It is simple to verify that $(Q_{n},\cup,\cdot)$
is a semiring for every $n\in{\rm Nature}$. We first show that the rating maps
$\tau_{n}$ are quasi-multiplicative for these multiplications.

######  Lemma 8.8.

The rating map $\tau_{0}:2^{A^{*}}\to Q_{0}$ is multiplicative. Moreover, for
every $n\geq 1$, the rating map $\tau_{n}:2^{A^{*}}\to Q_{n}$ is quasi-
multiplicative for the following endomorphism $\mu_{\tau_{n}}$ of
$(Q_{n},\cup)$:

$\mu_{\tau_{n}}(T)=\mathord{\downarrow_{Q_{n-1}}}\\{(r,\mu_{\tau_{n-1}}(V))\mid(r,V)\in
T\\}\quad\text{for every $T\in Q_{n}=2^{R\times Q_{n-1}}$}.$

###### Proof.

Recall that $Q_{0}=2^{R}$ and $\tau_{0}(K)=\\{\rho(w)\mid w\in K\\}$ for every
language $K$. It is immediate that $\tau_{0}$ is multiplicative since $\rho$
is multiplicative by hypothesis. The result for the rating maps $\tau_{n}$ for
$n\geq 1$ is then immediate from Lemma 6.8 using induction on $n$ since
${Pol\/}(\mathcal{C})$ is a quotient-closed lattice closed under concatenation
by Theorem 3.2. ∎

We complete this result with two lemmas which connect the hypothesis that the
rating maps $\tau_{n}$ are quasi-multiplicative with the maps $f_{n}:Q_{n}\to
2^{R}$.

######  Lemma 8.9.

For every $n\in{\rm Nature}$ and $T\in Q_{n}$, we have $f_{n}(T)\subseteq
f_{n}(\mu_{\tau_{n}}(T))$.

###### Proof.

We proceed by induction on $n\in{\rm Nature}$. When $n=0$, $\tau_{0}$ is
multiplicative and the endomorphism $\mu_{\tau_{0}}$ of $(Q_{0},\cup)$ is the
identity on $Q_{0}$. Hence, the lemma is immediate. Assume now that $n\geq 1$.
Consider $T\in Q_{n}$. Let $r\in f_{n}(T)\subseteq R$. By definition, we have
$(r_{1},T_{1}),\dots,(r_{k},T_{k})\in T$ such that $r_{1}+\cdots+r_{k}\in
f_{n-1}(T_{i})$ for every $i\leq k$ and $r\leq r_{1}+\cdots+r_{k}$. By
definition of $\mu_{\tau_{n}}$ in Lemma 8.8, we have
$(r_{1},\mu_{\tau_{n-1}}(T_{1})),\dots,(r_{k},\mu_{\tau_{n-1}}(T_{k}))\in\mu_{\tau_{n}}(T)$.
Moreover, since $r_{1}+\cdots+r_{k}\in f_{n-1}(T_{i})$ for every $i\leq k$ it
is immediate by induction hypothesis that $r_{1}+\cdots+r_{k}\in
f_{n-1}(\mu_{\tau_{n-1}}(T_{i}))$ for every $i\leq k$. Altogether, we obtain
that $r\in f_{n}(\mu_{\tau_{n}}(T))$, finishing the proof of the lemma. ∎

######  Lemma 8.10.

For every $n\in{\rm Nature}$ and $T,T^{\prime}\in Q_{n}$, we have
$f_{n}(T)\cdot f_{n}(T^{\prime})\subseteq f_{n}(T\cdot T^{\prime})$.

###### Proof.

We proceed by induction on $n\in{\rm Nature}$. We first handle the case $n=0$.
Let $r\in f_{0}(T)\cdot f_{0}(T^{\prime})$. We have $s\in f_{0}(T)$ and
$s^{\prime}\in f_{0}(T^{\prime})$ such that $r=ss^{\prime}$. By definition,
this yields $r_{1},\dots,r_{k}\in T$ and
$r^{\prime}_{1},\dots,r^{\prime}_{k^{\prime}}\in T^{\prime}$ such that $s\leq
r_{1}+\cdots+r_{k}$ and $s^{\prime}\leq
r^{\prime}_{1}+\cdots+r^{\prime}_{k^{\prime}}$. It follows that
$ss^{\prime}\leq\sum_{i\leq k}\sum_{j\leq k^{\prime}}r_{i}r^{\prime}_{j}$.
Since $r_{i}r^{\prime}_{j}\in T\cdot T^{\prime}$ for every $i\leq k$ and
$j\leq k^{\prime}$, it follows that $ss^{\prime}\in f_{0}(T\cdot T^{\prime})$.

Assume now that $n\geq 1$. Let $r\in f_{n}(T)\cdot f_{n}(T^{\prime})$. We have
$s\in f_{n}(T)$ and $s^{\prime}\in f_{n}(T^{\prime})$ such that
$r=ss^{\prime}$. By definition, this yields
$(r_{1},T_{1}),\dots,(r_{k},T_{k})\in T$ and
$(r^{\prime}_{1},T^{\prime}_{1}),\dots,(r^{\prime}_{k^{\prime}},T^{\prime}_{k^{\prime}})\in
T^{\prime}$ such that,

  * •

$s\leq r_{1}+\cdots+r_{k}$ and $r_{1}+\cdots+r_{k}\in f_{n-1}(T_{i})$ for
every $i\leq k$.

  * •

$s^{\prime}\leq r^{\prime}_{1}+\cdots+r^{\prime}_{k^{\prime}}$ and
$r^{\prime}_{1}+\cdots+r^{\prime}_{k^{\prime}}\in f_{n-1}(T^{\prime}_{j})$ for
every $j\leq k^{\prime}$.

Clearly, we have $ss^{\prime}\leq\sum_{i\leq k}\sum_{j\leq
k^{\prime}}r_{i}r^{\prime}_{j}$. Moreover, for every $i\leq k$ and $j\leq
k^{\prime}$, we have,

$\sum_{i\leq k}\sum_{j\leq k^{\prime}}r_{i}r^{\prime}_{j}\in
f_{n-1}(T_{i})\cdot f_{n-1}(T^{\prime}_{j}).$

By induction hypothesis, this yields,

$\sum_{i\leq k}\sum_{j\leq k^{\prime}}r_{i}r^{\prime}_{j}\in
f_{n-1}(T_{i}\cdot T^{\prime}_{j}).$

Finally, it is immediate that for every $i\leq k$ and $j\leq k^{\prime}$, we
have $(r_{i}r^{\prime}_{j},T_{i}\cdot T^{\prime}_{j})\in T\cdot T^{\prime}$.
Altogether, this yields $ss^{\prime}\in f_{n}(T\cdot T^{\prime})$ by
definition. ∎

Proof of Proposition 8.7. We now turn to the main argument. We fix
$S\subseteq({A^{*}}/{\sim_{\mathcal{C}}})\times R$ which is
${BPol\/}(\mathcal{C})$-saturated (for $\rho$). We have to show that
$S(D)\subseteq f_{n}(\tau_{n}(D))$ for every $n\in{\rm Nature}$ and
$D\in{A^{*}}/{\sim_{\mathcal{C}}}$. The argument is an induction on $n\in{\rm
Nature}$.

Base case. Assume that $n=0$ and let $D\in{A^{*}}/{\sim_{\mathcal{C}}}$. We
show that $S(D)\subseteq f_{0}(\tau_{0}(D))$. Let $r\in S(D)$, _i.e._ ,
$(D,r)\in S$. Since $S$ is ${BPol\/}(\mathcal{C})$-saturated, we get from (1)
that there exist $r_{1},\dots,r_{k}\in R$ such that $r\leq r_{1}+\cdots+r_{k}$
and $(D,r_{i},\\{r_{1}+\cdots+r_{k}\\})\in\mathcal{R}^{\rho}_{S}$ for every
$i\leq k$. For $i\leq k$, one may verify from the definition of
$\mathcal{R}^{\rho}_{S}$ that since
$(D,r_{i},\\{r_{1}+\cdots+r_{k}\\})\in\mathcal{R}^{\rho}_{S}$, we have
$w_{i}\in A^{*}$ such that $[w_{i}]_{\mathcal{C}}=D$ and $\rho(w_{i})=r_{i}$.
In particular, $w_{1},\dots,w_{k}\in D$ and it is therefore immediate from the
definition of $\tau_{0}$ that $r_{1},\dots,r_{k}\in\tau_{0}(D)$. Since $r\leq
r_{1}+\cdots+r_{k}$, we obtain $r\in f_{0}(\tau_{0}(D))$ by definition of
$f_{0}$, which concludes the proof.

Inductive step. We now assume that $n\geq 1$. The argument is based on the
following lemma, which is where we use the characterization of
${Pol\/}(\mathcal{C})$-optimal imprints (_i.e._ Theorem 7.1): we apply it to
the quasi-multiplicative rating map $\tau_{n-1}$. Moreover, this is also where
we apply induction on $n$ in Proposition 8.7.

######  Lemma 8.11.

For all $(D,q,U)\in\mathcal{R}^{\rho}_{S}$, we have $T\in Q_{n-1}$ satisfying
the following properties:

$\text{$T\in\mathcal{I}_{{Pol\/}(\mathcal{C})}\left[\rho_{*}^{-1}(q)\cap
D,\tau_{n-1}\right]$ and $U\subseteq f_{n-1}(T)$}.$

We start by explaining how Lemma 8.11 can be applied to complete the main
proof. Consider $D\in{A^{*}}/{\sim_{\mathcal{C}}}$. We show that
$S(D)\subseteq f_{n}(\tau_{n}(D))$. Let $r\in S(D)$ (_i.e._ , $(D,r)\in S$).
We prove that $r\in f_{n}(\tau_{n}(D))$.

Since $S$ is ${BPol\/}(\mathcal{C})$-saturated, we get from (1) that there
exist $r_{1},\dots,r_{k}\in R$ such that $r\leq r_{1}+\cdots+r_{k}$ and
$(D,r_{i},\\{r_{1}+\cdots+r_{k}\\})\in\mathcal{R}^{\rho}_{S}$ for every $i\leq
k$. Lemma 8.11 then yields $T_{i}\in Q_{n-1}$ such that
$T_{i}\in\mathcal{I}_{{Pol\/}(\mathcal{C})}\left[\rho_{*}^{-1}(r_{i})\cap
D,\tau_{n-1}\right]$ and $r_{1}+\cdots+r_{k}\in f_{n-1}(T_{i})$ for every
$i\leq k$. By definition, $\tau_{n}$ is the rating map
$\zeta_{{Pol\/}(\mathcal{C})}^{\rho_{*}}[\tau_{n-1}]:2^{A^{*}}\to 2^{R\times
Q_{n-1}}$. Thus, if we unravel the definition, the hypothesis that
$T_{i}\in\mathcal{I}_{{Pol\/}(\mathcal{C})}\left[\rho_{*}^{-1}(r_{i})\cap
D,\tau_{n-1}\right]$ exactly says that,

$(r_{i},U_{i})\in\tau_{n}(D).$

Since we also have $r\leq r_{1}+\cdots+r_{k}$ and $r_{1}+\cdots+r_{k}\in
f_{n-1}(T_{i})$ for every $i\leq k$, it is immediate by definition of $f_{n}$
that $r\in f_{n}(\tau_{n}(D))$, finishing the proof of Proposition 8.7. It
remains to prove Lemma 8.11.

###### Proof of Lemma 8.11.

We first apply Theorem 7.1 and then use the result to prove the lemma.
Clearly, the Cartesian product $({A^{*}}/{\sim_{\mathcal{C}}})\times R$ is a
monoid when equipped with the componentwise multiplication. Let
$\alpha:A^{*}\to({A^{*}}/{\sim_{\mathcal{C}}})\times R$ be the morphism
defined by $\alpha(w)=([w]_{\mathcal{C}},\rho(w))$. Clearly, $\alpha$ is a
$\mathcal{C}$-compatible morphism: for every pair
$(D,r)\in({A^{*}}/{\sim_{\mathcal{C}}})\times R$, it suffices to define
$[(D,r)]_{\mathcal{C}}=D$. Consequently, since we also know that $\tau_{n-1}$
is quasi-multiplicative by Lemma 8.8, we may apply Theorem 7.1 to obtain a
description of the set
$\mathcal{P}_{{Pol\/}(\mathcal{C})}^{\alpha}[\tau_{n-1}]\subseteq({A^{*}}/{\sim_{\mathcal{C}}})\times
R\times Q_{n-1}$. Consider the least ${Pol\/}(\mathcal{C})$-saturated subset
$X$ of $({A^{*}}/{\sim_{\mathcal{C}}})\times R\times Q_{n-1}$ for $\alpha$ and
$\tau_{n-1}$. Theorem 7.1 yields that,

$\mathcal{P}_{{Pol\/}(\mathcal{C})}^{\alpha}[\tau_{n-1}]=\mathord{\downarrow}_{Q_{n-1}}\big{\\{}(D,q,\mu_{\tau_{n-1}}(P))\mid(D,q,P)\in
X\big{\\}}.$ (4)

The proof of Lemma 8.11 is now based on the following lemma which we shall
prove by induction on the construction of an element of
$\mathcal{R}^{\rho}_{S}$.

######  Lemma 8.12.

For every $(D,q,U)\in\mathcal{R}^{\rho}_{S}$. There exists $P\in Q_{n-1}$ such
that $(D,q,P)\in X$ and $U\subseteq f_{n-1}(P)$.

Let us first use Lemma 8.12 to conclude the proof of Lemma 8.11. Let
$(D,q,U)\in\mathcal{R}^{\rho}_{S}$. We have to exhibit $T\in Q_{n-1}$ such
that $(D,q,T)\in\mathcal{P}_{{Pol\/}(\mathcal{C})}^{\alpha}[\tau_{n-1}]$ and
$U\subseteq f_{n-1}(T)$.

Using Lemma 8.12, we get $P\in Q_{n-1}$ such that $(D,q,P)\in X$ and
$U\subseteq f_{n-1}(P)$. In view of (4), this yields,

$(D,q,\mu_{\tau_{n-1}}(P))\in\mathcal{P}_{{Pol\/}(\mathcal{C})}^{\alpha}[\tau_{n-1}].$

By definition of the set
$\mathcal{P}_{{Pol\/}(\mathcal{C})}^{\alpha}[\tau_{n-1}]$ and of the morphism
$\alpha$, this exactly says that,

$\mu_{\tau_{n-1}}(P)\in\mathcal{I}_{{Pol\/}(\mathcal{C})}\left[\rho_{*}^{-1}(q)\cap
D,\tau_{n-1}\right].$

Moreover, Lemma 8.9 yields that $U\subseteq f_{n-1}(P)\subseteq
f_{n-1}(\mu_{\tau_{n-1}}(P))$. Hence, the desired property holds for
$T=\mu_{\tau_{n-1}}(P)$ and we are finished.

It remains to prove Lemma 8.12. Consider $(D,q,U)\in\mathcal{R}^{\rho}_{S}$.
By definition of $\mathcal{R}^{\rho}_{S}$, we know that
$(D,q,U)\in\mathcal{R}^{\rho}_{S}$ is built from trivial elements using three
operations: extended downset, multiplication and $S$-restricted closure. We
proceed by induction on this construction. There are four cases depending on
the last operation used to build $(D,q,U)\in\mathcal{R}^{\rho}_{S}$.

Base case: trivial elements. In that case, there exists $w\in A^{*}$ such that
$D=[w]_{\mathcal{C}}$, $q=\rho(w)$ and $U=\\{\rho(w)\\}$. Since $X$ is
${Pol\/}(\mathcal{C})$-saturated for $\rho_{*}$ and $\tau_{n-1}$, we know that
$([w]_{\mathcal{C}},\rho(w),\tau_{n-1}(w))\in X$. Hence, it remains to prove
that $U=\\{\rho(w)\\}\subseteq f_{n-1}(\tau_{n-1}(w))$. It will then be
immediate that Lemma 8.12 holds for $P=\tau_{n-1}(w)$. This is immediate from
the following fact.

######  Fact 8.13.

For every $m\in{\rm Nature}$, we have $\rho(w)\in f_{m}(\tau_{m}(w))$.

###### Proof.

This is a simple induction on $m$. When $m=0$, we have $\rho(w)\in\tau_{0}(w)$
by definition of $\tau_{0}$. It follows that $\rho(w)\in f_{0}(\tau_{0}(w))$
by definition of $f_{0}$. Assume now that $m\geq 1$. Recall that $\tau_{m}$ is
$\zeta_{{Pol\/}(\mathcal{C})}^{\rho_{*}}[\tau_{m-1}]:2^{A^{*}}\to 2^{R\times
Q_{m-1}}$ by definition. Therefore,

$\tau_{m}(w)(\rho(w))=\mathcal{I}_{{Pol\/}(\mathcal{C})}\left[\rho_{*}^{-1}(\rho(w))\cap\\{w\\},\tau_{m-1}\right]=\mathcal{I}_{{Pol\/}(\mathcal{C})}\left[\\{w\\},\tau_{m-1}\right]$

Thus, $\tau_{m-1}(w)\in\tau_{m}(w)(\rho(w))$. Moreover induction yields
$\rho(w)\in f_{m-1}(\tau_{m-1}(w))$. Therefore, by definition of $f_{m}$, we
have $\rho(w)\in f_{m}(\tau_{m}(w))$. ∎

Inductive case 1: extended downset. We have
$(D,r,U^{\prime})\in\mathcal{R}^{\rho}_{S}$ and such that
$U\subseteq\mathord{\downarrow_{R}}U^{\prime}$. By induction, we obtain $P\in
Q_{n-1}$ such that $(D,r,P)\in X$ and $U^{\prime}\subseteq f_{n-1}(P)$.
Clearly, this implies $U\subseteq\mathord{\downarrow_{R}}f_{n-1}(P)$.
Moreover, $\mathord{\downarrow_{R}}f_{n-1}(P)=f_{n-1}(P)$ by definition.
Therefore, $U\subseteq f_{n-1}(P)$ which concludes this case.

Inductive case 2: multiplication. We have
$(D_{1},r_{1},U_{1})\in\mathcal{R}^{\rho}_{S}$ and
$(D_{2},r_{2},U_{2})\in\mathcal{R}^{\rho}_{S}$ such that
$D=D_{1}\mathbin{\scriptscriptstyle\bullet}D_{2}$, $q=r_{1}r_{2}$ and
$U=U_{1}U_{2}$. By induction, we obtain $P_{1},P_{2}\in Q_{n-1}$ such that
$(D_{1},r_{1},P_{1}),(D_{2},r_{2},P_{2})\in X$, $U_{1}\subseteq
f_{n-1}(P_{1})$ and $U_{2}\subseteq f_{n-1}(P_{2})$. Since $X$ is
${Pol\/}(\mathcal{C})$-saturated, it is closed under multiplication and we
get,

$(D,q,P_{1}P_{2})=(D_{1}\mathbin{\scriptscriptstyle\bullet}D_{2},r_{1}r_{2},P_{1}P_{2})\in
X$

Moreover, Lemma 8.10 yields that,

$U=U_{1}U_{2}\subseteq f_{n-1}(P_{1})\cdot f_{n-1}(P_{2})\subseteq
f_{n-1}(P_{1}P_{2})$

Altogether, it follows that Lemma 8.12 holds for $P=P_{1}P_{2}$.

Inductive case 3: $S$-restricted closure. In that case, we have idempotents
$(E,f,F)\in\mathcal{R}^{\rho}_{S}$ such that $D=E$, $q=f$ and $U=F\cdot
S(E)\cdot F$.

By induction, we get $V\in Q_{n-1}$ such that $(E,f,V)\in X$ and $F\subseteq
f_{n-1}(V)$. Since $Q_{n-1}$ is a finite monoid, there exists a number $n\geq
1$ such that $V^{n}$ is a multiplicative idempotent of $Q_{n-1}$. Since
$(E,f,V)\in X$, and $X$ is closed under multiplication (as it is
${Pol\/}(\mathcal{C})$-saturated), we obtain $(E,f,V^{n})\in X$. We may again
use the hypothesis that $X$ is ${Pol\/}(\mathcal{C})$-saturated to apply
${Pol\/}(\mathcal{C})$-closure and obtain,

$(E,f,V^{n}\cdot\tau_{n-1}(E)\cdot V^{n})\in X.$

Since $D=E$ and $q=f$, it now remains to show that $U=F\cdot S(E)\cdot
F\subseteq f_{n-1}(V^{n}\cdot\tau_{n-1}(E)\cdot V^{n})$. It will then be
immediate that Lemma 8.12 holds for $P=V^{n}\cdot\tau_{n-1}(E)\cdot V^{n}$. It
is immediate by induction in Proposition 8.7 that,

$S(E)\subseteq f_{n-1}(\tau_{n-1}(E)).$

Moreover, since we already know that $F\subseteq f_{n-1}(V)$ is an idempotent
by definition, it follows from Lemma 8.10 that,

$F\cdot S(E)\cdot F=F^{n}\cdot S(E)\cdot F^{n}\subseteq
f_{n-1}(V^{n}\cdot\tau_{n-1}(E)\cdot V^{n}).$

This concludes the proof of Lemma 8.12. ∎

##  9\. Completeness in Theorem 5.3

In this section, we prove the completeness direction in Theorem 5.3. As for
the soundness proof, the section is divided in two parts. We start with a
preliminary result which applies to Boolean closure in general, _i.e._ , to
classes of the form ${Bool\/}(\mathcal{D})$ when $\mathcal{D}$ is an arbitrary
lattice. We then use it in the special case $\mathcal{D}={Pol\/}(\mathcal{C})$
to prove the completeness direction of Theorem 5.3.

###  9.1. Preliminary result

We fix a lattice $\mathcal{D}$. Moreover, we let $\rho:2^{A^{*}}\to R$ be a
nice rating map. Given a language $L\in\mathcal{D}$, we characterize the set
$\mathcal{I}_{{Bool\/}(\mathcal{D})}\left[L,\rho\right]\subseteq R$ using
$\mathcal{D}$-optimal $\xi_{{Bool\/}(\mathcal{D})}[\rho]$-imprints when
$\xi_{{Bool\/}(\mathcal{D})}[\rho]:2^{A^{*}}\to 2^{R}$ is the rating map
introduced in Proposition 6.2.

######  Proposition 9.1.

Let $L\in\mathcal{D}$. For all
$s\in\mathcal{I}_{{BPol\/}(\mathcal{C})}\left[L,\rho\right]$, we have
$r_{1},\dots,r_{k}\in R$ such that $s\leq r_{1}+\cdots+r_{k}$ and for every
$i\leq k$, we have
$\\{r_{1}+\cdots+r_{k}\\}\in\mathcal{I}_{\mathcal{D}}\left[L\cap\rho_{*}^{-1}(r_{i}),\xi_{{Bool\/}(\mathcal{D})}[\rho]\right]$.

###### Proof.

We fix $L\in\mathcal{D}$ and
$s\in\mathcal{I}_{{BPol\/}(\mathcal{C})}\left[L,\rho\right]$ for the proof.
For every $q\in R$, we let $\mathbf{H}_{q}$ as an optimal $\mathcal{D}$-cover
of $L\cap\rho_{*}^{-1}(q)$ for $\tau$. We have the following fact.

######  Fact 1.

There exists a ${Bool\/}(\mathcal{D})$-cover $\mathbf{K}$ of $L$ such that for
every $K\in\mathbf{K}$ and every $q\in R$ such that
$K\cap\rho_{*}^{-1}(q)\neq\emptyset$, there exists $H\in\mathbf{H}_{q}$ such
that $K\subseteq H$.

######  Proof 9.2.

Let $\mathbf{H}=\bigcup_{q\in R}\mathbf{H}_{q}$ and consider the following
equivalence $\sim$ defined on $L$. For every $u,v\in L$, we let $u\sim v$ if
and only if $u\in H\Leftrightarrow v\in H$ for every $H\in\mathbf{H}$. We let
$\mathbf{K}$ as the partition of $L$ into $\sim$-classes. By definition,
$\mathbf{K}$ is a cover of $L$. Moreover, it is a
${Bool\/}(\mathcal{D})$-cover. Indeed, by definition, $\mathbf{K}$ only
contains Boolean combinations of $L\in{Bool\/}(\mathcal{D})$ with languages in
$\mathbf{H}$ (which are in $\mathcal{D}$ and
${Bool\/}(\mathcal{D})={Bool\/}(\mathcal{D})$). It remains to show that
$\mathbf{K}$ satisfies the property described in the fact.

Let $q\in R$ and assume that there exists $w\in K\cap\rho_{*}^{-1}(q)$. By
definition of $\mathbf{K}$, we have $K\subseteq L$ which means that $w\in
L\cap\rho_{*}^{-1}(q)$. Therefore, since $\mathbf{H}_{q}$ is a cover of
$L\cap\rho_{*}^{-1}(q)$ by definition, we have $H\in\mathbf{H}_{q}$ such that
$w\in H$. Consequently, $K\cap H\neq\emptyset$. Finally, since $K$ is a
$\sim$-class by definition of $\mathbf{K}$, it follows from the definition of
$\sim$ that $K\subseteq H$.

We let $\mathbf{K}$ as the ${Bool\/}(\mathcal{D})$-cover of $L$ described in
Fact 1. By definition, we have $L\subseteq\bigcup_{K\in\mathbf{K}}K$. Hence,
it follows from Lemma 4.7 that,

$\mathcal{I}_{{Bool\/}(\mathcal{D})}\left[L,\rho\right]\subseteq\bigcup_{K\in\mathbf{K}}\mathcal{I}_{{Bool\/}(\mathcal{D})}\left[K,\rho\right]$

Consequently, since
$s\in\mathcal{I}_{{Bool\/}(\mathcal{D})}\left[L,\rho\right]$, we get some
$K\in\mathbf{K}$ such that
$s\in\mathcal{I}_{{Bool\/}(\mathcal{D})}\left[K,\rho\right]$. We let
$\mathbf{V}$ as an optimal ${Bool\/}(\mathcal{D})$-cover of $K$ for $\rho$.
Since $K\in{Bool\/}(\mathcal{D})$, we may choose $\mathbf{V}$ such that
$V\subseteq K$ for all $V\in\mathbf{V}$. By definition of $\mathbf{V}$, we
have and we obtain that $s\in$. Therefore, there exists $V\in\mathbf{V}$ such
that $s\leq\rho(V)$.

Since $\rho$ is nice by hypothesis, we have $w_{1},\dots,w_{k}\in V$ such that
$\rho(V)=\rho(w_{1})+\cdots+\rho(w_{k})$. We let $r_{i}=\rho(w_{i})$ for every
$i\leq k$. by definition, we have $s\leq r_{1}+\cdots+r_{k}$. Therefore, it
remains to show that
$\\{r_{1}+\cdots+r_{k}\\}\in\mathcal{I}_{\mathcal{D}}\left[L\cap\rho_{*}^{-1}(r_{i}),\xi_{{Bool\/}(\mathcal{D})}[\rho]\right]$
for every $i\leq k$.

We fix $i\leq k$ for the proof. By definition, $\rho(w_{i})=r_{i}$ and
$w_{i}\in V\subseteq K$. Hence, $w_{i}\in K\cap\rho_{*}^{-1}(r_{i})$ and it
follows from the definition of $\mathbf{K}$ in Fact 1 that there exists a
language $H\in\mathbf{H}_{r_{i}}$ such that $K\subseteq H$. Recall that
$r_{1}+\cdots+r_{k}=\rho(V)$. Moreover, $\rho(V)\in$ and we have .
Consequently, by Fact 4.6,

$r_{1}+\cdots+r_{k}\in\mathcal{I}_{{Bool\/}(\mathcal{D})}\left[K,\rho\right]\subseteq\mathcal{I}_{{Bool\/}(\mathcal{D})}\left[H,\rho\right]$

By definition of the rating map $\xi_{{Bool\/}(\mathcal{D})}[\rho]$, we have
$\mathcal{I}_{{Bool\/}(\mathcal{D})}\left[H,\rho\right]=\xi_{{Bool\/}(\mathcal{D})}[\rho](H)$.
Thus, the above can be rephrased as follows:

$\\{r_{1}+\cdots+r_{k}\\}\subseteq\xi_{{Bool\/}(\mathcal{D})}[\rho](H)$

Consequently, since $H\in\mathbf{H}_{r_{i}}$, we have
$\\{r_{1}+\cdots+r_{k}\\}\in$. Recall that we defined $\mathbf{H}_{r_{i}}$ as
an optimal $\mathcal{D}$-cover of $L\cap\rho_{*}^{-1}(r_{i})$ for
$\xi_{{Bool\/}(\mathcal{D})}[\rho]$. Hence, we obtain,

$\\{r_{1}+\cdots+r_{k}\\}\in\mathcal{I}_{\mathcal{D}}\left[L\cap\rho_{*}^{-1}(r_{i}),\xi_{{Bool\/}(\mathcal{D})}[\rho]\right]$

This concludes the proof of Proposition 9.1.

###  9.2. Completeness proof for Theorem 5.3

We may now come back to our main objective: completeness in Theorem 5.3. We
fix a finite quotient-closed Boolean algebra $\mathcal{C}$ and a nice
multiplicative rating map $\rho:2^{A^{*}}\to R$. We prove that
$\mathcal{P}_{{BPol\/}(\mathcal{C})}^{\mathcal{C}}[\rho]$ is
${BPol\/}(\mathcal{C})$-saturated for $\rho$. Since we already showed in the
previous section that every ${BPol\/}(\mathcal{C})$-saturated subset is
included in $\mathcal{P}_{{BPol\/}(\mathcal{C})}^{\mathcal{C}}[\rho]$, Theorem
5.3 will follow: $\mathcal{P}_{{BPol\/}(\mathcal{C})}^{\mathcal{C}}[\rho]$ is
the greatest ${BPol\/}(\mathcal{C})$-saturated subset of $R$. For the sake of
avoiding clutter, we write $S$ for
$\mathcal{P}_{{BPol\/}(\mathcal{C})}^{\mathcal{C}}[\rho]$.

######  Remark 9.3.

Contrary to the soundness direction, we do need the hypothesis that $\rho$ is
nice. This is required for applying the above preliminary result, Proposition
9.1.

By Theorem 3.2, ${Pol\/}(\mathcal{C})$ is a lattice. Therefore, we may
instantiate Proposition 9.1 for our nice multiplicative rating map
$\rho:2^{A^{*}}\to R$ in the special case when
$\mathcal{D}={Pol\/}(\mathcal{C})$. We complete it with another result
specific to this special case. As for the soundness direction, its proof if
based on Theorem 7.1: the characterization of ${Pol\/}(\mathcal{C})$-optimal
imprints. Together, these two results imply that
$S=\mathcal{P}_{{BPol\/}(\mathcal{C})}^{\mathcal{C}}[\rho]$ is
${BPol\/}(\mathcal{C})$-saturated for $\rho$.

######  Proposition 9.4.

Let $D\in{A^{*}}/{\sim_{\mathcal{C}}}$ and $r\in R$. For all
$U\in\mathcal{I}_{{Pol\/}(\mathcal{C})}\left[D\cap\rho_{*}^{-1}(r),\xi_{{BPol\/}(\mathcal{C})}[\rho]\right]$,
we have $(D,r,U)\in\mathcal{R}^{\rho}_{S}$.

We first combine Proposition 9.1 and Proposition 9.4 to prove that $S$ is
${BPol\/}(\mathcal{C})$-saturated for $\rho$ and finish the completeness
proof. We have to show that (1) is satisfied. That is, given $(D,r)\in S$, we
have to exhibit $r_{1},\dots,r_{k}\in R$ such that $r\leq r_{1}+\cdots+r_{k}$
and $(D,r_{i},\\{r_{1}+\cdots+r_{k}\\})\in\mathcal{R}^{\rho}_{S}$ for every
$i\leq k$.

Since $D\in{A^{*}}/{\sim_{\mathcal{C}}}$, we have
$D\in\mathcal{C}\subseteq{Pol\/}(\mathcal{C})$. Moreover, since
$S=\mathcal{P}_{{BPol\/}(\mathcal{C})}^{\mathcal{C}}[\rho]$ and $(D,r)\in S$,
we have $r\in\mathcal{I}_{{BPol\/}(\mathcal{C})}\left[D,\rho\right]$. Thus,
Proposition 9.1 yields $r_{1},\dots,r_{k}\in R$ such that $r\leq
r_{1}+\cdots+r_{k}$ and
$\\{r_{1}+\cdots+r_{k}\\}\in\mathcal{I}_{{Pol\/}(\mathcal{C})}\left[D\cap\rho_{*}^{-1}(r_{i}),\tau\right]$
for every $i\leq k$. Then, we obtain from Proposition 9.4 that
$(D,r,\\{r_{1}+\cdots+r_{k}\\})\in\mathcal{R}^{\rho}_{S}$ for every $i\leq k$.
Altogether, this is exactly the property stated in (1). This concludes the
completeness proof for Theorem 5.3. It remains to prove Proposition 9.4.

######  Proof 9.5 (Proof of Proposition 9.4).

The argument is based on Theorem 7.1. Clearly, the Cartesian product
$({A^{*}}/{\sim_{\mathcal{C}}})\times R$ is a monoid when equipped with the
componentwise multiplication. Let
$\alpha:A^{*}\to({A^{*}}/{\sim_{\mathcal{C}}})\times R$ be the morphism
defined by $\alpha(w)=([w]_{\mathcal{C}},\rho(w))$. Clearly, $\alpha$ is a
$\mathcal{C}$-compatible morphism: for every pair
$(D,r)\in({A^{*}}/{\sim_{\mathcal{C}}})\times R$, it suffices to define
$[(D,r)]_{\mathcal{C}}=D$.

Moreover, ${Pol\/}(\mathcal{C})$ is a quotient-closed lattice closed under
concatenation and $\rho:2^{A^{*}}\to R$ is a multiplicative rating map. Thus,
Lemma 6.9 yields a rating map $\tau:2^{A^{*}}\to 2^{R}$ which is quasi-
multiplicative for the endomorphism
$\mu_{\tau}:U\mapsto\mathord{\downarrow_{R}}U$ of $(2^{R},\cup)$ and such that
for every $K\in{Pol\/}(\mathcal{C})$, we have
$\tau(K)=\xi_{{BPol\/}(\mathcal{C})}[\rho](K)$. We have the following key
lemma which we need to apply Theorem 7.1.

######  Lemma 9.6.

The set $\mathcal{R}^{\rho}_{S}\subseteq({A^{*}}/{\sim_{\mathcal{C}}})\times
R\times 2^{R}$ is ${Pol\/}(\mathcal{C})$-saturated for $\alpha$ and $\tau$.

Let us first use the lemma to conclude the proof of Proposition 9.4. Let
$D\in{A^{*}}/{\sim_{\mathcal{C}}}$ and $r\in R$. Consider
$U\in\mathcal{I}_{{Pol\/}(\mathcal{C})}\left[D\cap\rho_{*}^{-1}(r),\xi_{{BPol\/}(\mathcal{C})}[\rho]\right]$.
We show that $(D,r,U)\in\mathcal{R}^{\rho}_{S}$. Since
$\tau(K)=\xi_{{BPol\/}(\mathcal{C})}[\rho](K)$ for every
$K\in{Pol\/}(\mathcal{C})$, it is immediate that,

$\mathcal{I}_{{Pol\/}(\mathcal{C})}\left[D\cap\rho_{*}^{-1}(r),\tau\right]=\mathcal{I}_{{Pol\/}(\mathcal{C})}\left[D\cap\rho_{*}^{-1}(r),\xi_{{BPol\/}(\mathcal{C})}[\rho]\right]$

Thus
$U\in\mathcal{I}_{{Pol\/}(\mathcal{C})}\left[D\cap\rho_{*}^{-1}(r),\tau\right]$
which means that $(D,r,U)\in\mathcal{P}_{{Pol\/}(\mathcal{C})}^{\alpha}[\tau]$
by definition. Moreover, $\mathcal{R}^{\rho}_{S}$ is
${Pol\/}(\mathcal{C})$-saturated for $\alpha$ and $\tau$ by Lemma 9.6. Hence,
since $\tau:2^{A^{*}}\to 2^{R}$ is a quasi-multiplicative rating map for
$\mu_{\tau}:U\mapsto\mathord{\downarrow_{R}}U$, it follows from Theorem 7.1
that,

$\mathcal{P}_{{Pol\/}(\mathcal{C})}^{\alpha}[\tau]\subseteq\mathord{\downarrow_{2^{R}}}\big{\\{}(D^{\prime},r^{\prime},\mathord{\downarrow_{R}}V)\mid(D^{\prime},r^{\prime},V)\in\mathcal{R}^{\rho}_{S}\big{\\}}.$

Altogether, we get $V\in 2^{R}$ such that $(D,r,V)\in\mathcal{R}^{\rho}_{S}$
and $U\subseteq\mathord{\downarrow_{R}}V$. By closure under extended downset
in the definition of $\mathcal{R}^{\rho}_{S}$, this yields
$(D,r,U)\in\mathcal{R}^{\rho}_{S}$ as desired.

It remains to prove Lemma 9.6: $\mathcal{R}^{\rho}_{S}$ is
${Pol\/}(\mathcal{C})$-saturated for $\alpha$ and $\tau$. We have four
conditions to check. We leave the trivial elements for last as we need the
other properties to handle them. It is immediate that $\mathcal{R}^{\rho}_{S}$
is closed under downset and multiplication as stated in the definition of
${Pol\/}(\mathcal{C})$-saturated subsets. This implied by the closure under
extended downset and multiplication in the definition of
$\mathcal{R}^{\rho}_{S}$. We turn to ${Pol\/}(\mathcal{C})$-closure. Let
$(E,f,F)\in\mathcal{R}^{\rho}_{S}$ be a triple of multiplicative idempotents.
We prove that $(E,f,F\cdot\tau(E)\cdot F)\in\mathcal{R}^{\rho}_{S}$. We may
use $S$-restricted closure in the definition of $\mathcal{R}^{\rho}_{S}$ to
obtain $(E,f,F\cdot S(E)\cdot F)\in\mathcal{R}^{\rho}_{S}$. Moreover,
$S=\mathcal{P}_{{BPol\/}(\mathcal{C})}^{\mathcal{C}}[\rho]$ by definition and
$\tau(E)=\xi_{{BPol\/}(\mathcal{C})}[\rho](E)=\mathcal{I}_{{BPol\/}(\mathcal{C})}\left[E,\rho\right]$
since $E\in\mathcal{C}\subseteq{Pol\/}(\mathcal{C})$. Thus, $S(E)=\tau(E)$. We
obtain $(E,f,F\cdot\tau(E)\cdot F)\in\mathcal{R}^{\rho}_{S}$ as desired.

We turn to the trivial elements. For $w\in A^{*}$, we show that
$([w]_{\mathcal{C}},\rho(w),\tau(w))\in\mathcal{R}^{\rho}_{S}$. First, we
consider the special case $w=\varepsilon$. By definition of
$\mathcal{R}^{\rho}_{S}$, we have
$([\varepsilon]_{\mathcal{C}},\rho(\varepsilon),\\{\rho(\varepsilon)\\})\in\mathcal{R}^{\rho}_{S}$.
Since this is a triple of multiplicative idempotents and we already
established that $\mathcal{R}^{\rho}_{S}$ satisfies
${Pol\/}(\mathcal{C})$-closure, this implies
$([\varepsilon]_{\mathcal{C}},\rho(\varepsilon),\tau([\varepsilon]_{\mathcal{C}}))\in\mathcal{R}^{\rho}_{S}$.
It then follows from closure under downset that
$([\varepsilon]_{\mathcal{C}},\rho(\varepsilon),\tau(\varepsilon))\in\mathcal{R}^{\rho}_{S}$
since $\tau(\varepsilon)\subseteq\tau([\varepsilon]_{\mathcal{C}})$. We now
consider the case when $w=a$ for some $a\in A$. We already established that
$([\varepsilon]_{\mathcal{C}},\rho(\varepsilon),\tau([\varepsilon]_{\mathcal{C}}))\in\mathcal{R}^{\rho}_{S}$.
Moreover, we know that
$([a]_{\mathcal{C}},\rho(a),\\{\rho(a)\\})\in\mathcal{R}^{\rho}_{S}$ by
definition of $\mathcal{R}^{\rho}_{S}$ Using closure under multiplication and
extended downset, we get,

$([a]_{\mathcal{C}},\rho(a),\mathord{\downarrow_{R}}\left(\tau([\varepsilon]_{\mathcal{C}})\cdot\\{\rho(a)\\}\cdot\tau([\varepsilon]_{\mathcal{C}})\right))\in\mathcal{R}^{\rho}_{S}$

We know that ${Pol\/}(\mathcal{C})$ is a quotient-closed lattice of regular
languages closed under marked concatenation by Theorem 3.2. Therefore, since
$\tau$ and $\xi_{{BPol\/}(\mathcal{C})}[\rho]$ coincide over languages in
${Pol\/}(\mathcal{C})$, we obtain from Lemma 6.11 that,

$\tau([\varepsilon]_{\mathcal{C}}a[\varepsilon]_{\mathcal{C}})=\mathord{\downarrow_{R}}\left(\tau([\varepsilon]_{\mathcal{C}})\cdot\\{\rho(a)\\}\cdot\tau([\varepsilon]_{\mathcal{C}})\right)$

Altogether, we obtain
$([a]_{\mathcal{C}},\rho(a),\tau([\varepsilon]_{\mathcal{C}}a[\varepsilon]_{\mathcal{C}}))\in\mathcal{R}^{\rho}_{S}$.
Moreover, we have
$\tau(a)\subseteq\tau([\varepsilon]_{\mathcal{C}}a[\varepsilon]_{\mathcal{C}})$.
Thus, we obtain from closure under downset that
$([a]_{\mathcal{C}},\rho(a),\tau(a))\in\mathcal{R}^{\rho}_{S}$ as desired.
Finally, assume that $w=a_{1}\cdots a_{n}$ for $n\geq 2$ and
$a_{1},\dots,a_{n}\in A$. We already established that
$([a_{i}]_{\mathcal{C}},\rho(a_{i}),\tau(a_{i}))\in\mathcal{R}^{\rho}_{S}$ for
every $i\leq n$. Therefore, closure under multiplication and extended downset
yield,

$([w]_{\mathcal{C}},\rho(w),\mathord{\downarrow_{R}}\left(\tau(a_{1})\cdots\tau(a_{n})\right))\in\mathcal{R}^{\rho}_{S}$

Since $\tau$ is quasi-multiplicative for the endomorphism
$\mu_{\tau}:U\mapsto\mathord{\downarrow_{R}}U$ of $(2^{R},\cup)$, this yields
$([w]_{\mathcal{C}},\rho(w),\tau(w))\in\mathcal{R}^{\rho}_{S}$, concluding the
proof.

##  10\. Conclusion

We proved that separation and covering are decidable for all classes of the
form ${BPol\/}(\mathcal{C})$ when $\mathcal{C}$ is a finite quotient-closed
Boolean algebra. This yields separation and covering algorithms for a whole
family of classes. Arguably, the most important one is the level two in the
Straubing-Thérien hierarchy (which corresponds to the logic
$\mathcal{B}\Sigma_{2}(<)$). Additionally, this result can be lifted to depth-
two using an effective reduction of [PZ20] to the level two in the Straubing-
Thérien hierarchy.

An interesting consequence of our results is that since we proved the
decidability of separation for the level two in the Straubing-Thérien
hierarchy, the main theorem of [PZ19] is an immediate corollary: membership
for this level is decidable. However, the algorithm of [PZ19] was actually
based on a characterization theorem: languages of level two in the Straubing-
Thérien hierarchy are characterized by a syntactic property of a canonical
recognizer (_i.e._ , their syntactic monoid). It turns out that one can also
deduce this characterization theorem from our results (this does require some
combinatorial work however). In fact, one may generalize it to all classes
${BPol\/}(\mathcal{C})$ when $\mathcal{C}$ is a finite quotient-closed Boolean
algebra.

Finally, the main and most natural follow-up question is much harder: can our
results be pushed to higher levels within concatenation hierarchies? For now,
we know that given any finite quotient-closed Boolean algebra $\mathcal{C}$,
${Pol\/}(\mathcal{C})$, ${BPol\/}(\mathcal{C})$ and
${Pol\/}({BPol\/}(\mathcal{C}))$ have decidable covering (the former and the
latter are results of [Pla18]). Consequently, the next relevant levels are
${BPol\/}({BPol\/}(\mathcal{C}))$ and
${Pol\/}({BPol\/}({BPol\/}(\mathcal{C})))$.

## References

  * [Arf87] Mustapha Arfi.  Polynomial operations on rational languages.  In Proceedings of the 4th Annual Symposium on Theoretical Aspects of Computer Science, STACS’87, pages 198–206, 1987.
  * [BC71] Janusz A. Brzozowski and Rina S. Cohen.  Dot-depth of star-free events.  Journal of Computer and System Sciences, 5(1):1–16, 1971.
  * [BK78] Janusz A. Brzozowski and Robert Knast.  The dot-depth hierarchy of star-free languages is infinite.  Journal of Computer and System Sciences, 16(1):37–55, 1978.
  * [CMM13] Wojciech Czerwiński, Wim Martens, and Tomáš Masopust.  Efficient separability of regular languages by subsequences and suffixes.  In Proceedings of the 40th International Colloquium on Automata, Languages, and Programming, ICALP’13, pages 150–161, 2013.
  * [DG08] Volker Diekert and Paul Gastin.  First-order definable languages.  In Logic and Automata: History and Perspectives, volume 2. Amsterdam Univ., 2008.
  * [DGK08] Volker Diekert, Paul Gastin, and Manfred Kufleitner.  A survey on small fragments of first-order logic over finite words.  International Journal on Foundations of Computer Science, 19(3), 2008.
  * [DKRH16] Volker Diekert, Manfred Kufleitner, Gerhard Rosenberger, and Ulrich Hertrampf.  Discrete Algebraic Methods: Arithmetic, Cryptography, Automata and Groups.  De Gruyter GmbH, 2016.
  * [GS00] Christian Glaßer and Heinz Schmitz.  Languages of dot-depth 3/2.  In Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science, (STACS’00), pages 555–566, 2000.
  * [Kna83] Robert Knast.  A semigroup characterization of dot-depth one languages.  RAIRO - Theoretical Informatics and Applications, 17(4):321–330, 1983.
  * [MP71] Robert McNaughton and Seymour A. Papert.  Counter-Free Automata.  MIT Press, 1971.
  * [Pin13] Jean-Éric Pin.  An explicit formula for the intersection of two polynomials of regular languages.  In DLT 2013, volume 7907 of Lect. Notes Comp. Sci., pages 31–45. Springer, 2013.
  * [Pin17a] Jean-Éric Pin.  The dot-depth hierarchy, 45 years later.  In The Role of Theory in Computer Science. Essays Dedicated to Janusz Brzozowski. World Scientific, 2017.
  * [Pin17b] Jean-Éric Pin.  Open problems about regular languages, 35 years later.  In The Role of Theory in Computer Science. Essays Dedicated to Janusz Brzozowski. World Scientific, 2017.
  * [Pla18] Thomas Place.  Separating regular languages with two quantifier alternations.  Logical Methods in Computer Science, 14(4), 2018.
  * [PP86] Dominique Perrin and Jean-Éric Pin.  First-order logic and star-free sets.  Journal of Computer and System Sciences, 32(3):393–406, 1986.
  * [PS85] Jean-Éric Pin and Howard Straubing.  Monoids of upper triangular boolean matrices.  In Semigroups. Structure and Universal Algebraic Problems, volume 39, pages 259–272. North-Holland, 1985.
  * [PvRZ13] Thomas Place, Lorijn van Rooijen, and Marc Zeitoun.  Separating regular languages by piecewise testable and unambiguous languages.  In Proceedings of the 38th International Symposium on Mathematical Foundations of Computer Science, MFCS’13, pages 729–740, 2013.
  * [PW97] Jean-Éric Pin and Pascal Weil.  Polynomial closure and unambiguous product.  Theory of Computing Systems, 30(4):383–422, 1997.
  * [PZ15] Thomas Place and Marc Zeitoun.  The tale of the quantifier alternation hierarchy of first-order logic over words.  SIGLOG news, 2(3):4–17, 2015.
  * [PZ17] Thomas Place and Marc Zeitoun.  Separation for dot-depth two.  In 32th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS’17, 2017.
  * [PZ18a] Thomas Place and Marc Zeitoun.  The covering problem.  Logical Methods in Computer Science, 14(3), 2018.
  * [PZ18b] Thomas Place and Marc Zeitoun.  Generic results for concatenation hierarchies.  Theory of Computing Systems (ToCS), 2018.  Selected papers from CSR’17.
  * [PZ19] Thomas Place and Marc Zeitoun.  Going higher in first-order quantifier alternation hierarchies on words.  Journal of the ACM, 66(2):12:1–12:65, 2019.
  * [PZ20] Thomas Place and Marc Zeitoun.  Adding successor: A transfer theorem for separation and covering.  ACM Trans. Comput. Log., 21(2):9:1–9:45, 2020.
  * [Sch65] Marcel Paul Schützenberger.  On finite monoids having only trivial subgroups.  Information and Control, 8(2):190–194, 1965.
  * [Sim75] Imre Simon.  Piecewise testable events.  In Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages, pages 214–222, 1975.
  * [Str81] Howard Straubing.  A generalization of the Schützenberger product of finite monoids.  Theoretical Computer Science, 13(2):137–150, 1981.
  * [Thé81] Denis Thérien.  Classification of finite monoids: The language approach.  Theoretical Computer Science, 14(2):195–208, 1981.
  * [Tho82] Wolfgang Thomas.  Classifying regular events in symbolic logic.  Journal of Computer and System Sciences, 25(3):360–376, 1982.


# Transmission of doughnut light through a bull’s eye structure

Lu-Lu Wang  Xi-Feng<EMAIL_ADDRESS>Rui Yang  Key Laboratory of
Quantum Information, University of Science and Technology of China, Hefei
230026, People’s Republic of China  Guang-Can Guo  Guo-Ping
<EMAIL_ADDRESS>Key Laboratory of Quantum Information, University of
Science and Technology of China, Hefei 230026, People’s Republic of China

###### Abstract

We experimentally investigate the extraordinary optical transmission of
doughnut light through a bull’s eye structure. Since the intensity is vanished
in the center of the beam, almost all the energy reaches the circular
corrugations (not on the hole), and excites surface plasmons which propagate
through the hole and reradiate photons. The transmitted energy is about 32
times of the energy input on the hole area. It is also interesting that the
transmitted light has a similar spatial shape with the input light even though
the diameter of the hole is much smaller than the wavelength of light.

###### pacs:

78.66.Bz,73.20.MF, 71.36.+c

The phenomenon of extraordinary optical transmission (EOT) through metallic
films which were perforated by nanohole arrays was first observed a decade
agoEbbesen98 . It is generally believed that surface plasmons (SPs) in metal
surface play a crucial role in this process, during which photons first
transform into SPs and then back to photons againMoreno ; liu . Such SPs are
involved in a wide range of applicationsBarnes03 ; Ozbay06 . The report of EOT
phenomenon attracts considerable attention because it shows that more light
than Bethe’s prediction could be transmitted through the holesBethe . This
stimulates much fundamental research and promotes subwavelength apertures as a
core element of new optical devices. For EOT in periodic hole arrays, not only
the polarization propertiesElli04 ; Koer04 ; RenAPL  but also the spatial mode
propertiesren06 ; ren062  are widely discussed. Even for a single aperture
surrounded by circular corrugations, we can also get high transmission
efficiencies and a well-defined spectrum since the periodic corrugations act
as an antenna to couple the incident light into SPsthio ; Lezec .

Usually, the light transmitted through the subwavelength holes can be divided
into two parts: one is the directly transmitted light and the other comes from
the surface plasmon assisted transmission process. Here we present a new
method to eliminate the influence of the first part in EOT phenomenon by using
a doughnut input light and a bull’s eye structure. Since the intensity is null
in the center of the beam, there is no light illuminating on the single hole
directly. Almost all the energy reaches the circular corrugations, and excites
SPs which propagate through the hole and reradiate photons (as shown in Fig.1
). It is also interesting that the transmitted light has a similar spatial
shape with the input light even though the diameter of the hole is much
smaller than the wavelength of light.

Figure 1: Sketch map of our protocol. The typical doughnut light has an
intensity null on the beam axis. Almost all the energy reaches the circular
corrugations, excite surface plasmons which propagate through the hole and
reradiate photons. Figure 2: (Color online) Transmission efficiency as a
function of wavelength for bull’s eye structure(Black square dots) and similar
structure without hole in center(Red round dots). Inset is a scanning electron
microscope picture of our bull’s eye structure(groove periodicity, 500 nm;
groove depth, 60 nm; hole diameter, 250nm; film thickness, 135 nm).

Inset of Fig. 2 is a scanning electron microscope picture of our bull’s eye
structure. The thickness of the gold layer is $135$ $nm$. The cylindrical
hole($250$ $nm$ diameter) and the grooves are produced by a Focused Ion Beam
Etching system (FIB, DB235 of FEB Co.). The grooves have a period of $500nm$
with the depth $60nm$ and width $250nm$. Transmission spectra of the hole
array are recorded by a Silicon avalanche photodiode (APD) single photon
detector coupled with a monochromator through a fiber. White light from a
stabilized tungsten-halogen source passes through a single mode fiber and a
polarizer (only vertical polarized light can pass), then illuminates on the
sample. The hole array is set between two lenses with the focus of $35mm$. The
light exiting from the hole array is launched into the monochromator. The
transmission spectra are shown in Fig. 2(Black square dots), in which the
transmission efficiency is determined by normalizing the intensity of
transmitted light over the intensity before the sample. At the resonant
frequency (632.8 nm in the experiment), the transmission efficiency is about
$2.55\%$, much higher than that of the non-resonant case. To verify the
phenomenon does not come from the direct transmitted light, we use another
sample as a comparison. The new sample also has a Bull’s eye geometry, but
without hole in center. The transmission efficiencies are all about $1.0\%$
and there is no transmission peak as shown in Fig. 2(Red round dots), which
verify that the transmission peak for bull’s eye structure come from the
surface plasmons assisted transmission process. In the following experiments,
we use the bull’s eye structure with a hole in center to investigate the
extraordinary optical transmission of doughnut light.

The typical doughnut light is produced by changing its orbital angular
momentum (OAM), which is associated with the transverse phase front of a light
beam. Light field of photons with OAM can be described by means of Laguerre-
Gaussian ($LG_{p}^{l}$) modes with two mode indices $p$ and $l$Allen92 . The
$p$ index gives the number of radial nodes and the $l$ index represents the
number of the $2\pi$-phase shifts along a closed path around the beam center.
Light with an azimuthal phase dependence $e^{-il\varphi}$ carries a well-
defined OAM of $l\hbar$ per photonAllen92 . When $l=0$, the light is in the
general Gaussian mode, while when $l\neq 0$, the associated phase
discontinuity produces an intensity null on the beam axis. If the mode
function is not a pure LG mode, each photon of this light is in a
superposition state, with the weights dictated by the contributions of the
comprised different $l$th angular harmonics. For the sake of simplification,
we can consider only LG modes with the index $p=0$. Computer generated
holograms (CGHs)ArltJMO ; VaziriJOB , a kind of transmission holograms, are
used to change the winding number $l$ of LG mode light. Inset of Fig. 3. shows
part of a typical CGH($n=1$) with a fork in the center. Corresponding to the
diffraction order $m$, the $n$ fork hologram can change the winding number $l$
of the input beam by $\Delta l_{m}=m*n$. In our experiment, we use the first
order diffraction light ($m=1$) and the efficiency of our CGHs is about
$30\%$. Superposition mode is produced using a displaced hologramVaziriJOB ,
which is particularly suitable for producing superposition states of
$LG_{0}^{l}$ mode with the Gaussian mode beam.

Figure 3: Experimental setup. A computer generated hologram(CGH) is used to
change the OAM of the laser beam. The polarized laser beam is directed into
the microscope and focused on the metal plate using a 100X objective lens
(Nikon, NA=0.90). Transmitted light is collected by another 100X objective
lens (Nikon, NA=0.80). Inset, pictures of part of a typical CGH($n=1$) and
produced light with the first order mode.

The experimental setup is shown in Fig. 3. The OAM of the laser light(632.8nm
wavelength) is changed by a CGH, while the polarization is controlled by a
polarization beam splitter (PBS, working wavelength 632.8 nm) followed by a
half wave plate (HWP, working wavelength 632.8 nm). The polarized laser beam
is directed into the microscope and focused on the metal plate using a 100X
objective lens (Nikon, NA=0.90) with a diameter about $3.8\mu m$. The CCD
camera before the objective lens is used to adjust the position of the hole
structure. Transmitted light is collected by another 100X objective lens
(Nikon, NA=0.80) and recorded by another CCD camera. The relative position of
the beam center to the hole is estimated as follows: We detect the
transmission of the Gaussian beam with the sample moved by a three-dimensional
stage (Suruga Seiki Co., Ltd. B71-80A). When the center of beam is coincided
with the center of hole, the maximum transmission is achieved. A CCD camera is
also used as an assistant to observe the picture directly. Since the doughnut
light is produced via the movement of hologram which does not affect the
optical path, we can realize the protocol that the position of zero electric
field coincides with the center of the hole.

Figure 4: CCD pictures of light beam before (upper) and after (lower) the
bull’s eye structure. The light power is decreased to give clear pictures. A,
B, C are the cases for light with Gaussian mode ($l=0$), the first order
mode($l=1$) and a typical superposition mode
$(a\left|0\right\rangle+b\left|1\right\rangle)/\sqrt{a^{2}+b^{2}}$ (where $a$
and $b$ are real numbers) respectively.

Transmission efficiencies are measured for light with Gaussian mode ($l=0$),
the first order mode($l=1$) and a typical superposition mode
$(a\left|0\right\rangle+b\left|1\right\rangle)/\sqrt{a^{2}+b^{2}}$, where $a$
and $b$ are real numbers). When the hologram is placed in the beam center, the
OAM of the first diffraction order light is $1$, while for hologram in the
beam edge, the OAM is $0$. In the middle part, the output light is in the
superposition mode of $0$ and $1$. The results for $0$ and $1$ order mode
light are $2.55\%$, and $2.28\%$ respectively. The transmission efficiency for
the superposition mode light is between the upper two cases and can be changed
with the ratio of $a$ and $b$ when we move the hologram. In all the cases,
transmission efficiency is much larger than the value obtained from the
classical theoryBethe . The reason is that the interaction of the incident
light and surface plasmon is made allowed by coupling through the grating
momentum and obeys conservation of the momentum

$\overrightarrow{k}_{sp}=\overrightarrow{k}_{0}\pm i\overrightarrow{G}_{x}\pm
j\overrightarrow{G}_{y},$ (1)

where $\overrightarrow{k}_{sp}$ is the surface plasmon wave vector,
$\overrightarrow{k}_{0}$ is the component of the incident wave vector that
lies in the plane of the sample, $\overrightarrow{G}_{x}$ and
$\overrightarrow{G}_{y}$ are the reciprocal lattice vectors, and i, j are
integers. $\overrightarrow{G}_{x,y}=2\pi/d_{x,y}$ are the lattice vectors in
the $x,y$ directions respectively, and $d_{x,y}$ are the periodicity of the
structure in the $x$ and $y$ direction. While in the practical experiments,
the Eq.1 can not be satisfied simply, because many parameters can influence
the resonant frequency, for example, the thickness of the metal film, the
width of the grooves, as mentioned in Degi . Due to the symmetry of the Bull’s
eye structure, the polarization of the light has no influence on the whole
process. We can see that the transmission efficiency for Gaussian mode light
is larger than that of the first order mode light. Although it is hard to give
a precise explanation, the possible factors may be that the additional
transmissions of Gaussian mode light from directly passing light, SPs excited
from the hole edge by scattering, and lower propagating loss in the hole. This
lower loss comes from the waveguide property of the hole in which the Gaussian
mode light has a higher transmission efficiency than that of other modes as
shown inMoreno ; ruan .

Calculation shows that the energy in the beam center (250nm diameter) is only
about $0.04\%$ of the whole doughnut light. Comparing with the SPs assisted
transmission efficiency $1.28\%$, we can find that the transmitted energy is
about 32 times of the directly illuminating light on the hole area. This can
be the evidence that the transmitted light in the case of doughnut mode
results from the surface plasmon assisted transmission process.

CCD pictures are also taken for the three cases as shown in Fig. 4. The light
power is decreased to give clear pictures. It is interesting that the spatial
shape of the light was still preserved after the plasmon assisted transmission
process, even though the hole diameter (250 nm) is much smaller than the light
wavelength (632.8 nm). Since the spatial shape of the light is determined by
its OAM, which is associated with the transverse phase front of a light beam,
we can conclude that the OAM of the photons are not influenced in this
process. It has been proven in many works that the phase of the photons can be
preserved in the surface plasmons assisted transmission process, here we show
that the helical wavefront of photons can also be transferred to SPs and
carried by themren06 .

In conclusion, we investigate the extraordinary optical transmission
phenomenon through a subwavelength aperture surrounded by circular
corrugations when the light is in the doughnut shape. Since all the energy
reaches the circular corrugations but not on the hole, the directly
transmitted light can be ignored. The present experiment could provide
intriguing prospects for both the exploiting of the surface plasmon based
devices and the study of fundamental physics issues.

This work was funded by the National Basic Research Programme of China (Grants
No.2009CB929600 and No. 2006CB921900), the Innovation funds from Chinese
Academy of Sciences, and the National Natural Science Foundation of China
(Grants No. 10604052 and No.10874163).

## References

  * (1) T.W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, Nature(London) 391, 667 (1998).
  * (2) L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, Phys. Rev. Lett. 86, 1114 (2001).
  * (3) Haitao Liu, Philippe Lalanne, Nature(London) 452, 728 (2008).
  * (4) W. L. Barnes, A. Dereux, T. W. Ebbesen, Nature 424, 824 (2003).
  * (5) E. Ozaby, Science 311, 189 (2006).
  * (6) H. A. Bethe, Phys. Rev. 66, 163 (1944).
  * (7) J. Elliott, I. I. Smolyaninov, N. I. Zheludev, and A. V. Zayats, Opt. Lett. 29, 1414 (2004).
  * (8) K. J. K. Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, Phys. Rev. Lett. 92, 183901 (2004).
  * (9) X. F. Ren, G. P. Guo, Y. F. Huang, Z. W. Wang, and G. C. Guo, Appl. Phys. Lett. 90, 161112 (2007).
  * (10) X. F. Ren, G. P. Guo, Y. F. Huang, Z. W. Wang, and G. C. Guo, Opt. Lett. 31, 2792, (2006).
  * (11) X. F. Ren, G. P. Guo, Y. F. Huang, C. F. Li, and G. C. Guo, Europhys. Lett. 76, 753 (2006).
  * (12) T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, Opt. Lett. 26, 1972 (2001).
  * (13) H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, T. W. Ebbesen, Science 297, 820 (2002).
  * (14) L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys.Rev.A. 45, 8185, (1992).
  * (15) J. Arlt, K. Dholokia, L. Allen, and M. Padgett. J. Mod. Opt. 45, 1231 (1998).
  * (16) A. Vaziri, G. Weihs, and A. Zeilinger. J. Opt. B: Quantum Semiclass. Opt 4, s47 (2002).
  * (17) A. Degiron and T.W. Ebbesen, Opt. Express 12, 3694 (2004).
  * (18) Zhichao Ruan, and Min Qiu, Phys. Rev. Lett. 96, 233901 (2006).


# Jittering jets by negative angular momentum feedback in cooling flows

Noam Soker  Department of Physics, Technion, Haifa, 3200003, Israel
<EMAIL_ADDRESS>
###### Abstract

I apply the jittering jets in cooling flow scenario to explain the
perpendicular to each other and almost coeval two pairs of bubbles in the
cooling flow galaxy cluster RBS 797, and conclude that the interaction of the
jets with the cold dense clumps that feed the supermassive black hole (SMBH)
takes place in the zone where the gravitational influence of the SMBH and that
of the cluster are about equal. According to the jittering jets in cooling
flow scenario jets uplift and entrain cold and dense clumps, impart the clumps
velocity perpendicular to the original jets’ direction, and ‘drop’ them closer
to the jets’ axis. The angular momentum of these clumps is at a very high
angle to the original jets’ axis. When these clumps feed the SMBH in the next
outburst the new jets’ axis is at a high angle to the axis of the first pair
of jets. I apply this scenario to recent observations that show the two
perpendicular pairs of bubbles in RBS 797 to have a small age difference of
$<10{~{}\rm Myr}$, and conclude that the jets-clumps interaction takes place
in a distance of about $\approx 10-100{~{}\rm pc}$ from the SMBH.
Interestingly, in this zone the escape velocity from the SMBH is about equal
to the sound speed of the intracluster medium (ICM). I discuss the
implications of this finding.

galaxies: clusters: individual: RBS 797; galaxies: clusters: intracluster
medium; galaxies: jets; Astrophysics - Astrophysics of Galaxies

##  1 Introduction

When the radiative cooling time of the hot, i.e., X-ray emitting, gas in a
galaxy, in a group of galaxies, or in a cluster of galaxies is shorter than
the age of the system some gas cools to very low temperatures and flows in.
This is a cooling flow. However, the mass cooling rate to low temperatures is
much smaller that the hot gas mass divided by the radiative cooling time. This
inequality requires a heating mechanism. Most studies agree that the heating
is by active galactic nucleus (AGN) jets that the central supermassive black
hole (SMBH) launches, and that the heating and cooling processes take place in
a negative feedback cycle (for reviews see, e.g., Fabian 2012; McNamara &
Nulsen 2012; Soker 2016; Werner et al. 2019).

The jet feedback mechanism contains two main parts. In the cooling part the
hot gas cools to form cold clumps that fall/stream inward to feed the SMBH via
an accretion disk. In the heating part the accretion disk launches jets that
heat the gas. If the cooling rate of the hot gas increases, then more cold
clumps fall-in to feed the SMBH leading to more energetic jets that in turn
increase the heating rate of the hot gas, therefore reducing cooling. This
closes the negative feedback cycle.

The process by which cold clumps carry the accretion flow onto the central
SMBH (rather than Bondi-type accretion of the hot gas) is the cold feedback
mechanism (Pizzolato & Soker 2005, 2010; Gaspari et al. 2013; Gaspari,
Brighenti, & Temi 2015; Voit & Donahue 2015; Voit et al. 2015; note that some
studies use other terms for this mechanism, like chaotic cold accretion, e.g.,
McKinley et al. 2021). There are many papers in recent years that support the
cold feedback mechanism (e.g., a small sample of recent papers, Babyk et al.
2018; Gaspari et al. 2018; Ji et al. 2018; Prasad et al. 2018; Pulido et al.
2018; Voit 2018a; Yang et al. 2018; Choudhury et al. 2019; Iani et al. 2019;
Rose et al. 2019; Russell et al. 2019; Stern et al. 2019; Storchi-Bergmann, &
Schnorr-Müller 2019; Vantyghem et al. 2019; Voit 2019; Hardcastle & Croston
2020; Martz et al. 2020; Prasad et al. 2020; Eckert et al. 2021; Maccagni et
al. 2021; Pasini et al. 2021; Qiu et al. 2021; Singh, Voit, & Nath 2021).

Although there is an agreement that jets heat the hot gas, there is no
consensus on the mechanism by which the jets and the bubbles they inflate
transfer energy to the hot gas. In what follows I will concentrate on the
intracluster medium (ICM), but arguments hold also for cooling flows in
galaxies. In one group of heating processes the jets and jet-inflated bubbles
do work on the ICM by (1) driving shock waves that propagate into the ICM
(e.g., Randall et al. 2015; Guo et al. 2018), and/or (2) exciting sound waves
(e.g., Fabian et al. 2006; Tang & Churazov 2018), and/or (3) generating
internal waves in the ICM by buoyantly rising bubbles (e.g., Zhang et al.
2018), and/or (4) powering turbulence in the ICM (e.g., De Young 2010;
Banerjee & Sharma 2014; Gaspari et al. 2014; Zhuravleva et al. 2018), and/or
(5) by uplifting gas from inner regions (e.g., Gendron-Marsolais et al. 2017).
The four first heating processes cause disturbances that expand to very large
distance from the jets and bubbles.

In the second group of processes the heating of the ICM is more local to the
surroundings of the jets and the bubbles. These processes are (6) heating by
cosmic rays that the jet-inflated bubbles accelerate and that stream into the
ICM and heat it (e.g. Fujita & Ohira 2013; Pfrommer 2013; Ehlert et al. 2018;
Ruszkowski et al. 2018; Kempski & Quataert 2020) and (7) heating by mixing hot
bubble gas (thermal gas and/or cosmic rays) with the ICM (e.g., Brüggen &
Kaiser 2002; Brüggen et al. 2009; Gilkis & Soker 2012; Hillel & Soker 2014;
Yang & Reynolds 2016). Vortexes that the inflation process of the bubbles
forms are responsible for this mixing (e.g., Gilkis & Soker 2012; Hillel &
Soker 2014). In Soker (2019) I argued that even if the content of the bubble
is cosmic rays, the main process that transfers the energy of the cosmic rays
to the ICM is mixing and not streaming.

I take the view that the most efficient heating mechanism is heating by mixing
although many do not accept it, e.g., Soker & Hillel (2021) and Brienza et al.
(2021) for the most recent dispute.

The most efficient heating of the ICM by mixing takes place when the jets’
axes change their directions (e.g., Soker 2018; Cielo et al. 2018). For that,
in Soker (2018) I proposed a processes by which the feedback cycle includes in
it a processes by which the jets’ axes change direction over time, the
jittering jets in cooling flow scenario. Namely, in addition to influencing
the energy and mass of the feedback cycle, the jets influence also the angular
momentum of the accreted cold clumps. I will elaborate on this in section 2,
and use new results by Ubertosi et al. (2021) to refine this scenario for the
case of the the cooling flow cluster RBS 797 (section 3).

In a recent thorough and enlightening study of the cooling flow cluster RBS
797, Ubertosi et al. (2021) find two pairs of bubbles with the two axes of the
pairs almost perpendicular to each other. Moreover, the ages of the two pairs
is within $\Delta t_{\rm p}\lesssim 10{~{}\rm Myr}$ from each other. They
mention that the almost coeval outbursts that form two perpendicular pairs of
bubbles resulted from two black holes (for a theoretical study see, e.g., Liu
2004) or from a reorientation event of the jets’ axis (see Gitti et al. 2013
for suggesting these two possibilities for RBS 797). Each SMBH of the binary
launched one pair of jets. Ubertosi et al. (2021) do not explain why the two
pairs of jets be perpendicular to each other. I take their results to support
the jittering jets in cooling flow scenario flow that I suggested (Soker,
2018). However, in the 2018 Research Note I did not specify the zone in the
cluster cooling flow where the jets interact with the clumps. The analysis of
Ubertosi et al. (2021) allows me to locate the zone where the interaction
takes place (section 3).

Previous observations already indicated the reorientation of active galactic
nucleus (AGN) jets’ axes on a time scale of few ${~{}\rm Myr}$ (e.g., Dennett-
Thorpe et al. 2002; Lal & Rao 2007). Dennett-Thorpe et al. (2002) attribute
the reorientation of the jets’ axis to the accretion disk interaction with the
SMBH spin as suggested by Natarajan & Pringle (1998). This process needs no
binary SMBH companion.

Here I rather explore a third possibility where one pair of jets influences
the orientation of the axis of the next pair of jets (section 2). This process
is a result of the jet-feedback mechanism that takes place in cooling flows in
galaxies, in groups of galaxies, and in clusters of galaxies, and from the
cold-feedback mechanism where cold clumps feed the AGN. In section 3 I point
to the interesting finding that the jets-clumps interaction zone in RBS 797
takes place in the region where the gravitational role of the SMBH and of the
cluster are about the same. I summarise in section 4.

##  2 Jittering jets in cooling flows

Consider an accretion episode where the central SMBH accretes cold clumps from
the ICM, i.e., the cold feedback mechanism. I define equatorial plane-1 to be
the plane where the clumps form the first accretion disk, disk-1, and polar-1
the polar directions along which the disk-1 launches the first pair of jets. I
follow and expand the arguments from Soker (2018). The jets and the bubbles
they inflate uplift dense clumps by pushing the gas out (e.g., Hillel & Soker
2018) or by entraining the clumps (e.g., Hillel & Soker 2014). The vortices
that the jets and the jet-inflated bubbles excite entrain the clumps, break
them, and drag them. The vortices are excited mainly because of the shear
between the jet and the ICM. This implies that the main velocity components of
the vortices are in planes that contain the jets’ axis and are perpendicular
to equatorial plane-1, i.e., meridional planes. Therefore, each dense and cold
clump will have its main velocity components after interacting with the jet in
the meridional plane that contains the clump and the jet’s axis. I present the
schematic flow structure in Fig. 1.

Figure 1: A schematic drawing in a meridional plane of the flow structure
where jet-inflated bubbles and jets (yellow) entrain and uplift dense cold gas
(in purple).

The clump’s specific angular momentum component along the original jet
direction (polar-1) is practically zero

$\left(\overrightarrow{j}_{\rm clump}\right)_{\rm
polar-1}=\left(\overrightarrow{r}_{\rm clump}\times\overrightarrow{v}_{\rm
clump}\right)_{\rm polar-1}\simeq 0.$ (1)

Moreover, even if a clump has a velocity component perpendicular to the
meridional plane its angular momentum will be at a high angle to polar-1
direction because the clump is close to the axis of the first pair of jets
(equation 1 in Soker 2018).

The conclusion is that because the clumps are close to the jets’ axis and
because the clumps acquire velocity mainly in the meridional plane, the
angular momentum direction of each clump is mainly perpendicular to the
angular momentum of the disk that launched the jets (polar-1 direction). When
the clumps merge to form the second accretion disk, the combined angular
momentum direction, polar-2 direction, will be at a very large angle
(perpendicular) to the polar-1 directions.

##  3 The jets-clumps interaction zone

I here try to estimate the zone where the jets interact with the clumps in the
case of RBS 797.

A clump falling from radius $R_{\rm int}$ at about the sound speed reaches the
center in a time of

$\displaystyle\tau_{f}\simeq 10^{4}\left(\frac{R_{\rm int}}{10{~{}\rm
pc}}\right)\left(\frac{C_{\rm s}}{1000{~{}\rm km}{~{}\rm
s}^{-1}}\right)^{-1}{~{}\rm yr},$ (2)

where I scale the sound speed as in Ubertosi et al. (2021). Hillel & Soker
(2014) perform hydrodynamical simulations of jets that interact with dense
clumps in the ICM. Although they place the dense clumps at $20{~{}\rm kpc}$
from the center, I can scale their findings to much closer clumps as long as
they are not too close to the accretion disk that launches the jets. They find
that the jets interact with the dense clump and entrain the clumps, break them
to smaller clumps, stretch the clumps, and drag them up within a time scale
that is about an order of magnitude longer than the time scale that equation
(2) gives. Hillel & Soker (2018) examine the uplifted cool ICM gas by jet-
inflated bubbles. However, they did not examine cold clumps that are the
subject of the present study. Therefore, I scale by the results of Hillel &
Soker (2014). The clumps will fall later (a process the simulations did not
follow). My crude estimate, an estimate that requires future simulations to
verify, is that the clumps will start falling at about few times the
entrainment time. Therefore, the time from the first jet activity to the
feeding of the SMBH with clumps that the first jets uplifted/entrained is

$\displaystyle\tau_{\rm feed,2}\approx{\rm few}\times 10\tau_{f}\approx{\rm
few}\times 10^{5}$ $\displaystyle\times\left(\frac{R_{\rm int}}{10{~{}\rm
pc}}\right)\left(\frac{C_{\rm s}}{1000{~{}\rm km}{~{}\rm
s}^{-1}}\right)^{-1}{~{}\rm yr}.$ (3)

For RBS-797 the age difference between the two outbursts (of the two pairs of
bubbles) is $\Delta t_{\rm p}\lesssim 10{~{}\rm Myr}$ or much shorter even.
Therefore, the requirement is $\tau_{\rm feed,2}\simeq\Delta t_{\rm p}\lesssim
10{~{}\rm Myr}$. This implies that in the case of RBS 797 the clumps interact
with the jets inside a radius of

$\displaystyle R_{\rm int}({\rm RBS}~{}797)\lessapprox 10{~{}\rm
pc}-100{~{}\rm pc}.$ (4)

Another requirement is that the time period between the two outburst be longer
than the lifetime of the accretion disk that feeds the first pair of jets
$\tau_{\rm feed,2}>\tau_{\rm disk}$. The reason is that we do not see the jets
to precess or change direction continuously. Namely, the first jet-activity
(outburst) ends, and only then the second jet activity starts. Let $R_{\rm
disk-1}$ be the radius of the disk that feeds the SMBH in the first outburst
(first jet-launching episode). I take the viscosity time to be about 100 times
the Keplerian orbital time in the disk and so the time to deplete the
accretion disk from mass is

$\displaystyle\tau_{\rm disk}\approx 10^{4}\left(\frac{M_{\rm
SMBH}}{10^{9}M_{\odot}}\right)^{1/2}\left(\frac{R_{\rm disk-1}}{0.1{~{}\rm
pc}}\right)^{-3/2}{~{}\rm yr}.$ (5)

For the scaling above the disk radius is about thousand times the
Schwarzschild radius of the SMBH, but the disk might be smaller even.

For the jets to interact with the clumps, the clumps should not be too close
to the equatorial plane-1. For that, the clumps should still be at a much
larger radius than the radius where they form the accretion disk, i.e., where
the centrifugal acceleration is still very weak. This condition is $R_{\rm
int}\gg R_{\rm disk-1}$.

We therefore derive two conditions on the zone of jets-clumps interaction for
the proposed scenario. The first is that the clumps that the jets interact
with be at $R_{\rm int}\lessapprox 10-100{~{}\rm pc}$ (equation 4) and the
second is that they be at

$R_{\rm int}\gg 0.1{~{}\rm pc}\gtrsim R_{\rm disk-1},$ (6)

for the above scaling of the SMBH mass (but see below). This claim for a
compact accretion disk, i.e., radius much smaller than $\simeq 1{~{}\rm kpc}$,
is compatible with the study of Prasad, Sharma, & Babul (2017). They conclude
from their simulations that a compact accretion flow with a short viscous time
ought to form.

Another relevant radius is the one where the gravity of the cluster and the
black hole are about equal, $R_{\rm C-BH}$, namely, where the escape velocity
from the SMBH is about equal to the sound speed $C_{\rm s}$. Its value is

$\displaystyle R_{\rm C-BH}=\frac{2GM_{\rm SMBH}}{C^{2}_{\rm
s}}=8.6\left(\frac{M_{\rm SMBH}}{10^{9}M_{\odot}}\right)$
$\displaystyle\times\left(\frac{C_{\rm s}}{1000{~{}\rm km}{~{}\rm
s}^{-1}}\right)^{-2}{~{}\rm pc}.$ (7)

I scale the mass of the SMBH as Cavagnolo et al. (2011) did, but note that
they give the possible range to be $0.6\times 10^{9}<M_{\rm SMBH}<7.8\times
10^{9}M_{\odot}$ for the SMBH in RBS 797. Namely, the value can be in the
range of $5{~{}\rm pc}\lesssim R_{\rm C-BH}\lesssim 70{~{}\rm pc}$.

The likely conclusion is that the jets-clumps interaction takes place in the
zone of

$R_{\rm int}\approx R_{\rm C-BH}.$ (8)

The volume inner to this zone should be almost depleted of clumps according
the the scenario I propose. The implication is that dense clumps evolve
relatively slowly while in the ICM, but when they come under the gravitational
influence of the SMBH they fall rapidly inward.

Equation (8) is the main finding of this study.

##  4 Summary

Motivated by the thorough study of Ubertosi et al. (2021) I applied the
general jittering jets in cooling flow scenario (Soker, 2018) to the cooling
flow cluster RBS 797. The thorough study of Ubertosi et al. (2021) allows me
to locate the zone in which the jets from the first outburst interact with the
clumps that later feed the SMBH to launch the jets of the second outburst.

My proposed scenario for the perpendicular and almost coeval two bubble-pairs
of RBS 797 is as follows.

  1. 1.

A group (or few groups) of clumps fall to the center and feed the SMBH of RBS
797 via an accretion disk of size $R_{\rm disk-1}\lesssim 0.1{~{}\rm pc}$
(equation 6). Not all clumps reach the center together, and while some clumps
reach the center and form the accretion disk that launches the first pair of
jets, some clumps are still further out.

  2. 2.

At large distances of $r\gtrsim R_{\rm int}$ the group (or few groups) of
clumps do not fall from the equatorial-1 direction, but rather from a large
angle to this plane. Therefore, the jets interact with many clumps, i.e.,
entrain and uplift the clumps. This weakens and terminates the first jet-
launching activity (first outburst). Equation (6) gives the lower bound on the
jets-clumps interaction zone.

  3. 3.

The jets and the bubbles they start to inflate drag and uplift the clumps. The
remnants of the clumps falls back and feed the SMBH at a time $\tau_{\rm
feed,2}$ as I estimated in equation (3).

  4. 4.

The time period of $\Delta t_{\rm p}<10{~{}\rm Myr}$ between the two outbursts
of RBS 797 (Ubertosi et al., 2021) implies the same limit on $\tau_{\rm
feed,2}$, which in turn bounds the jets-clumps interaction zone to $R_{\rm
int}({\rm RBS}~{}797)\lessapprox(10-100){~{}\rm pc}$ (equation 4). This allows
also the first jet-launching episode (outburst) to shut down,

  5. 5.

When the entrained/uplifted remnants of the clumps fall to the center and feed
the SMBH the second jet-launching episode (second outburst) occurs. The new
angular momentum direction is highly inclined to that of the first outburst
(equation 1).

This scenario might account for the almost coeval and perpendicular two pairs
of bubbles in RBS 797. It turned out that the jets-clumps interaction zone
that I derived from the observations of RBS 797, under the assumption that the
jittering jets in cooling flow scenario explains the perpendicular and almost
coeval two pairs of bubbles, is about equal to the region where the
gravitational influence of the SMBH and the cluster are about equal (equation
8).

In section 1 I listed the different ICM heating processes. The main challenge
of the heating by mixing is to heat the ICM in all directions. Jittering of
the jets ease this challenge. The jittering jets in cooling flow scenario
directly connects the heating by mixing process with the cold feedback
mechanism (section 1).

Equation (8) is the main finding of this study. I speculate hat it holds in
some other cooling flows, and I predict the concentration of dense-cold cloud
that feed the AGN in that zone.

## Acknowledgments

This research was supported by the Pazy Foundation.. Data availability The
data underlying this article will be shared on reasonable request to the
corresponding author.

## References

  * Babyk et al. (2018) Babyk, I. V., McNamara, B. R., Nulsen, P. E. J., Russell, H. R., Vantyghem, A. N., Hogan, M. T., & Pulido, F. A. 2018, ApJ, 862, 39
  * Banerjee & Sharma (2014) Banerjee, N., & Sharma, P. 2014, MNRAS, 443, 687
  * Brienza et al. (2021) Brienza M., Shimwell T. W., de Gasperin F., Bikmaev I., Bonafede A., Botteon A., Brüggen M., et al., 2021, NatAs.tmp. doi:10.1038/s41550-021-01491-0
  * Brüggen & Kaiser (2002) Brüggen, M., & Kaiser, C. R. 2002, Nature, 418, 301
  * Brüggen et al. (2009) Brüggen, M., Scannapieco, E., & Heinz, S. 2009, MNRAS, 395, 2210
  * Cavagnolo et al. (2011) Cavagnolo K. W., McNamara B. R., Wise M. W., Nulsen P. E. J., Brüggen M., Gitti M., Rafferty D. A., 2011, ApJ, 732, 71. doi:10.1088/0004-637X/732/2/71
  * Choudhury et al. (2019) Choudhury, P. P., Sharma, P., & Quataert, E. 2019, MNRAS, 488, 3195
  * Cielo et al. (2018) Cielo, S., Babul, A., Antonuccio-Delogu, V., Silk, J., & Volonteri, M. 2018, A&A, 617, A58
  * Dennett-Thorpe et al. (2002) Dennett-Thorpe J., Scheuer P. A. G., Laing R. A., Bridle A. H., Pooley G. G., Reich W., 2002, MNRAS, 330, 609. doi:10.1046/j.1365-8711.2002.05106.x
  * De Young (2010) De Young, D. S. 2010, ApJ, 710, 743
  * Ehlert et al. (2018) Ehlert, K., Weinberger, R., Pfrommer, C., Pakmor, R., & Springel, V. 2018, MNRAS, 481, 2878
  * Eckert et al. (2021) Eckert D., Gaspari M., Gastaldello F., Le Brun A. M. C., O’Sullivan E., 2021, Univ, 7, 142. doi:10.3390/universe7050142
  * Fabian (2012) Fabian, A. C. 2012, ARA&A,
  * Fabian et al. (2006) Fabian, A. C., Sanders, J. S., Taylor, G. B., Allen, S. W., Crawford, C. S., Johnstone, R. M., & Iwasawa, K. 2006, MNRAS, 366, 417
  * Fujita & Ohira (2013) Fujita, Y., & Ohira, Y. 2013, MNRAS, 428, 599
  * Gaspari, Brighenti, & Temi (2015) Gaspari M., Brighenti F., Temi P., 2015, A&A, 579, A62. doi:10.1051/0004-6361/201526151
  * Gaspari et al. (2014) Gaspari, M., Churazov, E., Nagai, D., Lau, E. T., & Zhuravleva, I. 2014, A&A, 569, A67
  * Gaspari et al. (2018) Gaspari, M., McDonald, M., Hamer, S. L., et al. 2018, ApJ, 854, 167
  * Gaspari et al. (2013) Gaspari, M., Ruszkowski, M., & Oh, S. P. 2013, MNRAS, 432, 3401
  * Gendron-Marsolais et al. (2017) Gendron-Marsolais, M., et al. 2017, ApJ, 848, 26
  * Gilkis & Soker (2012) Gilkis, A., & Soker, N. 2012, MNRAS, 427, 1482
  * Gitti et al. (2013) Gitti M., Giroletti M., Giovannini G., Feretti L., Liuzzo E., 2013, A&A, 557, L14. doi:10.1051/0004-6361/201322401
  * Guo et al. (2018) Guo, F., Duan, X., & Yuan, Y.-F. 2018, MNRAS, 473, 1332
  * Hardcastle & Croston (2020) Hardcastle M. J., Croston J. H., 2020, NewAR, 88, 101539. doi:10.1016/j.newar.2020.101539
  * Hillel & Soker (2014) Hillel S., Soker N., 2014, MNRAS, 445, 4161. doi:10.1093/mnras/stu2047
  * Hillel & Soker (2018) Hillel S., Soker N., 2018, RAA, 18, 081. doi:10.1088/1674-4527/18/7/81
  * Iani et al. (2019) Iani, E., Rodighiero, G., Fritz, J., et al. 2019, MNRAS, 487, 5593. doi:10.1093/mnras/stz1631
  * Ji et al. (2018) Ji, S., Oh, S. P., & McCourt, M. 2018, MNRAS, 476, 852
  * Kempski & Quataert (2020) Kempski P., Quataert E., 2020, MNRAS, 493, 1801. doi:10.1093/mnras/staa385
  * Lal & Rao (2007) Lal D. V., Rao A. P., 2007, MNRAS, 374, 1085. doi:10.1111/j.1365-2966.2006.11225.x
  * Liu (2004) Liu F. K., 2004, MNRAS, 347, 1357. doi:10.1111/j.1365-2966.2004.07325.x
  * Maccagni et al. (2021) Maccagni F. M., Serra P., Gaspari M., Kleiner D., Morokuma-Matsui K., Oosterloo T. A., Onodera M., et al., 2021, arXiv, arXiv:2108.05247
  * Martz et al. (2020) Martz C. G., McNamara B. R., Nulsen P. E. J., Vantyghem A. N., Gingras M.-J., Babyk I. V., Russell H. R., et al., 2020, ApJ, 897, 57. doi:10.3847/1538-4357/ab96cd
  * McKinley et al. (2021) McKinley B., Tingay S. J., Gaspari M., Kraft R. P., Matherne C., Offringa A. R., McDonald M., et al., 2021, arXiv, arXiv:2111.02683
  * McNamara & Nulsen (2012) McNamara B. R., Nulsen P. E. J., 2012, NJPh, 14, 055023. doi:10.1088/1367-2630/14/5/055023
  * Natarajan & Pringle (1998) Natarajan P., Pringle J. E., 1998, ApJL, 506, L97. doi:10.1086/311658
  * Pasini et al. (2021) Pasini T., Finoguenov A., Brüggen M., Gaspari M., de Gasperin F., Gozaliasl G., 2021, MNRAS, 505, 2628. doi:10.1093/mnras/stab1451
  * Pfrommer (2013) Pfrommer, C. 2013, ApJ, 779, 10
  * Prasad, Sharma, & Babul (2017) Prasad D., Sharma P., Babul A., 2017, MNRAS, 471, 1531. doi:10.1093/mnras/stx1698
  * Prasad et al. (2020) Prasad D., Voit G. M., O’Shea B. W., Glines F., 2020, ApJ, 905, 50. doi:10.3847/1538-4357/abc33c
  * Pizzolato & Soker (2005) Pizzolato, F., & Soker, N. 2005, ApJ, 632, 821
  * Pizzolato & Soker (2010) Pizzolato, F., & Soker, N. 2010, MNRAS, 408, 961
  * Prasad et al. (2018) Prasad, D., Sharma, P., & Babul, A. 2018, ApJ, 863, 62
  * Pulido et al. (2018) Pulido, F. A., McNamara, B. R., Edge, A. C., et al. 2018, ApJ, 853, 177
  * Qiu et al. (2021) Qiu Y., McNamara B. R., Bogdanovic T., Inayoshi K., Ho L. C., 2021, arXiv, arXiv:2103.06505
  * Randall et al. (2015) Randall, S. W., et al. 2015, ApJ, 805, 112
  * Rose et al. (2019) Rose T., Edge A. C., Combes F., Gaspari M., Hamer S., Nesvadba N., Peck A. B., et al., 2019, MNRAS, 489, 349. doi:10.1093/mnras/stz2138
  * Russell et al. (2017) Russell H. R., McNamara B. R., Fabian A. C., Nulsen P. E. J., Combes F., Edge A. C., Hogan M. T., et al., 2017, MNRAS, 472, 4024. doi:10.1093/mnras/stx2255
  * Russell et al. (2019) Russell H. R., McNamara B. R., Fabian A. C., Nulsen P. E. J., Combes F., Edge A. C., Madar M., et al., 2019, MNRAS, 490, 3025. doi:10.1093/mnras/stz2719
  * Ruszkowski et al. (2018) Ruszkowski, M., Yang, H.-Y. K., & Reynolds, C. S. 2018, ApJ, 858, 64
  * Singh, Voit, & Nath (2021) Singh P., Voit G. M., Nath B. B., 2021, MNRAS, 501, 2467. doi:10.1093/mnras/staa3827
  * Soker (2016) Soker, N. 2016, New Astronomy Reviews, 75, 1
  * Soker (2018) Soker N., 2018, RNAAS, 2, 48. doi:10.3847/2515-5172/aab9b5
  * Soker (2019) Soker, N. 2019, MNRAS, 482, 1883
  * Soker & Hillel (2021) Soker N., Hillel S., 2021, arXiv, arXiv:2110.10608
  * Stern et al. (2019) Stern, J., Fielding, D., Faucher-Giguère, C.-A., & Quataert, E. 2019, MNRAS, 488, 2549
  * Storchi-Bergmann, & Schnorr-Müller (2019) Storchi-Bergmann, T., & Schnorr-Müller, A. 2019, Nature Astronomy, 3, 48
  * Su et al. (2017) Su, Y., et al. 2017, ApJ, 847, 94
  * Tang & Churazov (2018) Tang, X., & Churazov, E. 2018, MNRAS, 477, 3672
  * Ubertosi et al. (2021) Ubertosi F., Gitti M., Brighenti F., Brunetti G., McDonald M., Nulsen P., McNamara B., et al., 2021, arXiv, arXiv:2111.03679
  * Vantyghem et al. (2019) Vantyghem, A. N., McNamara, B. R., Russell, H. R., et al. 2019, ApJ, 870, 57
  * Voit (2018a) Voit, G. M. 2018a, ApJ, 868, 102
  * Voit (2019) Voit, G. M. 2019, ApJ, 880, 139
  * Voit & Donahue (2015) Voit G. M., Donahue M., 2015, ApJL, 799, L1. doi:10.1088/2041-8205/799/1/L1
  * Voit et al. (2015) Voit, G. M., Donahue, M., Bryan, G. L., et al. 2015, Nature, 519, 203
  * Werner et al. (2019) Werner, N., McNamara, B. R., Churazov, E., & Scannapieco, E. 2019, Space Sci. Rev., 215, 5
  * Yang & Reynolds (2016) Yang, H.-Y. K., & Reynolds, C. S. 2016, ApJ, 829, 90
  * Yang et al. (2018) Yang, L., Tozzi, P., Yu, H., Lusso, E., Gaspari, M., Gilli, R., Nardini, E., & Risaliti, G. 2018, ApJ, 859, 65
  * Zhang et al. (2018) Zhang, C., Churazov, E., & Schekochihin, A. A. 2018, MNRAS, 478, 4785
  * Zhuravleva et al. (2018) Zhuravleva, I., Allen, S. W., Mantz, A. B., & Werner, N. 2018, ApJ, 865, 53