# Periodic frames

Lev D. Beklemishev111The corresponding author, e-mail<EMAIL_ADDRESS>Steklov
Mathematical Institute of Russian Academy of Sciences  National Research
University Higher School of Economics, Moscow  Yunsong Wang  Peking
University, Beiging, China

###### Abstract

Polymodal provability logic GLP is incomplete w.r.t. Kripke frames. It is
known to be complete w.r.t. topological semantics, where the diamond
modalities correspond to topological derivative operations. However, the
topologies needed for the completeness proof are highly non-constructive. The
question of completeness of GLP w.r.t. natural scattered topologies on
ordinals is dependent on large cardinal axioms of set theory and is still
open. So far, we are lacking a useable class of models for which GLP is
complete.

In this paper we define a natural class of countable general topological
frames on ordinals for which GLP is sound and complete. The associated
topologies are the same as the ordinal topologies introduced by Thomas Icard.
However, the key point is to specify a suitable algebra of subsets of an
ordinal closed under the boolean and topological derivative operations. The
algebras we define are based on the notion of a periodic set of ordinals
generalizing that of an ultimately periodic binary omega-word.

##  1 Introduction

Polymodal provability logic $\mathrm{GLP}$ was introduced by Giorgi Japaridze
[34, 23] as the logic of the extensions of Peano arithmetic by $n$-fold
applications of the omega-rule, for each $n<\omega$. Later, Konstantin
Ignatiev extended Japaridze’s work and showed that $\mathrm{GLP}$ is sound and
complete w.r.t. a wider class of arithmetical interpretations by any
sufficiently strong sequence of sound provability predicates [33]. He also
obtained normal forms and introduced a universal Kripke model for the
variable-free fragment of $\mathrm{GLP}$ that played a prominent role in later
studies.

More recently, $\mathrm{GLP}$ has found interesting applications in proof-
theoretic analysis of arithmetic [8, 9, 10, 35, 30, 37, 31, 18] which
stimulated further interest in the study of modal-logical properties of
$\mathrm{GLP}$ [17, 41, 32, 12, 27, 36, 40, 38].

One obstacle in the study of $\mathrm{GLP}$ is its lack of Kripke
completeness. It is easy to see that in any Kripke frame validating the axioms
of $\mathrm{GLP}$ the relations corresponding to all modalities but $[0]$ have
to be empty. Thereby, Kripke completeness of $\mathrm{GLP}$ fails in a very
strong way. Over the years there has been a steady effort to develop some
natural and usable models for this logic.

A sufficiently rich class of _Kripke models_ (rather than frames), for which
$\mathrm{GLP}$ is sound and complete, has been introduced in [12]. The
description of this class, although constructive, is rather complicated. In
particular, appropriate models are obtained from finite ones by a double limit
construction. Thereafter, attention of researchers was drawn to _neighborhood_
or _topological models_ of $\mathrm{GLP}$.

Under the standard topological interpretation of provability logic
$\mathrm{GL}$, propositional variables are interpreted as subsets of a
(scattered) topological space, boolean connectives correspond to boolean
operations, whereas the modality $\Diamond$ is interpreted as the topological
derivative operation mapping a set $A$ to the set of its limit points. This
interpretation originated in the work of H. Simmons [42] and L. Esakia [28]
who isolated the role of scatteredness, the topological counterpart of Löb’s
axiom. Later, Abashidze [1] and Blass [22] proved the completeness of
$\mathrm{GL}$ for the topological interpretation w.r.t. the interval topology
on any ordinal $\Omega$ provided $\Omega\geqslant\omega^{\omega}$. This result
served as a basis for various generalizations. We refer to [16] for a
comprehensive survey of topological models of provability logic.

Topological interpretation of $\mathrm{GLP}$ turned out to be a rich area of
study on its own with a number of significant contributions. Thomas Icard [32]
suggested natural topological models for the variable-free fragment of
$\mathrm{GLP}$ based on Ignatiev’s work. These models fail to characterize
full $\mathrm{GLP}$, however, they are somewhat simpler than Ignatiev’s Kripke
frame. Icard’s topologies on ordinals will play a major role in this paper.

In [11, 14] the notion of _$\mathrm{GLP}$ -space_, that is, a sound
topological model of $\mathrm{GLP}$, was introduced and it was shown that the
bimodal fragment of $\mathrm{GLP}$ is topologically complete. In [13], this
has been improved by showing that the bimodal fragment of $\mathrm{GLP}$ is
complete w.r.t. a natural pair of ordinal topologies (assuming the set-
theoretic hypotheses $V=L$ or Jensen’s square principles). This paper was
based on an earlier result by Andreas Blass [22] who showed that $\mathrm{GL}$
is sound and complete w.r.t. the club filter topology on $\aleph_{\omega}$.

The extension of this result to more than two modalities turned out to be a
significant problem. The ordinals bear a natural sequence of topologies making
it a $\mathrm{GLP}$-space, the so-called _ordinal $\mathrm{GLP}$-space_. This
model and the underlying topologies were introduced in [11, 14]. However,
whether $\mathrm{GLP}$ is complete for this model is still an open question,
even under set-theoretic assumptions such as $V=L$ and large cardinal axioms.
It was noticed that in order to guarantee that all the topologies in that
sequence are at least non-discrete — a necessary condition for completeness —
one really needs large cardinal assumptions. A natural sufficient condition
for the non-discreteness is that $\Pi_{n}^{1}$-indescribable cardinals exist,
for each $n<\omega$. More exactly, the limit points of the topologies of the
ordinal GLP-space correspond to the so-called _two-fold $d_{n}$-reflecting
ordinals_ [11, 16] or equivalently, using the terminology of [4, 7], to _$n$
-simultaneusly-stationary reflecting cardinals_, for $n<\omega$.

The question of consistency strength of $n$-(simultaneusly)-stationary
reflecting cardinals and its generalization to transfinite iterations turned
out to be fruitful from a set-theoretic point of view and stimulated a number
of papers starting from the work of Joan Bagaria [4], see [7, 6, 5]. See also
[16] for a survey of set-theoretic aspects of $\mathrm{GLP}$.

Despite these set-theoretic complications the topological completeness of
$\mathrm{GLP}$ was established in [15] on the basis of ZFC by a substantially
new construction. However, the constructed topological spaces were not natural
and, in fact, were obtained with the heavy use of the axiom of choice. The
technical difficulties involved were also significant. Later this result has
been extended and improved in various ways. In particular, Fernández-Duque
[29] extended it to a version of $\mathrm{GLP}$ in the language with countably
many modalities. For the uncountably many modalities the problem is still
open. Working towards its solution, Aguilera [2] proved the topological
completeness of $\mathrm{GLP}$ w.r.t. a model of generalized Icard topologies
over any sufficiently tall scattered space. Shamkanov [39] found an
alternative proof of topological completeness of $\mathrm{GLP}$ avoiding the
intermediate use of Kripke models and proceeding directly from the algebraic
models. He also established the _strong completeness_ of $\mathrm{GLP}$ w.r.t.
topological semantics generalizing an earlier result of Aguilera and
Fernández-Duque [3] for the case of Gödel–Löb logic.

In this paper we consider a more general kind of topological semantics for
$\mathrm{GLP}$ called _general topological (derivative) frames_. General
topological frames for a modal logic relate to its topological semantics in
the same way as general Kripke frames relate to Kripke semantics. These are
polytopological spaces $X$ equipped with an algebra of admissible subsets
${\mathcal{A}}\subseteq{\mathcal{P}}(X)$. The algebra is supposed to be closed
under the boolean operations and the derivative operations corresponding to
the topologies of the structure. Propositional formulas are then interpreted
as elements of ${\mathcal{A}}$.

Such structures have been considered in the modal logic literature before,
e.g., the authors of [20] consider general topological frames for S4 based on
topological closure rather than derivative operators. General topological
derivative frames have been introduced by David Fernández-Duque [29] under the
name _ambiences_. According to [29], a _provability ambience_ is a general
topological frame satisfying the axioms of $\mathrm{GLP}$ in which the
sequence of topologies is increasing and the topologies are scattered.222In
principle, a general topological frame satisfying the axioms of $\mathrm{GLP}$
need not be topologically scattered. However, in this paper as well as in [29]
we only deal with scattered topologies, that is, de facto ambiences. We will
call provability ambiences _$\mathrm{GLP}$ -ambiences_.

In this paper we define explicit and natural general $\mathrm{GLP}$-ambiences
for which the logic $\mathrm{GLP}$ is complete. We exhibit structures of this
kind in the polytopological space of the ordinal $\varepsilon_{0}$, equipped
with the sequence of Icard’s topologies. (In [29] such ambiences are called
_shifted Icard ambiences_.) In our treatment the algebra of admissible subsets
on $\varepsilon_{0}$ is defined by adopting a very natural idea of
periodicity.

Periodic transfinite binary words, or equivalently periodic subsets of the
ordinals, have appeared in the literature before. On the one hand, they are
considered in the context of the combinatorics of words, as a natural
generalization of periodic $\omega$-words (see, for example, [26]). On the
other hand, they appeared in the context of the studies of monadic second-
order theories of well-orders, starting from the work of J.R. Büchi [24, 25].
In fact, MSO definable sets on $\omega^{\omega}$ happen to be precisely the
_hereditarily periodic sets_ of this paper. (It seems to be the most natural
notion of a periodic set of ordinals that one can come up with.) However,
these sets are not enough for the completeness of $\mathrm{GLP}$ in the
language with more than one modality. So, we will generalize them to what we
call _$n$ -periodic sets_, for each $n<\omega$, and to arbitrary ordinals.

The main result of this paper is the completeness theorem for $\mathrm{GLP}$
w.r.t. the ordinal $\varepsilon_{0}$ (or any larger ordinal) equipped with
Icard’s topologies and the algebra of $n$-periodic sets. This structure gives
the best solution that we have at the moment to the problem of finding an
explicit and usable complete semantics for the provability logic
$\mathrm{GLP}$.

The paper is organized as follows. In Section 2 we present necessary
background results on $\mathrm{GLP}$, its algebraic and neighborhood
semantics, and general topological frames. In Section 3 we introduce periodic
transfinite words and sets of ordinals. We prove that periodic sets form a
Magari algebra. In Section 4 we introduce $n$-periodic sets and the
corresponding general topological frames. In Section 5 we borrow some
techniques from [15] reducing the question of completeness of $\mathrm{GLP}$
w.r.t. periodic frames to a covering construction. Some necessary intermediate
lemmas from [15] are presented in the appendix to this paper. Finally, in
Section 6 we prove the covering lemma and thereby complete the proof of our
main result.

The authors started working on this topic together in the context a Master’s
thesis project of Yunsong Wang at the ILLC of the University of Amsterdam. In
his Thesis [43] the notions of hereditarily periodic set and of 1-periodic set
were defined, and the corresponding completeness theorem for the bimodal
fragment of $\mathrm{GLP}$ was proved. After the completion of this work, the
first author of the paper reworked the whole setup and extended the result to
the full language of $\mathrm{GLP}$. The current paper is written up by Lev
Beklemishev.

We are grateful to Nick Bezhaishvili, a co-supervisor of the Master’s Thesis
of Yunsong Wang, and to David Fernández-Duque and Guram Bezhanishvili for
valuable comments.

##  2 Preliminaries

Here we introduce some of the background notions for this paper.

###  2.1 Provability logic $\mathrm{GLP}$

$\mathrm{GLP}$ is a propositional modal logic formulated in the language with
infinitely many modalities $[n]$, for all $n\in\omega$. As usual, we
abbreviate $\neg[n]\neg\varphi$ by $\langle n\rangle\varphi$. $\mathrm{GLP}$
is given by the following axiom schemata and rules:

Axioms:

  1. $\mathrm{(i)}$

Boolean tautologies;

  2. $\mathrm{(ii)}$

$[n](\varphi\rightarrow\psi)\rightarrow([n]\varphi\rightarrow[n]\psi)$;

  3. $\mathrm{(iii)}$

$[n]([n]\varphi\rightarrow\varphi)\rightarrow[n]\varphi$;

  4. $\mathrm{(iv)}$

$\langle m\rangle\varphi\rightarrow[n]\langle m\rangle\varphi$, for $m<n$.

  5. $\mathrm{(v)}$

$[m]\varphi\rightarrow[n]\varphi$, for $m\leqslant n$;

Rules:

modus ponens, $\varphi\vdash[n]\varphi$.

The first three axioms are the well-known axioms of the Gödel–Löb provability
logic $\mathrm{GL}$ (stated for modality $[n]$). We identify $\mathrm{GL}$
with the fragment of $\mathrm{GLP}$ restricted to the language with modality
$[0]$.

###  2.2 GLP-algebras

Algebraic models of $\mathrm{GLP}$ are called $\mathrm{GLP}$-algebras. These
are boolean algebras with operators $\langle n\rangle$, for all $n\in\omega$,
satisfying the following modal identities:

  1. $\mathrm{(i)}$

$\langle n\rangle(x\lor y)=\langle n\rangle x\lor\langle n\rangle y$;
$\neg\langle n\rangle\bot$;

  2. $\mathrm{(ii)}$

$\langle n\rangle x=\langle n\rangle(x\land\neg\langle n\rangle x)$;

  3. $\mathrm{(iii)}$

$\langle n\rangle x\land\langle m\rangle y=\langle n\rangle(x\land\langle
m\rangle y)$, for $m<n$;

  4. $\mathrm{(iv)}$

$\langle n\rangle x\land\langle m\rangle x=\langle n\rangle x$, for
$m\leqslant n$.

Identities (i) and (ii), for any modality $\langle n\rangle$, define the
variety of _Magari algebras_ , which are the algebraic models of
$\mathrm{GL}$. The last two identities correspond to the remaining axioms of
$\mathrm{GLP}$.

A _valuation_ on a $\mathrm{GLP}$-algebra ${\mathcal{M}}$ is a map $v$ from
propositional variables to ${\mathcal{M}}$. It is inductively extended to the
set of all $\mathrm{GLP}$-formulas by the rules:
$v(\varphi\land\psi)=v(\varphi)\land v(\psi)$, $v(\neg\varphi)=\neg
v(\varphi)$, and $v([n]\varphi)=\neg\langle n\rangle\neg v(\varphi)$.

The _logic_ of ${\mathcal{M}}$ is the set $\mathrm{Log}({\mathcal{M}})$ of all
formulas $\varphi$ such that $v(\varphi)=\top$, for all valuations $v$ in
${\mathcal{M}}$. We say that a logic $L$ is _complete_ for a class of algebras
$\mathbf{C}$, if
$L=\bigcap_{{\mathcal{M}}\in\mathbf{C}}\mathrm{Log}({\mathcal{M}})$.

On general grounds, the logic $\mathrm{GLP}$ is complete for the class of all
$\mathrm{GLP}$-algebras, and $\mathrm{GL}$ is complete for the class of all
Magari algebras.

###  2.3 Neighborhood frames and $\mathrm{GLP}$-spaces

Let $L$ be a normal modal logic in the language with modalities $[i]$, for
$i\in I$. We call a _neighborhood frame for $L$_ a structure
$(X,(\delta_{i})_{i\in I})$ where each $\delta_{i}$ is an operator
$\delta_{i}:{\mathcal{P}}(X)\to{\mathcal{P}}(X)$ and the algebra
$({\mathcal{P}}(X),(\delta_{i})_{i\in I})$ satisfies the axioms of $L$ under
the interpretation of boolean connectives as the usual set-theoretic
operations and $\langle i\rangle$ as $\delta_{i}$.

Esakia [28] has shown that in the case of Gödel–Löb logic $\mathrm{GL}$ every
such $\delta$ coincides with the topological derivative operation of a
(uniquely defined) scattered topology on $X$. Recall that a topological space
$(X,\tau)$ is _scattered_ if each nonempty subspace of $X$ has an isolated
point. Topological derivative operation $d_{\tau}$ maps a subset $A\subseteq
X$ to its set of limit points $d_{\tau}(A)$. Esakia’s result justifies us
calling neighborhood frames for $\mathrm{GL}$ _topological (derivative)
frames_.

The same applies to neighborhood frames for $\mathrm{GLP}$ which correspond to
_GLP-spaces_ [14].

###### Definition 1

$(X,(\tau_{i})_{i\in\omega})$ is a _$\mathrm{GLP}$ -space_ if, for each
$n\in\omega$:

  * (i)

$(X,\tau_{n})$ is a scattered topological space;

  * (ii)

$\tau_{n}\subseteq\tau_{n+1}$;

  * (iii)

For each $A\subseteq X$, $d_{\tau_{n}}(A)$ is $\tau_{n+1}$-open.

It is easy to see that $(X,(\tau_{i})_{i\in\omega})$ is a $\mathrm{GLP}$-space
iff $(X,(d_{\tau_{i}})_{i\in\omega})$ is a neighborhood frame for
$\mathrm{GLP}$ iff $({\mathcal{P}}(X),(d_{\tau_{i}})_{i\in\omega})$ is a
$\mathrm{GLP}$-algebra.

###  2.4 General topological frames and ambiences

Ambiences have been introduced by Fernández-Duque in [29] as a generalization
of $\mathrm{GLP}$-spaces. The same notion is called _general topological
frame_ in [43].

###### Definition 2

An _ambience_ is a tuple $(X,(\tau_{i})_{i\in I},\mathcal{A})$ where
$(X,(\tau_{i})_{i\in I})$ is a poly-topological space and
$\mathcal{A}\subseteq\mathcal{P}(X)$ is closed under the boolean operations
and $d_{\tau_{i}}$, for each $i\in I$. We call $\cal A$ the algebra of
_admissible sets_ of the ambience.

Similarly to neighborhood frames, with each ambience
$\mathcal{X}=(X,(\tau_{i})_{i\in I},\mathcal{A})$ we associate the boolean
algebra with operators $({\mathcal{A}},(d_{\tau_{i}})_{i\in I})$. Thus, an
ambience is essentially the same thing as a subalgebra of the algebra
associated with a topological derivative frame. By definition, the _logic of
$\mathcal{X}$_, denoted $\mathrm{Log}(\mathcal{X})$, is the logic of this
algebra.

###### Definition 3

An ambience $(X,(\tau_{n})_{n\in\omega},\mathcal{A})$ is a _$\mathrm{GLP}$
-ambience_ (or _provability ambience_ [29]) if, for each $n\in\omega$:

  * (i)

$(X,\tau_{n})$ is a scattered topological space;

  * (ii)

$\tau_{n}\subseteq\tau_{n+1}$;

  * (iii)

For each $U\in{\cal A}$, $d_{\tau_{n}}(U)$ is $\tau_{n+1}$-open.

If $\mathcal{X}$ is a $\mathrm{GLP}$-ambience then its algebra is a
$\mathrm{GLP}$-algebra and $\mathrm{Log}(\mathcal{X})$ contains
$\mathrm{GLP}$. $\mathrm{GLP}$-spaces are $\mathrm{GLP}$-ambiences in which
${\mathcal{A}}$ is the algebra of all subsets of $X$.

###### Remark 2.1

The requirements stated in items (i) and (ii) are actually stronger than is
necessary for the soundness of $\mathrm{GLP}$. For example, Löb’s axiom will
be satisfied if the topologies are ‘scattered’ w.r.t. the admissible subsets
only.

Let $(\mathcal{X}_{j})_{j\in J}$ be a family of ambiences,
$\mathcal{X}_{j}=(X_{j},(\tau_{ij})_{i\in I},{\mathcal{A}}_{j})$. The _sum_
$\mathcal{X}=\bigsqcup_{j\in J}\mathcal{X}_{j}$ is defined in a natural way.
We define $\mathcal{X}=(X,(\tau_{i})_{i\in I},{\mathcal{A}})$ so that $X$ is
the disjoint union of sets $X_{j}$. A subset $U\subseteq X$ is $\tau_{i}$-open
if $U\cap X_{j}\in\tau_{ij}$, for all $j\in J$, and ${\mathcal{A}}$ consists
of all sets $A\subseteq X$ such that $A\cap X_{j}\in{\mathcal{A}}_{j}$, for
all $j\in J$.

Clearly, in this case $({\mathcal{A}},(d_{\tau_{i}})_{i\in I})$ is isomorphic
to the product of the derivative algebras
$({\mathcal{A}}_{j},(d_{\tau_{ij}})_{i\in I})$. Since varieties of algebras
are closed under products, if all $\mathcal{X}_{j}$ are
$\mathrm{GLP}$-ambiences, then so is $\mathcal{X}$.

###### Definition 4

Let $\mathrm{GLP}_{n}$ denote the fragment of $\mathrm{GLP}$ in the language
with modalities $[i]$ for $i\leqslant n$. Thus, $\mathrm{GL}$ is a notational
variant of $\mathrm{GLP}_{0}$. The notions of $\mathrm{GLP}$-algebra,
$\mathrm{GLP}$-space, $\mathrm{GLP}$-ambience, etc. have obvious analogues for
$\mathrm{GLP}_{n}$, they will occasionally be used below.

###  2.5 Ranks and $d$-maps

Let $(X,\tau)$ be a scattered space. The _Cantor–Bendixson sequence_ of
subsets of $X$ is defined by transfinite iteration of the derivative operation
$d_{\tau}$ on $X$: $d_{\tau}^{0}X=X$;
$d_{\tau}^{\alpha+1}X=d_{\tau}(d_{\tau}^{\alpha}X)$ and
$d_{\tau}^{\alpha}X=\bigcap\limits_{\beta<\alpha}d_{\tau}^{\beta}X$ if
$\alpha\in\mathrm{Lim}$.

Notice that if $X$ is scattered, all $d_{\tau}^{\alpha}X$ are closed and hence
decreasing. In fact, Cantor proved that $(X,\tau)$ is scattered iff
$d_{\tau}^{\alpha}X=\varnothing$, for some $\alpha$.

The _rank function_ $\rho_{\tau}:X\to\mathrm{On}$ is defined by

$\rho_{\tau}(x):=\min\\{\alpha:x\notin d_{\tau}^{\alpha+1}(X)\\}.$

$\rho_{\tau}$ maps $X$ onto an ordinal $\rho_{\tau}(X)$ called the
Cantor–Bendixson rank of $X$.

Let $\Omega$ be an ordinal. Its _left topology_ $\tau_{\leftarrow}$ is given
by intervals of the form $[0,\alpha)$, for all $\alpha\in\Omega$. In this case
we have $\rho(\alpha)=\alpha$, for all $\alpha$.

The _interval topology_ on $\Omega$ is generated by $\\{0\\}$ and all
intervals of the form $(\alpha,\beta)$ for $\alpha<\beta\leqslant\Omega$. Its
rank function is the function $\ell$ defined by

$\ell(0)=0;\quad\ell(\alpha)=\beta\text{ if $\alpha=\gamma+\omega^{\beta}$,
for some $\gamma$, $\beta$.}$

Next we introduce $d$-maps that are very useful in the study of topological
derivative semantics. A map $f:X\to Y$ between topological spaces is called a
_d-map_ if $f$ is continuous, open and _pointwise discrete_ , that is,
$f^{-1}(y)$ is a discrete subspace of $X$ for each $y\in Y$. $d$-maps satisfy
the properties expressed in the following proposition (see [19]).

###### Proposition 2.2

  1. $\mathrm{(i)}$

$f^{-1}(d_{Y}(A))=d_{X}(f^{-1}(A))$, for any $A\subseteq Y$;

  2. $\mathrm{(ii)}$

$f^{-1}:({\mathcal{P}}(Y),d_{Y})\to({\mathcal{P}}(X),d_{X})$ is a homomorphism
of modal algebras;

  3. $\mathrm{(iii)}$

If $f$ is onto, then $f^{-1}$ is an embedding, and hence

  4. $\mathrm{(iv)}$

$\mathrm{Log}(({\mathcal{P}}(X),d_{X}))\subseteq\mathrm{Log}(({\mathcal{P}}(Y),d_{Y}))$.

When considering analogs of $d$-maps between ambiances, the relevant notion is
_algebra preserving $d$-map_.

Let $\mathcal{X}=(X,(\tau_{i})_{i\in I},{\mathcal{A}})$ and
$\mathcal{Y}=(Y,(\sigma_{i})_{i\in I},{\mathcal{B}})$ be ambiances. A map
$f:\mathcal{X}\to\mathcal{Y}$ is called _algebra preserving_ if
$f^{-1}(B)\in{\mathcal{A}}$, for each $B\in{\mathcal{B}}$. It is called a
$d$-map, if it is a $d$-map between $(X,\tau_{i})$ and $(Y,\sigma_{i})$, for
all $i\in I$.

###### Proposition 2.3

If $f:\mathcal{X}\to\mathcal{Y}$ is an algebra preserving $d$-map, then

  1. $\mathrm{(i)}$

$f^{-1}:({\mathcal{B}},(d_{\sigma_{i}})_{i\in
I})\to({\mathcal{A}},(d_{\tau_{i}})_{i\in I})$ is a homomorphism;

  2. $\mathrm{(ii)}$

If $f$ is onto, then $f^{-1}:{\mathcal{B}}\to{\mathcal{A}}$ is an embedding,
and hence

  3. $\mathrm{(iii)}$

$\mathrm{Log}(\mathcal{X})\subseteq\mathrm{Log}(\mathcal{Y})$.

The following proposition [16] states that the rank function, when the
ordinals are equipped with their left topology, becomes a $d$-map. It is also
uniquely characterized by this property.

###### Proposition 2.4

Let $\Omega:=\rho_{\tau}(X)$ taken with its left topology. Then

  1. $\mathrm{(i)}$

$\rho_{\tau}:X\twoheadrightarrow\Omega$ is an onto $d$-map;

  2. $\mathrm{(ii)}$

If $f:X\to\lambda$ is a $d$-map, where $\lambda$ is an ordinal with its left
topology, then $f(X)=\Omega$ and $f=\rho_{\tau}$.

###  2.6 Icard’s topologies

Let $\Omega$ be an ordinal. Icard’s topologies $\vartheta_{n}$, for each
$n\in\omega$, are defined as follows.333Our $\vartheta_{n}$ is Icard’s
topology $\theta_{n+1}$.

###### Definition 5

Let $\vartheta_{0}$ be the interval topology, and let $\vartheta_{n}$ be
generated as a subbase by $\vartheta_{0}$ and all sets of the form

$U^{m}_{\beta}(\Omega):=\\{\alpha\in\Omega:\ell^{m}(\alpha)>\beta\\},$

for $m\leqslant n$. Here $\ell^{m}$ denotes the $m$-th iterate of the $\ell$
function. We let $d_{n}$ and $\rho_{n}$ denote the derivative operator and the
rank function for $\vartheta_{n}$, respectively.

Clearly, $(\vartheta_{n})_{n\in\omega}$ is an increasing sequence of scattered
topologies. The following characterizations are from [16] and are easy to
establish on the basis of Proposition 2.4.

###### Proposition 2.5

  1. $\mathrm{(i)}$

$\ell:(\Omega,\vartheta_{n+1})\to(\Omega,\vartheta_{n})$ is a $d$-map;

  2. $\mathrm{(ii)}$

$\vartheta_{n+1}$ is the coarsest topology $\tau$ on $\Omega$ such that
$\ell:(\Omega,\tau)\to(\Omega,\vartheta_{n})$ is continuous;

  3. $\mathrm{(iii)}$

$\ell^{n+1}$ is the rank function of $\vartheta_{n}$, that is,
$\rho_{n}=\ell^{n+1}$;

  4. $\mathrm{(iv)}$

$\vartheta_{n+1}$ is generated by $\vartheta_{n}$ and
$\\{d_{n}^{\alpha+1}(\Omega):\alpha<\rho_{n}(\Omega)\\}$.

Notice that (iv) allows one to define the analogues of Icard’s topologies on
any scattered space.

##  3 Periodic sets of ordinals

###  3.1 Transfinite words and periodic sets

Here we develop some basic theory of transfinite words. We will only consider
binary words in this paper, although the notions can be easily generalized to
any alphabet.

###### Definition 6

A _(transfinite) word_ is map $A:\alpha\to\\{0,1\\}$ where $\alpha\in On$. The
ordinal $\alpha$ is called the length of $A$ and denoted $|A|$. We also call
$A$ an _$\alpha$ -word_. The empty word (the only word of length $0$) is
denoted $\Lambda$.

_Concatenation_ of transfinite words $A$ and $B$ is defined in a natural way
and is denoted $AB$. This operation obviously satisfies $|AB|=|A|+|B|$,
$A(BC)=(AB)C$.

A word $A$ is an _initial segment_ of $B$ if there is a $C$ such that $AC=B$.
Similarly, $A$ is an end segment of $B$ if there is a $C$ such that $CA=B$.

_Iteration_ $A^{\beta}$ of a transfinite word $A$ is defined by transfinite
induction on $\beta$: $A^{0}:=\Lambda$, $A^{\beta+1}:=A^{\beta}A$,
$A^{\lambda}:=\bigcup_{\beta<\lambda}A^{\beta}$ if $\lambda\in\mathrm{Lim}$.

Clearly, $|A^{\beta}|=|A|\cdot\beta$. We also have
$A^{\beta}A^{\gamma}=A^{\beta+\gamma}$ and
$(A^{\beta})^{\gamma}=A^{\beta\gamma}$.

Next we are going to introduce one of the main concepts in this paper, that of
a periodic word. We do not intend to treat this notion in full generality, but
we deal with a somewhat restricted concept suitable for our purposes. In
particular, in our version a periodic word must always contain $\lambda$-many
periods, for some limit ordinal $\lambda$. Thus, for example, a word of the
form $AAA$ is not always periodic, but $A^{\omega}$ is.

###### Definition 7

A word $X$ is _periodic_ if $X=A^{\lambda}$ for some $\lambda\in\mathrm{Lim}$
and some $A\neq\Lambda$. In this case, $A$ is called a _period_ of $X$ and
$\lambda$ is called the _exponent_.

$X$ is _ultimately periodic_ (u.p.) if $X=BY$ for some $Y,B$ with $Y$
periodic. Periods of $X$ are, by definition, periods of $Y$ for any such
representation. $B$ is called an _offset_ of $X$.

We remark that these definitions agree with the standard concepts of periodic
and ultimately periodic $\omega$-words.

Given an ordinal $\Omega$, there is a natural one-to-one correspondence
(depending on $\Omega$) between transfinite binary words of length $\Omega$
and subsets of $\Omega$: We associate with a set $X\subseteq\Omega$ its
characteristic function $\Omega\to\\{0,1\\}$, and with a word
$A:\Omega\to\\{0,1\\}$ the set $\\{\alpha\in\Omega:A(\alpha)=1\\}$. Thereby,
all the notions introduced so far for transfinite words are also meaningful
for sets.

###### Definition 8

We call a set $X$ _periodic/u.p. in_ $\Omega$ if so is the $\Omega$-word
associated with the subset $X\cap\Omega$ of $\Omega$.

Thus, we extend the terminology to sets $X$ not necessarily contained in
$\Omega$. Clearly, this makes sense only for limit ordinals $\Omega$
(otherwise there are no periodic sets in $\Omega$).

###  3.2 Periods

Our next goal is to show that u.p. subsets of $\Omega$ form a Boolean algebra.
To this end, we need to look into the structure of periods of u.p. sets and to
establish a useful _synchronization lemma_.

In general, period of a word, if it exists, is not unique. For example, we
obviously have

###### Lemma 3.1

If $A$ is a period of $X$, then so is $A^{n}$, for all $n<\omega$.

Proof. $(A^{n})^{\lambda}=A^{n\lambda}=A^{\lambda}$, since
$\lambda\in\mathrm{Lim}$. $\Box$

An offset of an u.p. word is also not unique. In fact, if $B$ is an offset of
$X$, then so is the extension of $B$ by any number of periods as long as it is
a proper initial segment of $X$. We can always assume the exponent of an u.p.
word to be a power of $\omega$. Indeed, if $\lambda=\beta+\omega^{\alpha}$
with $\alpha>0$ then $A^{\lambda}=A^{\beta}A^{\omega^{\alpha}}$ and one can
extend the offset by $A^{\beta}$. We will use this principle and Lemma 3.1
without reference.

###### Lemma 3.2

If $X$ is u.p., then $X$ has a period of length $\omega^{\beta}\cdot n$, for
some $\beta,n$ with $0<n<\omega$.

Proof. Let $X=BA^{\alpha}$, $\alpha\in\mathrm{Lim}$. Consider two cases:

Case 1: $\alpha=\omega\alpha^{\prime}$ with $\alpha^{\prime}\in\mathrm{Lim}$.
Then $A^{\alpha}=(A^{\omega})^{\alpha^{\prime}}$ and
$|A^{\omega}|=|A|\cdot\omega=\omega^{\beta}$, for some $\beta$.

Case 2: $\alpha=\alpha^{\prime\prime}+\omega$. Then $A^{\alpha}=CA^{\omega}$,
for some $C$. Assume $|A|=\omega^{\alpha_{1}}\cdot n+\gamma$ in Cantor normal
form and $A=A_{1}A_{2}$ such that $|A_{1}|=\omega^{\alpha_{1}}\cdot n$ and
$|A_{2}|=\gamma$. Then
$A^{\omega}=(A_{1}A_{2})^{\omega}=A_{1}(A_{2}A_{1})^{\omega}$ where
$|A_{2}A_{1}|=\gamma+\omega^{\alpha_{1}}n=\omega^{\alpha_{1}}n$. $\Box$

We say that $\alpha$ is a (right) multiple of $\beta$ if
$\alpha=\beta\cdot\gamma$, for some $\gamma>0$.

###### Lemma 3.3

Multiples of $\omega^{\beta}$ are precisely the ordinals $\alpha$ such that
$\ell(\alpha)\geqslant\beta$.

Proof. Let $\alpha=\omega^{\alpha_{1}}+\cdots+\omega^{\alpha_{n}}$ in Cantor
normal form with $\alpha_{1}\geqslant\cdots\geqslant\alpha_{n}$. If
$\ell(\alpha)=\alpha_{n}\geqslant\beta$ then there are
$\alpha_{1}^{\prime}\geqslant\cdots\geqslant\alpha_{n}^{\prime}$ such that
$\alpha_{i}=\beta+\alpha_{i}^{\prime}$, for all $1\leqslant i\leqslant n$.
Hence, $\alpha=\omega^{\beta}\alpha^{\prime}$, where
$\alpha^{\prime}=\omega^{\alpha_{1}^{\prime}}+\cdots+\omega^{\alpha_{n}^{\prime}}$.
So, $\ell(\alpha)\geqslant\beta$ implies
$\alpha=\omega^{\beta+1}\alpha^{\prime}$, for some $\alpha^{\prime}>0$. The
opposite implication also holds, since
$\ell(\omega^{\beta}\alpha^{\prime})=\beta+\ell(\alpha^{\prime})$. $\Box$

###### Corollary 3.4

$\alpha_{1}+\alpha_{2}$ is a multiple of $\omega^{\beta}$ iff $\alpha_{2}$ is
a multiple of $\omega^{\beta}$.

######  Lemma 3.5 (synchronization)

Assume $X$, $Y$ are u.p. and $|X|=|Y|$. Then $X$ and $Y$ synchronize, that is,
they can be represented as $X=CA^{\alpha}$, $Y=DB^{\beta}$ with $|C|=|D|$,
$|A|=|B|$ and $\alpha=\beta$.

Proof. By Lemma 3.2 we can assume $|A|=\omega^{\alpha_{1}}n$,
$|B|=\omega^{\beta_{1}}m$. Consider three cases.

Case 1: $\alpha_{1}<\beta_{1}$, say $\alpha_{1}+\gamma=\beta_{1}$ with
$\gamma>0$. Then
$\omega^{\beta_{1}}=\omega^{\alpha_{1}}\omega^{\gamma}=\omega^{\alpha_{1}}n\omega^{\gamma}$.
Hence, for some $\delta\in\mathrm{Lim}$,
$\omega^{\beta_{1}}=|A|\cdot\delta=|A^{\delta}|$.

We can select $D$ at least as long as $C$. Then $|DB|=|C|+\mu$, for some
$\mu=\nu+\omega^{\beta_{1}}$ in Cantor normal form. Such a $\mu$ is a multiple
of $\omega^{\beta_{1}}$, hence $\mu=|A^{\gamma_{1}}|$, for some $\gamma_{1}$.
It follows that $|DB|=|CA^{\gamma_{1}}|$. This means that $X$ and $Y$
synchronize with offset length $|DB|$ and period length $|B|$.

Case 2: $\beta_{1}=\alpha_{1}$. There are two subcases.

Case 2.1: $\alpha>\omega$. We may assume $|C|\geqslant|D|$. We claim that $X$
and $Y$ synchronize with offset length $|CA^{\omega}|$ and period length
$\omega^{\alpha_{1}}\cdot\mathrm{lcm}(n,m)$. Since $\alpha>\omega$ the word
$CA^{\omega}$ is a proper initial segment of $X$.

Similarly to Case 1, we have $|CA^{\omega}|=|D|+\nu+\omega^{\alpha_{1}+1}$,
for some $\nu$. The ordinal $\mu=\nu+\omega^{\alpha_{1}+1}$ is a multiple of
$\omega^{\alpha_{1}+1}$, hence $\mu=|B^{\lambda}|$, for some
$\lambda\in\mathrm{Lim}$. It follows that $|CA^{\omega}|=|DB^{\lambda}|$.
Clearly, $\omega^{\alpha_{1}}\cdot\mathrm{lcm}(n,m)$ is a multiple of both
$|A|$ and $|B|$, hence a common period length.

Case 2.2: $\alpha=\omega$. We may select offsets $C$ and $D$ such that $|C|$
and $|D|$ are multiples of $\omega^{\alpha_{1}}$. Indeed,
$|CA|=\omega^{\alpha_{k}}+\cdots+\omega^{\alpha_{2}}+\omega^{\alpha_{1}}$ in
Cantor normal form, in particular $\alpha_{i}\geqslant\alpha_{1}$, for all
$i$. Then $|CA|=\omega^{\alpha_{1}}\cdot\alpha^{\prime}$, for some
$\alpha^{\prime}>0$. Since $|CA|<|X|=\omega^{\alpha_{1}+1}$ we have
$\alpha^{\prime}<\omega$.

Furthermore, we can assume $|C|\leqslant|D|<|CA|$. Let $A=A_{1}A_{2}$ with
$|D|=|CA_{1}|$, then $X=CA_{1}(A_{2}A_{1})^{\omega}=DB^{\omega}$. Since $|D|$
is a multiple of $\omega^{\alpha_{1}}$ so is $|A_{1}|$. Since $|A|$ is a
multiple of $\omega^{\alpha_{1}}$ so is $|A_{2}|$. Then we have
$|A_{2}A_{1}|=|A_{1}A_{2}|=|A|$ and we can select the common period length
$\omega^{\alpha_{1}}\cdot\mathrm{lcm}(n,m)$ again.

Case 3: $\beta_{1}<\alpha_{1}$. This is symmetric to Case 1. $\Box$

###### Corollary 3.6

The set $\\{X\subseteq\Omega:\text{$X$ is u.p.\ in $\Omega$}\\}$ is closed
under the boolean operations.

Proof. The closure under complement is easy: the inversion of an u.p.
$\Omega$-word is ultimately periodic. To show the closure under union first
represent the two sets $X,Y$ as synchronized $\Omega$-words. Then $X\cup Y$
corresponds to the bitwise maximum of the two words. Its period is the bitwise
maximum of the corresponding periods. $\Box$

###  3.3 Hereditarily periodic sets

$A$ is _hereditarily periodic_ (h.p.) in $\Omega$ if, for any limit ordinal
$\beta\leqslant\Omega$, $A$ is u.p. in $\beta$. Let $\mathcal{H}_{0}(\Omega)$
denote the set of all $X\subseteq\Omega$ such that $X$ is h.p. in $\Omega$. We
summarize basic closure properties of h.p. sets in the following lemma.

###### Lemma 3.7

  1. $\mathrm{(i)}$

$\mathcal{H}_{0}(\beta)$ is closed under boolean operations;

  2. $\mathrm{(ii)}$

Any interval contained in $\beta$ is h.p. in $\beta$;

  3. $\mathrm{(iii)}$

If $A\in\mathcal{H}_{0}(\beta)$ then
$(A\cap\alpha)\in\mathcal{H}_{0}(\alpha)$, for all $\alpha<\beta$;

  4. $\mathrm{(iv)}$

$A\in\mathcal{H}_{0}(\beta)$ iff $\alpha+A\in\mathcal{H}_{0}(\alpha+\beta)$;

  5. $\mathrm{(v)}$

If $A\in\mathcal{H}_{0}(\alpha)$ and $B\in\mathcal{H}_{0}(\beta)$ then
$AB\in\mathcal{H}_{0}(\alpha+\beta)$;

  6. $\mathrm{(vi)}$

If $A\in\mathcal{H}_{0}(\beta)$ then
$A^{\alpha}\in\mathcal{H}_{0}(\beta\alpha)$ for all $\alpha>0$.

Proof. Claim (i) follows from Corollary 3.6. Claims (ii) and (iii) are
obvious. Claim (iv) holds because the map $x\mapsto\alpha+x$ preserves
ultimate periodicity. Claim (v) follows from (iv), since the concatenation
$AB$ is the union of sets $A$ and $|A|+B$. Claim (vi) is proved by transfinite
induction on $\alpha$ using (v) for the successor step. $\Box$

Let $d_{0}$ denote the derivative operation w.r.t. the interval topology on
$\Omega$.

###### Proposition 3.8

$\mathcal{H}_{0}(\Omega)$ is closed under $d_{0}$.

Proof. It is sufficient to show that if $X$ is u.p. in an
$\Omega\in\mathrm{Lim}$ then so is $d_{0}(X)$. Let $X=BA^{\alpha}$ with
$\alpha\in\mathrm{Lim}$ and $|X|=\Omega$. We can assume $\alpha$ to be a power
of $\omega$. Let $\beta:=|A|$ and $\lambda:=\beta\alpha=|A^{\alpha}|$. By the
distributivity of $d_{0}$ over finite unions it is sufficient to show that
$d_{0}(A^{\alpha})$ is u.p. in $\lambda$. ($d_{0}(B)$ is bounded in $\Omega$,
hence u.p. in $\Omega$.) Let $E:=\\{\beta\gamma:\gamma<\alpha\\}$.

If $A=\varnothing$, then $d_{0}(A^{\alpha})=\varnothing$ and there is nothing
to prove.444Pay attention that $A=\varnothing$ corresponds to a word of length
$\beta$, not to the empty word. Assume $A\neq\varnothing$. If $\beta\in
d_{0}(A)$ then $E\setminus\\{0\\}\subseteq d_{0}(A)$. Letting
$C:=d_{0}(A)\cap\beta$ and $C_{0}:=C\cup\\{0\\}$ we obtain
$d_{0}(A^{\alpha})=CC_{0}^{\alpha}$ which is u.p. in $\lambda$.

If $\beta\notin d_{0}(A)$ then we let $E^{\prime}:=d_{0}(E)$ and observe that
$d_{0}(A^{\alpha})=C^{\alpha}\cup E^{\prime}$. If $\alpha=\omega$ then
$E^{\prime}$ is empty. Otherwise, $\alpha=\omega^{1+\alpha^{\prime}}$ with
$\alpha^{\prime}>0$. Then
$E^{\prime}=\\{\beta\omega\gamma:\omega\gamma<\alpha,\gamma\neq 0\\}$ is u.p.
with period $\beta\omega$ and exponent $\omega^{\alpha^{\prime}}$. $\Box$

###### Corollary 3.9

$(\mathcal{H}_{0}(\Omega),d_{0})$ is a Magari algebra, hence its logic
contains $\mathrm{GL}$.

In the following we will show that the logic of this algebra is exactly
$\mathrm{GL}$ provided $\Omega\geqslant\omega^{\omega}$. However, for the
completeness proof for $\mathrm{GLP}$ this algebra of sets is insufficient.
Therefore, in the next section we introduce a hierarchy of more general
classes called _$n$ -periodic sets_.

##  4 $n$-periodic sets

###### Definition 9

Define a sequence of subsets $\mathcal{V}_{n}(\Omega)$ of
$\mathcal{P}(\Omega)$ by

$\mathcal{V}_{0}(\Omega):=\mathcal{H}_{0}(\Omega),\quad\mathcal{V}_{n+1}(\Omega):=\\{B\cap\ell^{-1}(A):B\in\mathcal{H}_{0}(\Omega),A\in\mathcal{V}_{n}(\Omega)\\}.$

Let $\mathcal{H}_{n}(\Omega)$ denote the closure of $\mathcal{V}_{n}(\Omega)$
under finite unions. We call sets in $\mathcal{H}_{n}(\Omega)$ _$n$ -periodic
in $\Omega$_.

Intuitively, $(n+1)$-periodic sets are generated by $n$-periodic sets and all
sets whose ranks are $n$-periodic. The sets $\mathcal{V}_{n}(\Omega)$ give a
more convenient base for $n$-periodic sets.

###### Lemma 4.1

For each $\Omega$,

  1. $\mathrm{(i)}$

$\mathcal{V}_{n}(\Omega)$ is closed under $\cap$;

  2. $\mathrm{(ii)}$

$\mathcal{V}_{n}(\Omega)\subseteq\mathcal{V}_{n+1}(\Omega)$;

  3. $\mathrm{(iii)}$

$A\in\mathcal{H}_{n}(\Omega)$ implies
$\ell^{-1}(A)\in\mathcal{H}_{n+1}(\Omega)$;

  4. $\mathrm{(iv)}$

$\mathcal{H}_{n}(\Omega)$ is closed under Boolean operations.

For a class of sets $\mathcal{H}$, let $\mathcal{H}|_{\Omega}$ denote the
class $\\{A\cap\Omega:A\in\mathcal{H}\\}$.

###### Lemma 4.2

If $\Omega_{1}<\Omega_{2}$ then
$\mathcal{V}_{n}(\Omega_{1})=\mathcal{V}_{n}(\Omega_{2})|_{\Omega_{1}}$ and
$\mathcal{H}_{n}(\Omega_{1})=\mathcal{H}_{n}(\Omega_{2})|_{\Omega_{1}}.$

Proof. Induction on $n$. If $n=0$ then the claim is obvious by the property of
hereditarity. For $A,B\in\mathcal{V}_{n}(\Omega_{2})$ we have that

$B\cap\ell^{-1}(A)\cap\Omega_{1}=(B\cap\Omega_{1})\cap\ell^{-1}(A\cap\Omega_{1})\in\mathcal{V}_{n+1}(\Omega_{1}).$

Hence,
$\mathcal{V}_{n+1}(\Omega_{2})|_{\Omega_{1}}\subseteq\mathcal{V}_{n+1}(\Omega_{1})$.
$\Box$

In view of this lemma, we can consider $\mathcal{H}_{n}(\Omega)$ as the
restriction of a fixed class-size algebra $\mathcal{H}_{n}$ to $\Omega$. This
allows us to write $\mathcal{H}_{n}$ for $\mathcal{H}_{n}(\Omega)$ whenever
$\Omega$ is understood from context.

Recall that $d_{n}$ denotes the derivative operation w.r.t. Icard’s topology
$\vartheta_{n}$ on $\Omega$. Our next goal is to show that $\mathcal{H}_{n}$
is closed under $d_{0},\dots,d_{n}$, for $n>0$. As a base case we are going to
prove that $d_{n}(A)\in\mathcal{H}_{n}$, for any $A\in\mathcal{H}_{0}$. We
need a few auxiliary lemmas.

###### Lemma 4.3

For any set $A\subseteq\Omega$, $d_{0}(\ell^{-1}(A))=\ell^{-1}(\min A,\Omega)$
in $(\Omega,\vartheta_{0})$.

Proof. Recall that $\ell:(\Omega,\vartheta_{0})\to(\Omega,\tau_{\leftarrow})$
is the rank function of $\vartheta_{0}$, hence a $d$-map. Hence,
$d_{0}(\ell^{-1}(A))=\ell^{-1}(d_{\leftarrow}(A))=\ell^{-1}(\min A,\Omega)$.
$\Box$

Recall that $\ell^{-1}(\min A,\Omega)$ is the set $U^{1}_{\beta}(\Omega)$,
where $\beta=\min A$. Corollary 4.5 below shows that sets
$U^{1}_{\beta}(\Omega)$ are h.p. in $\Omega$.

If $A$ is a class of ordinals, we use notation $\alpha\cdot A$ for
$\\{\alpha\beta:\beta\in A\\}$.

###### Lemma 4.4

$\omega^{\beta+1}\cdot[1,\lambda)=U^{1}_{\beta}(\omega^{\beta+1}\lambda)$.

Proof. The inclusion $(\subseteq)$ clearly holds: If $\alpha\in[1,\lambda)$
then
$\ell(\omega^{\beta+1}\cdot\alpha)=\beta+1+\ell(\alpha)\geqslant\beta+1>\beta$.
For the inclusion $(\supseteq)$ write $\alpha\in U^{1}_{\beta}$ in Cantor
normal form $\alpha=\omega^{\alpha_{1}}+\cdots+\omega^{\alpha_{n}}$. We have
$\alpha_{1}\geqslant\cdots\geqslant\alpha_{n}$. Since
$\ell(\alpha)=\alpha_{n}>\beta$ there are
$\alpha_{1}^{\prime}\geqslant\cdots\geqslant\alpha_{n}^{\prime}$ such that
$\alpha_{i}=\beta+1+\alpha_{i}^{\prime}$, for all $1\leqslant i\leqslant n$.
Hence, $\alpha=\omega^{\beta+1}\alpha^{\prime}$, where
$\alpha^{\prime}=\omega^{\alpha_{1}^{\prime}}+\cdots+\omega^{\alpha_{n}^{\prime}}$.
So, $\ell(\alpha)>\beta$ implies $\alpha=\omega^{\beta+1}\alpha^{\prime}$, for
some $\alpha^{\prime}>0$. Moreover, $\alpha^{\prime}<\lambda$, otherwise
$\alpha=\omega^{\beta+1}\alpha^{\prime}\geqslant\omega^{\beta+1}\lambda$.
$\Box$

###### Corollary 4.5

For any $\beta$, $U^{1}_{\beta}$ is hereditarily periodic in any limit
ordinal.

Proof. Let $\lambda$ be a limit ordinal. The set
$U^{1}_{\beta}:=\omega^{\beta+1}\cdot[1,\lambda)$ can be represented as the
word $BA^{\lambda}$ where $B$ corresponds to the empty subset of
$\omega^{\beta+1}$ and $A\subseteq\omega^{\beta+1}$ is the singleton
$\\{0\\}$. Both $A$ and $B$ are h.p. in $\omega^{\beta+1}$. Hence,
$U^{1}_{\beta}$ is h.p. in $\omega^{\beta+1}\lambda$. It follows that it is
also h.p. in $\lambda$. $\Box$

Let $E$ denote the set
$\\{\beta\alpha:\alpha\in\mathrm{Lim}\cup\\{0\\}\\}\cap\Omega$. We note that
$E\in\mathcal{H}_{0}(\Omega)$.

###### Lemma 4.6

  1. $\mathrm{(i)}$

$d_{0}(E)=U^{1}_{\nu+1}(\Omega)$, for some $\nu$;

  2. $\mathrm{(ii)}$

$d_{n}(E)=U^{n+1}_{0}(\Omega)\cap E$, if $n>0$;

  3. $\mathrm{(iii)}$

For any $A\subseteq\Omega$, $d_{0}(E\cap\ell^{-1}(A))=U^{1}_{\mu}(\Omega)$,
for some $\mu>0$.

Notice that the mentioned sets can also be empty.

Proof. Since $\alpha\in\mathrm{Lim}\cup\\{0\\}$ we have (letting
$\alpha=\omega\alpha^{\prime}$)

$E=\\{\beta\omega\alpha^{\prime}:\alpha^{\prime}\in\mathrm{On}\\}\cap\Omega.$

The ordinal $\beta\omega$ has the form $\omega^{\nu+1}$, for some $\nu$.
Hence, $E=U^{1}_{\nu}(\Omega)$.

(i) $d_{0}(E)=d_{0}(U^{1}_{\nu}(\Omega))=U^{1}_{\nu+1}(\Omega)$.

(ii) Assume $n>0$, then $E=U^{1}_{\nu}(\Omega)$ is open in $\vartheta_{n}$.
Therefore, $d_{n}(E)=d_{n}(\Omega)\cap E$. Since $\ell^{n+1}$ is the rank
function of $\vartheta_{n}$ we obtain
$d_{n}(\Omega)=\ell^{-(n+1)}(0,\Omega)=U^{n+1}_{0}(\Omega),$ hence the result.

(iii) We have

$U^{1}_{\nu}(\Omega)\cap\ell^{-1}(A)=\ell^{-1}(\nu,\Omega)\cap\ell^{-1}(A)=\ell^{-1}(A\cap(\nu,\Omega)).$

Hence,
$d_{0}(E)=d_{0}(\ell^{-1}(A\cap(\nu,\Omega)))=\ell^{-1}(\min(A\cap(\nu,\Omega)),\Omega)=U^{1}_{\mu}(\Omega)$
with $\mu=\min(A\cap(\nu,\Omega))$. $\Box$

The following lemma strengthens Proposition 3.8. Its proof follows the same
pattern, however we will refer to Proposition 3.8 to simplify some arguments
in the proof.

###### Lemma 4.7

Suppose $X$ is u.p. in $\Omega$. Then so is $d_{0}(X\cap\ell^{-1}(A))$.

Proof. Let $B$ be the period of $X$, so that $X=B_{0}B^{\alpha}$, for some
$\alpha\in\mathrm{Lim}$ (which we will also assume to be a power of $\omega$).
Let $\alpha_{0}=|B_{0}|$ and $\beta:=|B|=\omega^{\nu}(n+1)$.

The subset $X$ is naturally split in two subsets of $\Omega$ corresponding to
the subwords $B_{0}$ and $B^{\alpha}$. Hence, by the additivity of $d_{0}$ it
is sufficient to show that $d_{0}(B_{0}\cap\ell^{-1}(A))$ u.p. in $\Omega$ and
$d_{0}(B^{\alpha}\cap\ell^{-1}(A))$ is u.p. in $\lambda:=\beta\alpha$. (The
bound $\Omega$ changes to $\beta\alpha$, since by considering $B^{\alpha}$ we
shift the set leftwards by $\alpha_{0}$. This operation does not change the
rank555That is, the value of the $\ell$-function. of any point except for the
first bit in $B^{\alpha}$. Changing a set at one point has no effect on the
derivative.)

The set $B_{0}\cap\ell^{-1}(A)$ is bounded in $\Omega$, hence so is
$d_{0}(B_{0}\cap\ell^{-1}(A))$, and therefore it is u.p. in $\Omega$.

To analyse $d_{0}(B^{\alpha}\cap\ell^{-1}(A))$ we consider the set
$E:=\\{\beta\gamma:\gamma<\alpha\\}.$ These are the coordinates of the first
bits of occurrences of period $B$. We split $B^{\alpha}$ into sets
$B^{\alpha}\cap E$ and $B^{\alpha}\setminus E$. By periodicity, if $0\in B$
then $E\cap B^{\alpha}=E$, otherwise $E\cap B^{\alpha}=\varnothing$.

Using the additivity of $d_{0}$ we now deal separately with the sets
$d_{0}((B^{\alpha}\setminus E)\cap\ell^{-1}(A))$ and $d_{0}((B^{\alpha}\cap
E)\cap\ell^{-1}(A))$ on $\lambda$.

For the former, we observe that the ranks of points in $B^{\alpha}\setminus E$
are the same as of the corresponding points in
$B\setminus\\{0\\}\subseteq\beta$. Hence, $d_{0}((B^{\alpha}\setminus
E)\cap\ell^{-1}(A))=d_{0}(C^{\alpha}),$ where
$C:=(B\setminus\\{0\\})\cap\ell^{-1}(A)$ with $|C|=\beta$. Therefore, this set
is the derivative of an u.p. set in $\lambda$, hence u.p. itself by
Proposition 3.8.

For the latter, we consider two cases. If $0\notin B$ then
$d_{0}((B^{\alpha}\cap E)\cap\ell^{-1}(A))$ is empty. If $0\in B$ then
$d_{0}((B^{\alpha}\cap E)\cap\ell^{-1}(A))=d_{0}(E\cap\ell^{-1}(A))$. Here we
consider two subcases according to the value of the exponent $\alpha$ (which
is a power of $\omega$). If $\alpha=\omega$ then
$d_{0}(E\cap\ell^{-1}(A))\cap\lambda\subseteq
d_{0}(E)\cap\lambda=\varnothing$. If $\alpha=\omega^{\nu}$ with $\nu>1$, then
by Lemma 4.6 (iii) we obtain $d_{0}(E\cap\ell^{-1}(A))=U^{1}_{\mu}(\lambda)$,
for some $\mu$. Hence, it is also u.p. in $\lambda$. $\Box$

###### Corollary 4.8

If $B\in\mathcal{H}_{0}(\Omega)$ then
$d_{0}(B\cap\ell^{-1}(A))\in\mathcal{H}_{0}(\Omega)$.

###### Corollary 4.9

If $V\in\mathcal{H}_{n}$ then $d_{0}(V)\in\mathcal{H}_{0}$.

Proof. It is sufficient to prove the lemma for $V\in\mathcal{V}_{n}$. If $n=0$
the statement is in Lemma 3.8. If $n>0$, then $V=B\cap\ell^{-1}(A)$, for some
$B\in\mathcal{H}_{0}$ and $A\in\mathcal{V}_{n-1}$. By Lemma 4.8
$d_{0}(V)\in\mathcal{H}_{0}$. $\Box$

###### Lemma 4.10

If $d$ is a derivative operator of a topology $\tau$ extending
$\vartheta_{1}$, then, for all $B\in\mathcal{H}_{0}$,

$d(B\cap\ell^{-1}(A))=d(B)\cap d(\ell^{-1}(A)).$

Proof. Let $\lambda\leqslant\Omega$, $\lambda\in d(B)\cap d(\ell^{-1}(A)).$
Then $B$ is u.p. in $\lambda$ that is $B=B_{0}B_{1}^{\alpha}$ with
$\alpha\in\mathrm{Lim}$. As before, without loss of generality we may assume
$B_{0}$ to be empty.

Since $\lambda\in d(B)\subseteq d_{1}(B)$ we have $\ell^{2}(\lambda)>0$ and we
can assume that $|B_{1}|=\omega^{\nu}$ with $\nu>0$.

As before, we let $E:=\\{\omega^{\nu}\gamma:\gamma<\alpha\\}$. If $0\in B_{1}$
then $E\subseteq B_{1}^{\alpha}$, otherwise $B_{1}^{\alpha}\cap
E=\varnothing$.

For any point $x<\omega^{\nu}$ we have $\ell(x)<\nu$, whereas
$\ell(\lambda)=\nu+\alpha>\nu$. By the periodicity and preservation of ranks
it follows that $\ell(x)<\nu$ for all $x\in\lambda\setminus E$.

The set $U^{1}_{\nu}=\ell^{-1}(\nu,\lambda)\cap\lambda$ is a $\tau$-open
punctured neighbourhood of $\lambda$. By the above, $U^{1}_{\nu}\subseteq E$.
To show that $\lambda\in d(B_{1}^{\alpha}\cap\ell^{-1}A)$ consider any
$\tau$-open punctured neighbourhood $U$ of $\lambda$. We may assume that
$U\subseteq U^{1}_{\nu}$, otherwise consider their intersection.

Since $\lambda\in d(\ell^{-1}A)$ there is a point $x\in U\cap\ell^{-1}A$. Then
$x\in E$ and $\ell(x)\in A$. On the other hand, $\lambda\in
d(B_{1}^{\alpha})$, hence there is a point $y\in U\subseteq E$ such that $y\in
B_{1}^{\alpha}$. Therefore, $0\in B_{1}$ and $E\subseteq B_{1}^{\alpha}$, so
$x\in B_{1}^{\alpha}\cap\ell^{-1}A$ and $x\in U$, as required. $\Box$

###### Lemma 4.11

If $X\in\mathcal{H}_{0}(\Omega)$ and $n>0$, then
$d_{n}(X)\in\mathcal{H}_{n}(\Omega)$.

Proof. The proof generalizes that of Lemma 3.8. We argue by transfinite
induction on $\Omega$. So, we assume the claim to be true for all limit
$\lambda<\Omega$.

Let $X=BA^{\alpha}$ with $\alpha\in\mathrm{Lim}$ and $|X|=\Omega$. Let
$\beta:=|A|$ and $\lambda:=\beta\alpha=|A^{\alpha}|$. By the distributivity of
$d_{n}$ over finite unions it is sufficient to show that
$d_{n}(B)\in\mathcal{H}_{n}(\Omega)$ and
$d_{n}(A^{\alpha})\in\mathcal{H}_{n}(\lambda)$. The former is equivalent to
$d_{n}(B)\in\mathcal{H}_{n}(\mu)$ for $\mu:=|B|$. Since $\mu<\Omega$ we have
the claim by the transfinite induction hypothesis. To deal with the latter we
consider two sets.

Let $I_{\gamma}$ denote the $\vartheta_{0}$-clopen interval
$[\beta\gamma+1,\beta(\gamma+1)]$. Then $\lambda$ is the disjoint union of
sets $I:=\bigcup_{\gamma<\alpha}I_{\gamma}$ and
$E:=\\{\beta\gamma:\gamma<\alpha,\gamma\in\mathrm{Lim}\cup\\{0\\}\\}$. We have
$d_{n}(A^{\alpha})\cap\lambda=(d_{n}(A^{\alpha})\cap
I)\cup(d_{n}(A^{\alpha})\cap E)$, and we consider the two parts $I$ and $E$
separately.

Case $I$. Since $I_{\gamma}$ is clopen, $d_{n}(A^{\alpha})\cap
I_{\gamma}=d_{n}(A^{\alpha}\cap I_{\gamma})$. By periodicity, the latter is
isomorphic to $C:=d_{n}(A)\cap I_{0}$ by the map $f_{\gamma}:I_{0}\to
I_{\gamma}$, $x\mapsto\beta\gamma+x$. So,

$d_{n}(A^{\alpha})\cap I=\bigcup_{\gamma<\alpha}f_{\gamma}(C).$

Let $C_{0}:=(C\cap\beta)\cup\\{0\\}$ if $\beta\in C$, and $C_{0}:=C\cap\beta$,
otherwise. Then $d_{n}(A^{\alpha})\cap I=C_{0}^{\alpha}\cap I$. As both
$E,I\in\mathcal{H}_{0}(\lambda)$ it is now sufficient to prove that
$C_{0}^{\alpha}\in\mathcal{H}_{n}(\lambda)$. If $n=0$ and
$C_{0}\in\mathcal{H}_{0}(\lambda)$, then $C_{0}^{\alpha}$ is u.p. in any
ordinal $\mu\leqslant\lambda$. If $n>0$ then $C_{0}$ has the form
$\bigcup_{i=1}^{k}(B_{i}\cap\ell^{-1}A_{i})$ with
$B_{i}\in\mathcal{H}_{0}(\beta)$ and $A_{i}\in\mathcal{V}_{n-1}(\beta)$.
Notice that $f_{\gamma}$ preserves the ranks of points, hence

$f_{\gamma}(B_{i}\cap\ell^{-1}(A_{i}))=f_{\gamma}(B_{i})\cap\ell^{-1}(A_{i}).$

Then

$\bigcup_{\gamma<\alpha}f_{\gamma}(C)=\bigcup_{\gamma<\alpha}\bigcup_{i=1}^{k}(f_{\gamma}(B_{i})\cap\ell^{-1}(A_{i}))=\bigcup_{i=1}^{k}(\bigcup_{\gamma<\alpha}f_{\gamma}(B_{i})\cap\ell^{-1}(A_{i})).$

Letting $D_{i}:=\bigcup_{\gamma<\alpha}f_{\gamma}(B_{i})$ thus we obtain

$d_{n}(A^{\alpha})\cap I=\bigcup_{i=1}^{k}(D_{i}\cap\ell^{-1}(A_{i})).$

The sets $D_{i}\in\mathcal{H}_{0}(\lambda)$ by the same argument as for the
case $n=0$ (with $D_{i}$ playing the role of $C$). Hence,
$D_{i}\cap\ell^{-1}(A_{i})\in\mathcal{V}_{n}(\lambda)$ and then
$d_{n}(A^{\alpha})\cap I\in\mathcal{H}_{n}(\lambda)$.

Case $E$. Since $n>0$ the set $E$ is clopen in $\vartheta_{n}$. Therefore,
$d_{n}(A^{\alpha})\cap E=d_{n}(A^{\alpha}\cap E)$. However, $A^{\alpha}\cap E$
is empty, if $0\notin A$, and $A^{\alpha}\cap E=E$, otherwise. In the first
case $d_{n}(A^{\alpha}\cap E)$ is empty. In the second case, Lemma 4.6 (ii) is
applicable and we obtain $d_{n}(A^{\alpha}\cap
E)=d_{n}(E)=U_{0}^{n+1}(\lambda)\cap E$. We have
$U_{0}^{n+1}(\lambda)=\ell^{-n}(\lambda\cap\mathrm{Lim})$. Since
$\lambda\cap\mathrm{Lim}\in\mathcal{H}_{0}(\lambda)$ we conclude that
$U_{0}^{n+1}$ and $d_{n}(E)$ are in $\mathcal{H}_{n}(\lambda)$, as required.
$\Box$

###### Lemma 4.12

If $V\in\mathcal{H}_{n}$, $n>0$, then $d_{i}(V)\in\mathcal{H}_{i}$ for all
$i\leqslant n$.

Proof. It is sufficient to prove it for $V\in\mathcal{V}_{n}$. We use
induction on $i\leqslant n$. For $i=0$ the statement follows from Lemma 4.9.
For $i>0$ we have $d_{i}(V)=d_{i}(B\cap\ell^{-1}(A))$ with
$B\in\mathcal{H}_{0}$ and $A\in\mathcal{V}_{n-1}$. Then by Lemma 4.10
$d_{i}(V)=d_{i}(B)\cap d_{i}(\ell^{-1}(A))=d_{i}(B)\cap\ell^{-1}(d_{i-1}A)$.
By Lemma 4.11 $d_{i}(B)\in\mathcal{H}_{i}$ and by the induction hypothesis
$d_{i-1}A\in\mathcal{H}_{i-1}$, hence $\ell^{-1}(d_{i-1}A)\in\mathcal{H}_{i}$.
$\Box$

##  5 Logic $J_{n}$ and $J_{n}$-morphisms

This section presents the technical notion of $J_{n}$-morphism from [15]. This
notion allows us to map general topological frames to Kripke models of logic
$J_{n}$ in such a way that the validity of formulas is preserved. It is
essentially the same as the notion of $J_{n}$-morphism from [15] with the
exception that we now also require backwards preservation of admissible sets.
This requirement is relatively minor, so our presentation is almost literally
taken from [15] and all the results in this section are from that paper.

Let $\mathcal{L}_{n}$ denote the modal language with modalities
$[0],[1],\dots,[n]$. Recall that we denote by $\mathrm{GLP}_{n}$ the
restriction of $\mathrm{GLP}$ to $\mathcal{L}_{n}$. While $\mathrm{GLP}$ and
$\mathrm{GLP}_{n}$ for $n>0$ are Kripke incomplete, the following subsystem
$\mathrm{J}$ serves as a best approximation to $\mathrm{GLP}$ which is. An
axiomatization of $\mathrm{J}$ is obtained by replacing Axiom (v) by two
theorems of $\mathrm{GLP}$:

  1. $\mathrm{(i)}$

$[m]\varphi\rightarrow[n][m]\varphi$, for $m\leqslant n$;

  2. $\mathrm{(ii)}$

$[m]\varphi\to[m][n]\varphi$, for $m\leqslant n$.

We denote by $\mathrm{J}_{n}$ the fragment of $\mathrm{J}$ in
${\mathcal{L}}_{n}$.

###### Definition 10

The class of _$J_{n}$ -trees_, which are finite Kripke frames of the form
$(W,R_{0},\dots,R_{n})$, is defined inductively as follows.

  1. 1.

$J_{0}$-trees are finite irreflexive transitive tree-like Kripke frames
$(W,R_{0})$;

  2. 2.

If ${\mathcal{W}}=(W,R_{0},\dots,R_{n})$ is a $J_{n}$-tree, then
${\mathcal{W}}^{+}:=(W,\varnothing,R_{0},\dots,R_{n})$ is a $J_{n+1}$-tree;

  3. 3.

If ${\mathcal{W}}_{1},\dots,{\mathcal{W}}_{k}$ are $J_{n}$-trees and
${\mathcal{V}}$ is a $J_{n-1}$-tree, then the structure
${\mathcal{W}}=(W,R_{0},\dots,R_{n})$ is a $J_{n}$-tree, where ${\mathcal{W}}$
is obtained from the disjoint sum of structures ${\mathcal{V}}^{+}$ and
${\mathcal{W}}_{1},\dots,{\mathcal{W}}_{k}$ by extending $R_{0}$ with all
pairs $(v,w)$ for $v\in V$ and $w\in\bigcup_{i=1}^{k}W_{i}$.

We call _$J$ -trees_ the structures that are $J_{n}$-trees for some
$n\in\omega$. We can view them as Kripke frames $(W,(R_{i})_{i\in\omega})$
where all relations $R_{i}$ for $i>n$ are empty.

It is easy to check that the relations of a $J$-tree are irreflexive partial
orders and satisfy the following properties:

  * •

$R_{k}R_{m}\subseteq R_{\min(k,m)}$ (polytransitivity);

  * •

$R_{m}^{-1}R_{k}\subseteq R_{k}$, for $k<m$.

These conditions ensure that $\mathrm{J}_{n}$ is sound w.r.t. $J_{n}$-trees.
The following proposition is from [12].

###### Proposition 5.1

  1. $\mathrm{(i)}$

$\mathrm{J}_{n}$ is complete w.r.t. the class of $J_{n}$-trees;

  2. $\mathrm{(ii)}$

$\mathrm{J}$ is complete w.r.t. the class of $J$-trees.

Let ${\mathcal{T}}=(T,R_{0},\dots,R_{n})$ be a $J_{n}$-tree and let
$R^{*}_{k}$ denote $\bigcup_{i=k}^{n}R_{i}$. By the polytransitivity,
$R^{*}_{k}$ is transitive. An $R^{*}_{k}$-minimal node $w\in T$ is called a
_hereditary $k$-root_. Since ${\mathcal{T}}$ is a $J_{n}$-tree, for each $w\in
T$ and each $k\leqslant n$ there exists a hereditary $k$-root $v\in T$ such
that $v=w$ or $vR_{k}w$. We call the _root_ of ${\mathcal{T}}$ its hereditary
$0$-root.

As any Kripke model, we can view ${\mathcal{T}}$ as a poly-topological space
${\mathcal{T}}=(T,\sigma_{0},\dots,\sigma_{n})$ by considering all
$R_{i}$-upsets to be $\sigma_{i}$-open.

###### Definition 11

Given a poly-topological space ${\mathcal{X}}=(X,\tau_{0},\dots,\tau_{n})$ and
a map $f:{\mathcal{X}}\to{\mathcal{T}}$ we say that $f$ is a _$J_{n}$
-morphism_ if:

  * ($j_{1}$)

$f:(X,\tau_{n})\to(T,\sigma_{n})$ is a $d$-map;

  * ($j_{2}$)

$f:(X,\tau_{k})\to(T,\sigma_{k})$ is an open map for all $k\leqslant n$;

  * ($j_{3}$)

For each $k<n$ and each hereditary $(k+1)$-root $w\in T$, the sets
$f^{-1}(R_{k}^{*}(w))$ and $f^{-1}(R_{k}^{*}(w)\cup\\{w\\})$ are open in
$\tau_{k}$;

  * ($j_{4}$)

For each $k<n$ and each hereditary $(k+1)$-root $w\in T$, the set $f^{-1}(w)$
is a $\tau_{k}$-discrete subspace of ${\mathcal{X}}$.

Here $R_{k}^{*}(w)$ denotes the set $\\{u\in T:wR_{k}^{*}u\\}$. Notice that
the sets $R_{k}^{*}(w)$ and $R_{k}^{*}(w)\cup\\{w\\}$ are $\sigma_{i}$-open
for all $i\geqslant k$, therefore condition ($j_{3}$) is a weakening of the
continuity of $f:(X,\tau_{k})\to(T,\sigma_{k})$.

If $f:{\mathcal{X}}\to{\mathcal{T}}$ is a $J_{n}$-morphism and
$({\mathcal{X}},{\mathcal{A}})$ is an ambience on ${\mathcal{X}}$, then $f$ is
called _algebra preserving_ if $f^{-1}(A)\in{\mathcal{A}}$ for all $A\subseteq
T$. (We consider ${\mathcal{T}}$ as an ambience where all subsets are
admissible.) Since $T$ is finite, this requirement is equivalent to
$f^{-1}(x)\in{\mathcal{A}}$, for all $x\in T$.

Let $\varphi$ be a formula in ${\mathcal{L}}_{n}$. Define:

$M(\varphi):=\bigwedge_{i<s}\bigwedge_{k=m_{i}+1}^{n}([m_{i}]\varphi_{i}\rightarrow[k]\varphi_{i}),$

where $[m_{i}]\varphi_{i}$, $i<s$, are all subformulas of $\varphi$ of the
form $[m]\psi$ and $n:=\max_{i<s}m_{i}$. Also, let
$M^{+}(\varphi):=M(\varphi)\land\bigwedge_{m\leqslant n}[m]M(\varphi)$. These
formulas provide a well-known translation from $\mathrm{GLP}_{n}$ to
$\mathrm{J}_{n}$. Clearly, $\mathrm{GLP}_{n}\vdash M^{+}(\varphi)$.

The following theorem allows one to reduce the completeness proof for
$\mathrm{GLP}$ to a covering construction. It is obtained by the same proof as
in [15] for the situation of ambiences. For the reader’s convenience, we
reproduce the proof of this theorem in the Appendix.

###### Theorem 1

Let ${\mathcal{X}}$ be an ambience, ${\mathcal{T}}$ a ${J}_{n}$-tree,
$f:{\mathcal{X}}\twoheadrightarrow{\mathcal{T}}$ an algebra preserving
surjective ${J}_{n}$-morphism. Then, for all $\mathcal{L}_{n}$-formulas
$\varphi$, ${\mathcal{X}}\vDash\varphi$ implies ${\mathcal{T}}\vDash
M^{+}(\varphi)\to\varphi$.

##  6 Covering lemma

Recall that $\varepsilon_{0}$ is the supremum of the countable ordinals
$\omega_{k}$ recursively defined by $\omega_{0}=1$ and
$\omega_{k+1}=\omega^{\omega_{k}}$. The crucial point of the completeness
proof is the following lemma.

######  Lemma 6.1 (covering)

For each ${J}_{n}$-tree ${\mathcal{T}}$ there exist an ordinal
$\lambda<\omega_{n+2}$ and an onto algebra preserving ${J}_{n}$-morphism
$f:{\mathcal{X}}\twoheadrightarrow{\mathcal{T}}$ where
${\mathcal{X}}=([1,\lambda],\vartheta_{0},\dots,\vartheta_{n},\mathcal{H}_{n})$.

Of course, the ambience ${\mathcal{X}}$ is isomorphic to
$(\Omega,\vartheta_{0},\dots,\vartheta_{n},\mathcal{H}_{n})$, where
$\Omega=\lambda+1$ if $\lambda$ is infinite, otherwise $\Omega=\lambda$, but
it will be more convenient in the inductive construction below to deal with
the intervals of the form $[1,\lambda]$. For the proof we need a few
preparatory lemmas. The first one covers the case $n=0$.

###### Lemma 6.2

For each finite tree ${\mathcal{T}}=(T,R_{0})$ with the root $r$ there is an
ordinal $\lambda<\omega^{\omega}$ and an onto algebra preserving $d$-map
$f:{\mathcal{X}}\twoheadrightarrow{\mathcal{T}}$ where
${\mathcal{X}}=([1,\lambda],\vartheta_{0},\mathcal{H}_{0})$ and
$f^{-1}(r)=\\{\lambda\\}$.

Proof. The construction of such a $d$-map is well-known, see e.g. [21]. The
essentially new claim is that the $d$-map is algebra preserving. We construct
$f$ by induction on the height of ${\mathcal{T}}$. If $ht({\mathcal{T}})=0$,
then we put $\lambda=1$. If $ht({\mathcal{T}})>0$ let
${\mathcal{T}}_{1},\dots,{\mathcal{T}}_{k}$ denote the immediate subtrees of
${\mathcal{T}}$, their roots are denoted $r_{1},\dots,r_{k}$. By the induction
hypothesis there are algebra preserving $d$-maps
$g_{i}:([1,\lambda_{i}],\vartheta_{0},\mathcal{H}_{0})\twoheadrightarrow{\mathcal{T}}_{i}$
such that $g_{i}^{-1}(r_{i})=\lambda_{i}$, $i=1,\dots,k$. We define
$\mu:=\lambda_{1}+\cdots+\lambda_{k}$ and a $d$-map
$g:[1,\mu]\twoheadrightarrow\mathcal{S}$ where $\mathcal{S}$ is the disjoint
sum of the trees ${\mathcal{T}}_{i}$. Obviously,
$([1,\mu],\vartheta_{0},\mathcal{H}_{0})$ is isomorphic to the sum of
ambiences $([1,\lambda_{i}],\vartheta_{0},\mathcal{H}_{0})$, so $g$ is defined
as the sum of maps $f_{i}$ and is an algebra preserving $d$-map. Then we put
$\lambda:=\mu\omega$ and define $f(\lambda):=r$ and $f(\alpha):=g(\beta)$ if
$\alpha=\mu n+\beta$, where $n<\omega$ and $1\leqslant\beta\leqslant\mu$.

Obviously, $f:[1,\lambda]\twoheadrightarrow T$. The interval $[1,\lambda)$ is
isomorphic to the sum of $\omega$-many copies of $[1,\mu]$, and the
restriction of $f$ to each copy is the same as $g$. Hence, the restriction of
$f$ to $[1,\lambda)$ is a $d$-map onto $\mathcal{S}$.

To verify that $f$ is an open map consider an open set
$U\subseteq[1,\lambda]$. If $\lambda\notin U$ then $f(U)$ is open by the
above. Otherwise, $U$ contains an interval $[\mu n,\lambda]$, for some $n$. In
this case $f(U)=T$ and is also open. Similarly, it is easy to see that $f$ is
continuous and pointwise discrete.

We show that $f$ is algebra preserving, that is,
$f^{-1}(w)\in\mathcal{H}_{0}$, for each $w\in T$. If $w=r$ the claim trivially
holds. If $w\in T$ and $w\neq r$, then $f^{-1}(w)$ can be represented as a
transfinite word $A^{\omega}$ where $A=g^{-1}(w)\subseteq[1,\mu]$. By the
induction hypothesis $A\in\mathcal{H}_{0}$. It follows that $f^{-1}(w)$ is
hereditarily periodic in each $\alpha<\lambda$. On the other hand,
$A^{\omega}$ is u.p. in $\lambda$, hence $A^{\omega}$ is hereditarily periodic
in $\lambda$. $\Box$

We remark that the $d$-map constructed in Lemma 6.2 is also algebra preserving
for each of the larger algebras $\mathcal{H}_{n}$, for $n>0$, and
$\mathcal{H}$.

###### Lemma 6.3

For all $\lambda$, $n$, the function $\ell$ is an algebra preserving onto
$d$-map

$\ell:([1,\omega^{\lambda}],\vartheta_{n+1},\mathcal{H}_{n+1})\to([0,\lambda],\vartheta_{n},\mathcal{H}_{n}).$

Proof. We already know that $\ell$ is an onto $d$-map. We show that it is
algebra preserving. Every set in $\mathcal{H}_{n}$ is a finite union of sets
in ${\mathcal{V}}_{n}$. Since $\ell^{-1}$ preserves finite unions, it is
sufficient to show that $\ell^{-1}(A)\in{\mathcal{V}}_{n+1}$ if
$A\in{\mathcal{V}}_{n}$. But $\ell^{-1}(A)\in{\mathcal{V}}_{n+1}$ by
definition. $\Box$

An immediate corollary of this lemma is that $\ell$ is also an algebra
preserving $d$-map

$([1,\omega^{\lambda}],\vartheta_{i+1},\mathcal{H}_{n})\to([0,\lambda],\vartheta_{i},\mathcal{H}_{n-1}),$

for each $i<n$.

Proof of covering lemma. For each $J_{n}$-tree
${\mathcal{T}}=(T,R_{0},\dots,R_{n})$ with a root $r$ we are going to build a
periodic frame
${\mathcal{X}}=([1,\lambda],\vartheta_{0},\dots,\vartheta_{n},\mathcal{H}_{n})$
and an algebra preserving onto $J_{n}$-morphism
$f:{\mathcal{X}}\twoheadrightarrow{\mathcal{T}}$. The construction goes by
induction on $n$ with a subordinate induction on the $R_{0}$-height of
${\mathcal{T}}$, which is denoted $ht_{0}({\mathcal{T}})$. For the induction
to work it will be convenient for us to ensure two additional properties:
$f^{-1}(r)=\\{\lambda\\}$ and, for each $w\in T$, $f^{-1}(w)$ is a
$\vartheta_{0}$-discrete subset of $X$. Such $J_{n}$-morphisms will be called
_suitable_.

The case $n=0$ is Lemma 6.2. Assume $n>0$ and the claim holds for all $m<n$.
If $ht_{0}({\mathcal{T}})=0$ then $R_{0}=\varnothing$. Consider the
$J_{n-1}$-tree $\mathcal{S}:=(T,R_{1},\dots,R_{n})$. By the induction
hypothesis, for some $\mu$, there is a suitable $J_{n-1}$-morphism

$g:([0,\mu],\vartheta_{0},\dots,\vartheta_{n-1},\mathcal{H}_{n-1})\twoheadrightarrow\mathcal{S}.$

(Here we also use the obvious isomorphism between the general frames $[0,\mu]$
and $[1,1+\mu]$.) By Lemma 6.3, $\ell$ is an algebra preserving onto $d$-map,
for each $i<n$,

$\ell:([1,\omega^{\mu}],\vartheta_{i+1},\mathcal{H}_{n})\twoheadrightarrow([0,\mu],\vartheta_{i},\mathcal{H}_{n-1}).$

For this map $\ell^{-1}(\mu)=\\{\omega^{\mu}\\}$. Hence, it is easy to verify
that

$g\circ\ell:([1,\omega^{\mu}],\vartheta_{0},\dots,\vartheta_{n},\mathcal{H}_{n})\twoheadrightarrow(T,\varnothing,R_{1},\dots,R_{n})$

is a suitable $J_{n}$-morphism.

Assume $ht_{0}({\mathcal{T}})>0$. Let
${\mathcal{T}}_{1},\dots,{\mathcal{T}}_{m}$ denote the immediate subtrees of
${\mathcal{T}}$ with the roots $r_{1},\dots,r_{m}$. By the induction
hypothesis there are suitable $J_{n}$-morphisms
$g_{i}:([1,\lambda_{i}],\vartheta_{0},\dots,\vartheta_{n},\mathcal{H}_{n})\twoheadrightarrow{\mathcal{T}}_{i}$
such that $g_{i}^{-1}(r_{i})=\lambda_{i}$, $i=1,\dots,m$. We define
$\nu:=\lambda_{1}+\cdots+\lambda_{m}$ and a map
$g:[1,\nu]\twoheadrightarrow\mathcal{S}$ where $\mathcal{S}$ is the disjoint
sum of the trees ${\mathcal{T}}_{i}$. Obviously,
$([1,\nu],\vartheta_{0},\dots,\vartheta_{n},\mathcal{H}_{n})$ is isomorphic to
the sum of ambiences
$([1,\lambda_{i}],\vartheta_{0},\dots,\vartheta_{n},\mathcal{H}_{n})$, so we
define $g$ as the sum of maps $f_{i}$. We have that $g$ is algebra preserving
and satisfies the conditions ($j_{1}$)–($j_{4}$).

Further, as in base case, consider the subset $Q:=R^{*}_{1}(r)\cup\\{r\\}$ and
the $J_{n-1}$-tree $(Q,R_{1},\dots,R_{n})$. Notice that the restriction of
$R_{0}$ to $Q$ is empty. By the induction hypothesis, for some $\mu$, there is
a suitable $J_{n-1}$-morphism

$h:([1,\mu],\vartheta_{0},\dots,\vartheta_{n-1},\mathcal{H}_{n-1})\twoheadrightarrow(Q,R_{1},\dots,R_{n}).$

By Lemma 6.3, $\ell$ is an algebra preserving onto $d$-map, for each $i<n$,

$\ell:([1,\omega^{\mu}],\vartheta_{i+1},\mathcal{H}_{n})\twoheadrightarrow([0,\mu],\vartheta_{i},\mathcal{H}_{n-1}).$

Since $\ell^{-1}(0)$ is the set of successor ordinals in $[1,\omega^{\mu}]$,
$\ell$ maps $Y:=[1,\omega^{\mu}]\cap\mathrm{Lim}$ onto $[1,\mu]$. For this map
$\ell^{-1}(\mu)=\\{\omega^{\mu}\\}$. Hence, it is easy to verify that

$h\circ\ell:(Y,\vartheta_{0},\dots,\vartheta_{n},\mathcal{H}_{n})\twoheadrightarrow(T,\varnothing,R_{1},\dots,R_{n})$

is a suitable $J_{n}$-morphism. Then we put $\lambda:=\nu\cdot\omega^{\mu}$
and define for $1\leqslant\alpha\leqslant\lambda$

$f(\alpha):=\begin{cases}h(\ell(\gamma)),&\text{if $\alpha=\nu\gamma$,
$\gamma\in\mathrm{Lim}$, $\gamma\leqslant\omega^{\mu}$,}\\\ g(\beta),&\text{if
$\alpha=\nu\gamma+\beta$, $\gamma<\omega^{\mu}$,
$1\leqslant\beta\leqslant\nu$.}\end{cases}$

We claim that $f$ is a suitable $J_{n}$-morphism
$([1,\lambda],\vartheta_{0},\dots,\vartheta_{n},\mathcal{H}_{n})\twoheadrightarrow{\mathcal{T}}$.

Notice that $[1,\lambda]$ is the disjoint sum of two sets:
$L:=\\{\nu\alpha:\alpha\in Y\\}=\nu\cdot Y$ and
$M:=\\{\nu\alpha+\beta:\beta\in[1,\nu],\alpha<\omega^{\mu}\\}$. Clearly,
$L=f^{-1}(Q)$ and $M=f^{-1}(S)$. We also have
$f^{-1}(r)=\nu\cdot\ell^{-1}(h^{-1}(r))=\\{\nu\cdot\omega^{\mu}\\},$ as
expected.

The subspace $(L,\vartheta_{i})$ is homeomorphic to $(Y,\vartheta_{i})$ by the
map $\alpha\mapsto\nu\alpha$. The subspace $(M,\vartheta_{i})$ is the disjoint
sum of clopens $I_{\alpha}:=[\nu\alpha+1,\nu\alpha+\nu]$, each of which is
homeomorphic to $([1,\nu],\vartheta_{i})$. In particular, $M$ is
$\vartheta_{0}$-open in $[1,\lambda]$.

By Lemma 4.4 $L$ is $\vartheta_{1}$-open, hence $\vartheta_{i}$-open in
$[1,\lambda]$, for all $i>0$. (We have
$[1,\omega^{\mu}]\cap\mathrm{Lim}=\\{\nu\omega\alpha:0<\alpha\leqslant\omega^{\mu}\\}$.
However, $\nu\omega=\omega^{\beta+1}$, for some $\beta$, so Lemma 4.4
applies.)

Claim 1: $f$ is open for each $\vartheta_{i}$, $i\leqslant n$.

Consider a $\vartheta_{i}$-open set $A\subseteq[1,\lambda]$. Then
$f(A)=f(A\cap M)\cup f(A\cap L)$. The set $A\cap I_{\alpha}$ is
$\vartheta_{i}$-open, therefore so is
$B:=\\{\beta\in[1,\nu]:\nu\alpha+\beta\in A\\}$. Since $g:[1,\nu]\to S$ is a
$J_{n}$-morphism, $f(A\cap I_{\alpha})=g(B)$ is $\sigma_{i}$-open. This holds
for all $\alpha<\omega^{\mu}$, hence $f(A\cap
M)=\bigcup_{\alpha<\omega^{\mu}}A\cap I_{\alpha}$ is $\sigma_{i}$-open.

If $A$ is $\vartheta_{i}$-open, then $A\cap L$ is $\vartheta_{i}$-open in the
relative topology of $L$, hence $B:=\\{\gamma:\nu\gamma\in A\\}$ is open in
$([1,\omega^{\mu}]\cap\mathrm{Lim},\vartheta_{i})$. Then $\ell(B)$ is
$\vartheta_{i-1}$-open and $f(A\cap L)=h(\ell(B))$ is $\sigma_{i}$-open in
$(Q,\sigma_{1},\dots,\sigma_{n})$, as $h$ is a $J_{n}$-morphism from
$([0,\mu],\vartheta_{0},\dots,\vartheta_{n-1})$ to $(Q,R_{1},\dots,R_{n})$. We
have shown that $f(A\cap M)$ and $f(A\cap L)$ are both $\sigma_{i}$-open,
hence so is $f(A)$.

Claim 2: The map $f:([1,\lambda],\vartheta_{n})\to(T,\sigma_{n})$ is
continuous.

If $A\subseteq T$ is $\sigma_{n}$-open, then so are $A\cap Q$ and $A\cap S$
($S$ is $\sigma_{i}$-open for all $i\leqslant n$, and $Q$ is $\sigma_{i}$-open
for $i>0$.) Since $h\circ\ell:Y\twoheadrightarrow Q$ is a $J_{n}$-morphism,
the set $B:=\ell^{-1}(h^{-1}(A\cap Q))$ is $\vartheta_{n}$-open in $Y$. Then
$\nu\cdot B$ is $\vartheta_{n}$-open in $L$. Since $L$ is a
$\vartheta_{1}$-open subset of $[1,\lambda]$ and $n\geqslant 1$, we conclude
that $f^{-1}(A\cap Q)=\nu\cdot B$ is also $\vartheta_{n}$-open.

Similarly, the set $C:=g^{-1}(A\cap S)\subseteq[1,\nu]$ is
$\vartheta_{i}$-open, since $g$ is continuous. Hence, so is the set
$\nu\alpha+C\subseteq I_{\alpha}$, for each $\alpha$, and therefore
$f^{-1}(A\cap S)=\bigcup_{\alpha<\omega^{\mu}}(\nu\alpha+C)$ is
$\vartheta_{n}$-open. Thus, we have proved that $f^{-1}(A)=f^{-1}(A\cap S)\cup
f^{-1}(A\cap Q)$ is $\vartheta_{n}$-open.

Claim 3: For $k<n$, if $w$ is a $(k+1)$-root of ${\mathcal{T}}$, then the sets
$f^{-1}(R_{k}^{*}(w))$ and $f^{-1}(R_{k}^{*}(w)\cup\\{w\\})$ are
$\vartheta_{k}$-open.

If $w\in S$, then $R_{k}^{*}(w)\subseteq S$, and since
$g:[1,\nu]\twoheadrightarrow S$ is a $J_{n}$-morphism, $g^{-1}(R_{k}^{*}(w))$
and $g^{-1}(R_{k}^{*}(w)\cup\\{w\\})$ are $\vartheta_{k}$-open in $[1,\nu]$.
It follows that $f^{-1}(R_{k}^{*}(w))$ and $f^{-1}(R_{k}^{*}(w)\cup\\{w\\})$
are $\vartheta_{k}$-open in $M$ and in $[1,\lambda]$, as $M$ is
$\vartheta_{0}$-open in $[1,\lambda]$.

If $w\in Q$ we consider two subcases. If $k=0$ then $w=r$ (the only 1-root of
$T$ in $Q$). Then $R_{0}^{*}(w)=T\setminus\\{r\\}$ and hence
$f^{-1}(R_{0}^{*}(w))=[1,\lambda)$, which is $\vartheta_{0}$-open. The set
$f^{-1}(R_{0}^{*}(w)\cup\\{w\\})=f^{-1}(T)=[1,\lambda]$ is also
$\vartheta_{0}$-open.

If $k>0$, then $R^{*}_{k}(w)\subseteq Q$. Since
$h\circ\ell:[1,\omega^{\mu}]\twoheadrightarrow Q$ is a $J_{n}$-morphism, we
obtain that $A:=(h\circ\ell)^{-1}(R^{*}_{k}(w))$ and
$B:=(h\circ\ell)^{-1}(R^{*}_{k}(w)\cup\\{w\\})$ are $\vartheta_{k}$-open in
$Y$. Since $\alpha\mapsto\nu\cdot\alpha$ is a $\vartheta_{k}$-open map, for
$k>0$, we conclude that $f^{-1}(R^{*}_{k}(w))=\nu\cdot A$ and
$f^{-1}(R^{*}_{k}(w)\cup\\{w\\})=\nu\cdot B$ are $\vartheta_{k}$-open.

Claim 4: For each $w\in T$ the set $f^{-1}(w)$ is $\vartheta_{0}$-discrete.

Let $w\in T$. If $w\in S$ then $f^{-1}(w)\subseteq M$ is
$\vartheta_{0}$-discrete in the relative topology of $M$. This yields that
$f^{-1}(w)$ is $\vartheta_{0}$-discrete in $[1,\lambda]$. If $w\in Q$ then
$h^{-1}(w)$ is $\vartheta_{0}$-discrete in $[1,\mu]$. By Lemma 2.4 (i) $\ell$
is $\vartheta_{0}$-pointwise discrete, therefore $B=\ell^{-1}(h^{-1}(w))$ is a
$\vartheta_{0}$-discrete subset of $Y$. Then $f^{-1}(w)=\nu\cdot B$ is
$\vartheta_{0}$-discrete in $L\subseteq[1,\lambda]$, hence in $[1,\lambda]$.

Finally, we need to show

Claim 5: $f$ is algebra preserving.

We show that $A:=f^{-1}(w)\in\mathcal{H}_{n}([1,\lambda])$, for all $w\in T$.
Again, we consider two cases.

Case 1: $w\in S$. Then $A\subseteq M$ and
$B:=g^{-1}(w)\in\mathcal{H}_{n}([1,\nu])$ by the induction hypothesis and
preservation under the sum. Hence, $A\cap
I_{\alpha}=\nu\alpha+B\in\mathcal{H}_{n}(I_{\alpha})$, for each
$\alpha<\omega^{\mu}$. We need to show that $A$ is u.p. in $\beta$ for each
$\beta\leqslant\lambda$.

If $\beta\in M$ then $\beta\in I_{\alpha}$, for some $\alpha<\omega^{\mu}$,
and $\beta=\nu\alpha+\beta^{\prime}$, for some $\beta^{\prime}\in[1,\nu]$. $B$
is u.p. in $\beta^{\prime}$, hence $A\cap I_{\alpha}=\nu\alpha+B$ is u.p. in
$\beta$. Therefore $A$ is u.p. in $\beta$.

If $\beta\notin M$, then $\beta\in L$, that is, $\beta=\nu\alpha$, for some
$\alpha\in\mathrm{Lim}$. For some $\delta>0$ there holds
$\alpha=\omega\delta$. If $\delta$ is a successor than
$\beta=\nu(\gamma+\omega)=\nu\gamma+\nu\omega$, for some $\gamma$. Then
$A\cap\beta$ can be represented as a transfinite word $CB^{\omega}$, for some
$C$. Hence, $A$ is u.p. in $\beta$.

Suppose $\delta$ is a limit ordinal. Then $A\cap\nu\omega$ can be represented
as the word $0B^{\omega}$. Since $A\cap L=\varnothing$ we obtain that
$A\cap\beta$ is $(0B^{\omega})^{\delta}$, which periodic.

Case 2: $w\in Q$. By the induction hypothesis
$h^{-1}(w)\in\mathcal{H}_{n-1}([1,\mu])$. Hence, by definition,
$B:=\ell^{-1}(h^{-1}(w))\in\mathcal{H}_{n}([1,\omega^{\mu}])$. It follows that
$\nu\cdot B\in\mathcal{H}_{n}([1,\lambda])$. $\Box$

##  7 Completeness

Now we are ready to prove our main theorem.

###### Theorem 2

  1. $\mathrm{(i)}$

For each $n<\omega$ and $\Omega\geqslant\omega_{n+1}$,

$\mathrm{Log}(\Omega;\vartheta_{0},\dots,\vartheta_{n};\mathcal{H}_{n}(\Omega))=\mathrm{GLP}_{n};$

  2. $\mathrm{(ii)}$

For $\Omega\geqslant\varepsilon_{0}$,
$\mathrm{Log}(\Omega;(\vartheta_{n})_{n<\omega};\mathcal{H}_{n}(\Omega))=\mathrm{GLP}$.

Proof. Suppose $\mathrm{GLP}_{n}\nvdash\varphi$. Obviously,
$\mathrm{J}_{n}\nvdash M^{+}(\varphi)\to\varphi$. Then there exists a finite
$J_{n}$-tree ${\mathcal{T}}$ such that ${\mathcal{T}}\nvDash
M^{+}(\varphi)\to\varphi$. By Lemma 6.1 there is an ordinal
$\lambda<\omega_{n+1}$ and a $J_{n}$-morphism
$f:{\mathcal{X}}\twoheadrightarrow{\mathcal{T}}$, where
${\mathcal{X}}=([1,\lambda],\vartheta_{0},\dots,\vartheta_{n};\mathcal{H}_{n}([1,\lambda]))$.
By Theorem 1 we have ${\mathcal{X}}\nvDash\varphi$. Since ${\mathcal{X}}$ is
isomorphic to a subframe of
$(\Omega;\vartheta_{0},\dots,\vartheta_{n};\mathcal{H}_{n}(\Omega))$ we have
the required. $\Box$

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##  8 Appendix: Proof of Theorem 1

Let $\tilde{d}(A)$ abbreviate $X\setminus d(X\setminus A)$. Obviously,
$x\in\tilde{d}(A)$ iff $A$ contains some punctured neighborhood of $x$.

###### Lemma 8.1

Assume $f:{\mathcal{X}}\to{\mathcal{T}}$ is a $J_{n}$-morphism. Then, for any
hereditary $(k+1)$-root $w\in T$,

$f^{-1}(R_{k}^{*}(w)\cup\\{w\\})\subseteq\tilde{d}_{k}(f^{-1}(R_{k}^{*}(w))).$
$None$

Proof. Assume $a\in f^{-1}(R_{k}^{*}(w)\cup\\{w\\})$. We have to construct a
punctured neighborhood of $a$ contained in $f^{-1}(R_{k}^{*}(w))$. Consider

$U:=f^{-1}(R_{k}^{*}(w)\cup\\{w\\})=f^{-1}(R_{k}^{*}(w))\cup f^{-1}(\\{w\\}).$

By the first part of $(j_{3})$, $U$ is a neighborhood of $a$. If $a\in
f^{-1}(R_{k}^{*}(w))$ then $V:=f^{-1}(R_{k}^{*}(w))$ is a neighborhood of $a$
by the second part of $(j_{3})$, so $V-\\{a\\}$ is as required. If $a\in
f^{-1}(w)$ then by $(j_{4})$ there is a neighborhood $V_{a}$ such that
$V_{a}\cap f^{-1}(w)=\\{a\\}$. Then,

$V_{a}\cap U=(V_{a}\cap f^{-1}(R_{k}^{*}(w)))\cup\\{a\\}$

is a neighborhood of $a$. Then, $(V_{a}\cap U)\setminus\\{a\\}$ is a punctured
neighborhood of $a$ contained in $f^{-1}(R_{k}^{*}(w))$. $\Box$

#### Proof of Theorem 1.

Let ${\mathcal{X}}$ be an ambience, ${\mathcal{T}}$ a ${J}_{n}$-tree,
$f:{\mathcal{X}}\twoheadrightarrow{\mathcal{T}}$ an algebra preserving
surjective ${J}_{n}$-morphism. We show that, for all
$\mathcal{L}_{n}$-formulas $\varphi$, ${\mathcal{X}}\vDash\varphi$ implies
${\mathcal{T}}\vDash M^{+}(\varphi)\to\varphi$.

Suppose ${\mathcal{T}}\nvDash M^{+}(\varphi)\to\varphi$. Then for some
valuation $\nu$ on ${\mathcal{T}}$ and some point $w\in T$ (assume without
loss of generality that $w$ is the root of ${\mathcal{T}}$) we have that
$w\in\nu(M^{+}(\varphi))$ but $w\not\in\nu(\varphi)$. Consider a valuation
$\nu^{\prime}$ on ${\mathcal{X}}$ by taking $\nu^{\prime}(p):=f^{-1}(\nu(p))$.

###### Lemma 8.2

For all subformulas $\theta$ of $\varphi$, we have
$\nu^{\prime}(\theta)=f^{-1}(\nu(\theta))$.

Proof. We argue by induction on the complexity of $\theta$. If $\theta$ is a
propositional letter, the claim follows from the definition of $\nu^{\prime}$.
The case of propositional connectives is easy.

If $\theta=[n]\psi$, then the claim follows by condition $(j_{1})$ of $f$
being a $J_{n}$-morphism.

Suppose $\theta=[k]\psi$ for some $k<n$. To show that
$\nu^{\prime}(\theta)\subseteq f^{-1}(\nu(\theta))$ assume
$x\in\nu^{\prime}(\theta)$. Then there exists a $U\subseteq X$ such that
$\\{x\\}\cup U\in\tau_{k}$ and $U\subseteq\nu^{\prime}(\psi)$. By IH we obtain
$U\subseteq f^{-1}(\nu(\psi))$. Hence $f(U)\subseteq
f(f^{-1}(\nu(\psi)))=\nu(\psi)$. By $(j_{2})$, the set $f(\\{x\\}\cup
U)=\\{f(x)\\}\cup f(U)$ is an $R_{k}$-upset and so $R_{k}(f(x))\subseteq
f(U)\subseteq\nu(\psi)$. It follows that $f(x)\in\nu([k]\psi)$. In other
words, $x\in f^{-1}(\nu(\theta))$.

For the converse inclusion suppose $x\in f^{-1}(\nu(\theta))$, that is,
$f(x)\vDash[k]\psi$. We must show $x\in\nu^{\prime}(\theta)$. By the induction
hypothesis,

$\nu^{\prime}(\theta)=\tilde{d}_{k}(\nu^{\prime}(\psi))=\tilde{d}_{k}(f^{-1}(\nu(\psi))).$

Let $v\in T$ be a hereditary $(k+1)$-root such that $v=f(x)$ or
$vR_{k+1}f(x)$. Since ${\mathcal{T}}$ is a $J_{n}$-tree,
$R_{k}(v)=R_{k}(f(x))$. Thus $v\vDash[k]\psi$. We also have $v\vDash
M^{+}(\varphi)$. In particular, $v\vDash[k]\psi\to[k^{\prime}]\psi$ for any
$k^{\prime}$ with $k\leqslant k^{\prime}\leqslant n$ and hence
$v\vDash[k^{\prime}]\psi$. It follows that for each $k^{\prime}$ between $k$
and $n$ we have $R_{k^{\prime}}(v)\subseteq\nu(\psi)$. Therefore
$R^{*}_{k}(v)\subseteq\nu(\psi)$ and hence $f^{-1}(R^{*}_{k}(v))\subseteq
f^{-1}(\nu(\psi))$. By the construction of $v$, $x\in
f^{-1}(R^{*}_{k}(v)\cup\\{v\\})$. Hence, by Lemma 8.1,

$x\in\tilde{d}_{k}(f^{-1}(R^{*}_{k}(v)))\subseteq\tilde{d}_{k}(f^{-1}(\nu(\psi))),$

as required. $\Box$

From this lemma we obtain $y\notin\nu^{\prime}(\varphi)=f^{-1}(\nu(\varphi))$,
for any $y$ with $f(y)=w$. Consequently, ${\mathcal{X}}\nvDash\varphi$. $\Box$