# A Proof of The Tree Alternative Conjecture Under the Topological Minor
Relation

Jorge Bruno and Paul J. Szeptycki

###### Abstract.

In 2006 Bonato and Tardif posed the Tree Alternative Conjecture (TAC): the
equivalence class of a tree under the embeddability relation is, up to
isomorphism, either trivial or infinite. In 2022 Abdi, et al. provided a
rigorous exposition of a counter-example to TAC developed by Tetano in his
2008 PhD thesis. In this paper we provide a positive answer to TAC for a
weaker type of graph relation: the topological minor relation. More precisely,
letting $[T]$ denote the equivalence class of $T$ under the topological minor
relation we show that

  1. (1)

$|[T]|=1$ or $|[T]|\geq\aleph_{0}$ and

  2. (2)

$\forall r\in V(T)$, $|[(T,r)]|=1$ or $|[(T,r)]|\geq\aleph_{0}$.

In particular, by means of curtailing trees, we show that for any tree $T$
with at least one ray with infinitely many vertices with degree at least $3$:
$|[T]|\geq 2^{\aleph_{0}}$.

##  1\. Introduction

In [4] Bonato and Tardif proved that $|[T]|=1$ or $\infty$ for any rayless
tree and conjectured the same must be true of any tree: the Tree Alternative
Conjecture (TAC) stated that the number of isomorphism classes of trees
mutually embeddable with a given tree $T$ is either 1 or infinite. Since
mutually embeddable finite trees are necessarily isomorphic, TAC is becomes
true for finite trees, and it has been confirmed for a number of nontrivial
classes of infinite trees - [4], [11] and [15]. In particular, in [15] it is
shown that TAC holds for all rooted trees, where a rooted tree $(T,r)$ is one
with a distinguished vertex $r\in V(T)$. In 2022 a counterexample to TAC was
found by Abdi et al. [1] where for each $n\in\mathbb{N}$ an unrooted and
locally finite tree $T_{n}$ is constructed with $|[T_{n}]|=n$.

Herewith we present a proof of TAC with respect to the topological minor
relation $\leq^{\sharp}$, where for trees $T,S$ we have $T\leq^{\sharp}S$ if
some subdivision of the tree $T$ is isomorphic to a subgraph of $S$. The
rooted topological minor notion is defined analogously to the unrooted case.
In this paper we prove the following theorem, where $[T]$ represents the
equivalence class of $T$ under the topological minor relation.

Theorem 1. For any tree $T$,

  1. (1)

$|[T]|=1$ or $|T|\geq\aleph_{0}$ and

  2. (2)

$\forall r\in V(T)$, $|[(T,r)]|=1$ or $|[(T,r)]|\geq\aleph_{0}$.

The techniques developed and employed to prove Theorem 1 heavily rely on the
fact the trees are, as discussed in the Background, well-quasi-ordered (wqo)
under the topological minor relation. Although Tyomkyn in [15] establishes TAC
for all rooted trees under the embeddability relation, there appears to be a
strong connection between wqo and TAC for unrooted trees and other similar
structures.

The layout of the paper is quite simple: in Section 3 we prove Theorem 1 for
large trees by employing the method of curtailing \- developed by the present
authors; and Section 4 deals with all small trees by adapting the techniques
developed by Bonato and Tardif in [4], Halin in [6], and Polat and Sabidussi
in [13] to the topological minor relation.

##  2\. Background

###  2.1. Graph-theoretic background

Most notation is standard and can be found in [5] but we present a brief
summary of key concepts and non-standard terminology employed in this paper. A
rooted tree $(T,r)$ is composed of a tree $T$ and a distinguished vertex $r\in
V(T)$. The isomorphism relation on rooted and unrooted trees is denoted by
$\simeq$ where, in the rooted case, an isomorphism must send root to root. The
splitting number of a vertex $v$, $\operatorname{sp}(v)$, in a rooted tree
$(T,r)$ is simply $\deg(v)-1$ if $v\not=r$ and $\deg(v)$ otherwise. Any rooted
tree $(T,r)$ generates a partial ordering $\leq_{T}$ on its set of vertices by
establishing that $s\leq_{T}t$ provided that the unique path from $r$ to $t$
contains $s$; if $s<_{T}t$ then we say that $s$ is below $t$ and that $t$ is
above $s$. This is called the tree order on $(T,r)$ and defines a meet
semilattice order on all vertices of $(T,r)$; $(T,r)$ is closed under all non-
empty meets. We denote the meet of a collection $C\subseteq V(T)$ by
$\wedge_{T}C$ and $v_{1}\wedge_{T}\ldots\wedge_{T}v_{n}$ if $C$ is finite. A
tree $(T,r)$ is a rooted subtree of a tree $(S,s)$ if $T$ is a subtree of $S$
and $v\leq_{S}w\Longleftrightarrow v\leq_{T}w$ for all $v,w\in V(T)$. Given a
vertex $v\in V(T)$ the full subtree of $(T,r)$ rooted at $v$, $(T_{v},v)$, is
the rooted subtree of $(T,r)$ induced by all vertices $w$ with $v\leq_{T}w$. A
subdivision of a tree $T$ (resp. of a rooted tree $(T,r)$) is any tree (resp.
rooted tree) obtained by subdividing any number of its edges. A path,
$p:v_{1},\ldots,v_{n}$, is a sequence of distinct pairwise adjacent vertices
(i.e., $v_{i}$ adjacent to $v_{i+1}$) and we demand $v_{i}<_{T}v_{i+1}$ for
all $i\leq n-1$ when $p$ is in a rooted tree $(T,r)$. The length of a path
$p$, $|p|$, is simply the number of vertices in it. A path
$p:v_{1},\ldots,v_{n}$ is bare if $\deg(v_{i})\leq 2$ for all $i\leq n$, and
maximal bare if $p$ is bare and not contained within any other bare path. For
vertices $v\leq_{T}w$ in a rooted tree $(T,r)$, $d_{T}(v,w)=|p|-1$ where
$p:v,\ldots,w$ is the unique path joining said vertices. For the case where
$v=r$ we denote $d_{T}(v,w)=l_{T}(w)$ and refer to it as the level of $w$. For
a vertex $v$ we use $\operatorname{pred}(v)$ to denote the unique vertex
immediately preceding $v$ and $\operatorname{succ}(v)$ to denote the set of
immediate successors of $v$. If $\operatorname{sp}(v)=1$ then we use
$\operatorname{succ}(v)$ to denote the unique vertex succeeding $v$ instead of
the set containing it. Given a ray $R=v_{1}v_{2}\ldots$ we use the notation
$R_{n}=v_{n}$ and $R_{n}^{\uparrow}=v_{n}v_{n+1}\ldots$. A ray $R$ is
eventually bare if for some $N\in\mathbb{N}$, $\deg(R_{n})=2$ for all $n\geq
N$, and bare if $N=1$. A rooted tree is said to be large if it contains at
least one ray that is not eventually bare. A rooted tree is then small if it’s
not large. We also define an unrooted tree $T$ to be large (resp. small) if
$(T,r)$ is large (resp. small) for any $r\in V(T)$.

###  2.2. Topological minor relation and wqos

Given trees $T$ and $S$, an injective map $\phi:V(T)\to V(S)$ is a minor
embedding if $\phi$ can be extended to an isomorphism between a subdivision of
$T$ and the smallest subtree $S^{\prime}$ of $S$ containing all vertices in
$\phi(V(T))$. If there exists such a minor embedding $\phi$ then we say that
$T$ is a topological minor of $S$ and we write $T\leq^{\sharp}S$. We use the
shorter term embedding when referring to a minor embedding provided there is
no danger of ambiguity. If $T\leq^{\sharp}S$ and $T\geq^{\sharp}S$ then we
write $T\equiv^{\sharp}S$ and say that they are topologically equivalent or of
the same topological type. The equivalence class of topological types of a
tree is denoted by $[T]$ and its size by $|[T]|$. These topological notions
are defined in a similar way for rooted trees. An injective map $\phi:V(T)\to
V(S)$ between rooted trees $(T,r)$ and $(S,s)$ is a rooted minor embedding if
$\phi$ can be extended to an isomorphism between a subdivision of $(T,r)$ and
the smallest rooted subtree $(S^{\prime},s^{\prime})$ of $(S,s)$ containing
all vertices in $\phi(V(T))$. A tree $(T,r)$ is self-similar provided that
there exists $v>_{T}r$ with $(T,r)\leq^{\sharp}(T_{v},v)$. An equivalent
definition for a minor embedding $\phi:V(T)\to V(S)$ is that of a meet
semilattice homomorphism: for any pair $w,v\in V(T)$,
$\phi(v\wedge_{T}w)=\phi(v)\wedge_{S}\phi(w)$. If there exists an embedding
$\phi:V(T)\to V(S)$ then we say that $(T,r)$ is a rooted topological minor of
$(S,s)$ and we write $(T,r)\leq^{\sharp}(S,s)$. If $(T,r)\leq^{\sharp}(S,s)$
and $(T,r)\geq^{\sharp}(S,s)$ then we write $(T,r)\equiv^{\sharp}(S,s)$ and
say that they are topologically equivalent. The equivalence class of
topological types of a tree is denoted by $[(T,r)]$ and its cardinality by
$|[(T,r)]|$.

A quasi-ordered set $(X,\leq)$ is well-quasi-ordered (wqo) if it is well-
founded and all antichains are finite. The rooted topological minor relation
is a wqo on the collection of trees ([9] and [14]). We make use of the
following equivalent characterisations for a quasi-ordered set $(X,\leq)$ to
be a wqo [12]:

  * •

any sequence $x_{1},x_{2},\ldots$ in $X$ contains a pair $x_{i}\leq x_{j}$
with $i<j$; and

  * •

any sequence $x_{1},x_{2},\ldots$ in $X$ has an increasing subsequence
$x_{n_{1}}\leq x_{n_{2}}\leq\ldots$.

######  Remark 2.1.

Since the rooted topological minor relation well-quasi orders rooted trees,
for trees distinct from the ray or double ray, the notion of being large is
equivalent to having a self-similar subtree. In particular, not having a self-
similar subtree is equivalent to being small.

###  2.3. Set-theoretic background

We use some basic set-theoretic notation and terminology which can be found in
[7] or [8]. Recall that $\aleph_{0}$ is the cardinality of the natural numbers
and we use the set-theoretic notation $\omega$ to denote the the set of
natural numbers, $\mathbb{N}$, with $0$ included. The continuum $\mathfrak{c}$
or $2^{\aleph_{0}}$ denotes the cardinality of the set of real numbers (which
is also equal to the cardinality of $\mathbb{N}^{\mathbb{N}}$, the set of all
functions $f:\mathbb{N}\rightarrow\mathbb{N}$). An almost disjoint family
(a.d.f.) $\mathcal{D}$ is a collection of infinite subsets of $\mathbb{N}$ so
that for any pair $X,Y\in\mathcal{D}$, $|X\cap Y|\in\mathbb{N}$. It is well
know that there exists an almost disjoint family of size $\mathfrak{c}$ (pg.
159, [8]).

##  3\. Large Trees

In this section all trees are assumed to be large. We begin by introducing the
notion of curtailing and subsequently employ it to prove Theorem 1 for all
large trees. The outline is as follows: for a fixed tree $T$ and $r\in V(T)$
we create - by means of curtailing $(T,r)$ \- a tree $(T^{\prime},r)$ of the
same topological type as $(T,r)$ and $2^{\aleph_{0}}$ subdivisions of
$(T^{\prime},r)$, $(T^{\prime}_{\kappa},r)$ with $\kappa\in 2^{\aleph_{0}}$,
of the same topological type as $(T^{\prime},r)$ but
$T^{\prime}_{\kappa}\not\simeq T_{\gamma}^{\prime}$ for any pair
$\kappa\not=\gamma$. Since $T^{\prime}_{\kappa}\equiv^{\sharp}T$ for all
$\kappa$ we obtain the following result.

######  Theorem 3.1.

For any large tree $T$:

  1. (1)

$|[T]|\geq 2^{\aleph_{0}}$ and

  2. (2)

$\forall r\in V(T)$, $|[(T,r)]|\geq 2^{\aleph_{0}}$.

###  3.1. Curtailing

Fix a rooted tree $(T,r)$ and a self-similar tree $(S,s)$ distinct from the
ray. For any $u\in V(T)$ with $(S,s)\equiv^{\sharp}(T_{u},u)$ and
$\operatorname{sp}(u)=1<\operatorname{sp}(\operatorname{pred}(u))$ let
$p_{u}:u,\ldots,u_{n}$ be the maximal bare path starting from $u$; such a path
must always exist since $(S,s)$ is not small. Now define $(T^{\prime},r)$ as
the tree obtained from $(T,r)$ by simultaneously replacing any such path
$p_{u}$ with an edge
$e_{u}=\\{\operatorname{pred}(u),\operatorname{succ}(u_{n})\\}$. That is,
simultaneously suppress all vertices in all paths $p_{u}$. We say that $(T,r)$
is curtailed into $(T^{\prime},r)$ via $(S,s)$. By design, all vertices of
splitting number greater than $1$ in $(T,r)$ are also present in
$(T^{\prime},r)$. It is entirely possible that $(T,r)\equiv^{\sharp}(S,s)$ and
that $\operatorname{sp}(r)=1$. In which case, the above construction would
delete $r$. By Lemma 3.2 below the tree obtained from deleting the maximal
bare path starting at $r$ yields a topologically equivalent tree to $(T,r)$.
Hence, without loss of generality, we assume that $\operatorname{sp}(r)>1$. Of
course, if $(S,s)\not\equiv^{\sharp}(T_{u},u)$ for any $u\in V(T)$ then
curtailing $(T,r)$ via $(S,s)$ results in $(T,r)$ itself and, trivially, Lemma
3.3 and Theorem 3.4 are true. We begin by proving Lemma 3.2 .

$(T_{w},w)$

$r$

$pred(u)$$u$$u_{n}$$w$

$(T^{\prime}_{w},w)$

$r$

$pred(u)$$w$$e_{u}$ Figure 1. Turning a maximal bare path $p_{u}$ into an edge
$e_{u}$.

######  Lemma 3.2.

Let $(T,r)$ be self-similar. It follows that the tree $(T^{\prime},r_{1})$
that results from adding a bare path $p:r_{1},\ldots,r_{n}$ below $r$ is of
the same topological type as $(T,r)$.

###### Proof.

We need only show that $(T^{\prime},r_{1})\leq^{\sharp}(T,r)$. Since $(T,r)$
is self-similar we can create a sequence
$r<_{T}\phi(r)<_{T}\phi^{2}(r)<_{T}\ldots$ where $\phi:V(T)\to V(T)$ witnesses
$(T,r)$’s is self-similarity. Choose any $k$ so that given the path
$p^{\prime}:w_{1}=r,\ldots,w_{j}=\phi^{k}(r)$ we have
$j>n=|p:r_{1},\ldots,r_{n}|$. Let $\psi:V(T^{\prime})\to V(T)$ with
$\psi(v)=\phi(v)$ for all $v\geq_{T^{\prime}}r$ and $\psi(r_{i})=w_{j}$. Then
$\psi$ is clearly a minor embedding. ∎

######  Lemma 3.3.

Let $(T^{\prime},r)$ be $(T,r)$ curtailed via $(S,s)$. For any $v\in V(T)$
with $\operatorname{sp}(v)>1$ if $(T_{v},v)\equiv^{\sharp}(S,s)$, then
$(T^{\prime}_{v},v)\equiv^{\sharp}(S,s)$.

###### Proof.

Since $(T_{v},v)$ is a subdivision of $(T^{\prime}_{v},v)$ then
$(T^{\prime}_{v},v)\leq^{\sharp}(S,s)$ whenever $(T_{v},v)\leq^{\sharp}(S,s)$.
For the reverse inequality, let $v\in V(T)$ with $\operatorname{sp}(v)>1$,
$(T_{v},v)\leq^{\sharp}(S,s)$ witnessed by a $\phi:V(S)\to V(T_{v})$ and, by
Lemma 3.2, assume that $\operatorname{sp}(s)>1$. Next, we construct a rooted
minor embedding $\psi:V(S)\to V(T^{\prime}_{v})$ witnessing
$(T^{\prime}_{v},v)\geq^{\sharp}(S,s)$. In order to so do, fix a minor rooted
embedding $\eta:V(S)\to V(S)$ witnessing $(S,s)$’s self-similarity (i.e.,
$\eta(s)>_{S}s)$), set

$L_{n}=\\{w\in V(S)\mid l_{S}(w)=n\\}$

and put $(S_{n},s)$ as the full subtree of $(S,s)$ generated from all vertices
in $L_{0}\cup\ldots\cup L_{n}$. We construct $\psi$ by inducting on the
$(S_{n},s)$’s. More precisely, for any $k\in\mathbb{N}$ we focus on the least
$n_{k}\in\mathbb{N}$ for which $\psi(u)$ is not defined for some $u\in
V(S_{n_{k}})$. By the minimality of $n_{k}$,
$\psi\upharpoonright(S_{n_{k}},s)$ will be a rooted topological minor
embedding.

Base Case: for any $u\in L_{1}$ if $\phi(u)\in V(T^{\prime}_{v})$, then we let
$\psi(u)=\phi(u)$; since $\operatorname{sp}(s)>1$ then $\phi(s)\in
V(T^{\prime}_{v})$ and $\psi(s)=\phi(s)$. Otherwise,
$\operatorname{sp}(\phi(u))=1$ and
$(S,s)\equiv^{\sharp}(T_{\phi(u)},\phi(u))$. That is, $\phi(u)$ was removed
when constructing $(T^{\prime},r)$ because
$(S,s)\equiv^{\sharp}(T_{\phi(u)},\phi(u))$. Let $\phi_{u}:V(S)\to
V(T_{\phi(u)})$ witness $(S,s)\leq^{\sharp}(T_{\phi(u)},\phi(u))$ and consider
the increasing sequence
$\phi_{u}\circ\eta(s)<_{T}\phi_{u}\circ\eta^{2}(s)<_{T}\ldots$. If $(S_{u},u)$
is a bare ray $u=u_{1}u_{2}\ldots$ then extend $\psi$ by letting
$\psi(u_{i})=\phi_{u}\circ\eta^{i}(s)$. Otherwise, let
$p_{u}:u=u_{1},\ldots,u_{m}$ be the maximal bare path in $(S,s)$ starting with
$u$ and extend $\psi$ by letting $\psi(u_{i})=\phi_{u}\circ\eta^{i}(s)$ and
$\psi(u_{m+1})=\phi_{u}\circ\eta^{m}(u_{m+1})$. It is simple to verify that
$\psi$ is indeed a meet semilattice homomorphism on $(S_{1},s)$

Inductive Case: at stage $k\in\mathbb{N}$, let $n_{k}\in\mathbb{N}$ be the
smallest so that there exists a vertex $u\in V(S_{n_{k}})$ where $\psi$ is
undefined and so that $\psi:(S_{i},s)\to(T,r)$ defines a meet semilattice
homomorphism for all $i<n_{k}$. By construction, either

  * •

$\psi(\operatorname{pred}(u))=\phi_{w}\circ\eta^{j}(\operatorname{pred}(u))$,
for some $w<_{S}u$ and $j\in\mathbb{N}$, or

  * •

$\psi(\operatorname{pred}(u))=\phi(\operatorname{pred}(u))$.

Set $u^{*}$ to be $\phi(u)$ or $\phi_{w}\circ\eta^{j}(u)$ depending on the
case above that is true. If $u^{*}\in V(T^{\prime})$ then set $\psi(u)=u^{*}$.
Otherwise, as with the base case, it must be that $\operatorname{sp}(u^{*})=1$
and $(S,s)\leq^{\sharp}(T_{u^{*}},u^{*})$ with embedding $\phi_{u}:V(S)\to
V(T_{u^{*}})$. In which case, consider the increasing sequence
$\phi_{u}\circ\eta(s)<_{T}\phi_{u}\circ\eta^{2}(s)<_{T}\ldots$. If $(S_{u},u)$
is a bare ray $u=u_{1}u_{2}\ldots$ then extend $\psi$ by letting
$\psi(u_{i})=\phi_{u}\circ\eta^{i}(s)$. Otherwise, let
$p_{u}:u=u_{1},\ldots,u_{m}$ be the maximal bare path in $(S,s)$ starting with
$u$ and extend $\psi$ by letting $\psi(u_{i})=\phi_{u}\circ\eta^{i}(s)$ and
$\psi(u_{m+1})=\phi_{u}\circ\eta^{m}(u_{m+1})$.

To show that $\psi$ is a meet semilattice homomorphism on $(S_{n},s)$ we must
verify a few cases separately. Firstly, observe that by construction for any
pair $w,w^{\prime}\in V(S_{n_{k}})$ it follows that $w\leq_{S}w^{\prime}$ if,
and only if, $\psi(w)\leq_{T^{\prime}}\psi(w^{\prime})$. If
$\operatorname{pred}(w)\not=\operatorname{pred}(w^{\prime})$, then

$\psi(w\wedge_{S}w^{\prime})=\psi(\operatorname{pred}(w)\wedge_{S}\operatorname{pred}(w^{\prime}))=\psi(\operatorname{pred}(w))\wedge_{T^{\prime}}\psi(\operatorname{pred}(w^{\prime}))$

by the inductive hypothesis and

$\psi(\operatorname{pred}(w))\wedge_{T^{\prime}}\psi(\operatorname{pred}(w^{\prime}))=\psi(w)\wedge_{T^{\prime}}\psi(w^{\prime})$

since $\psi(w)\bot_{T^{\prime}}\psi(w^{\prime})$,
$\psi(\operatorname{pred}(w))\leq_{T^{\prime}}\psi(w)$ and
$\psi(\operatorname{pred}(w^{\prime}))\leq_{T^{\prime}}\psi(w^{\prime})$.
Otherwise we have that
$\operatorname{pred}(w)=\operatorname{pred}(w^{\prime})$. If
$\psi(\operatorname{pred}(w))=\phi(\operatorname{pred}(w))$, then, by
construction, it follows that $\psi(w)\geq_{T}\phi(w)$ and
$\psi(w^{\prime})\geq_{T}\phi(w^{\prime})$. Hence,

$\psi(w)\wedge_{T^{\prime}}\psi(w^{\prime})=\phi(w)\wedge_{T}\phi(w^{\prime})=\phi(w\wedge_{S}w^{\prime})=\psi(w\wedge_{S}w^{\prime}).$

Finally, if
$\psi(\operatorname{pred}(w))=\phi_{t}\circ\eta^{j}(\operatorname{pred}(w))$
for some $t\leq_{S}\operatorname{pred}(w)$ and $j\in\mathbb{N}$, then
$\psi(w)\geq_{T}\phi_{t}\circ\eta^{j}(w)$ and
$\psi(w^{\prime})\geq_{T}\phi_{t}\circ\eta^{j}(w^{\prime})$. Therefore,
$\psi(w)\wedge_{T^{\prime}}\psi(w^{\prime})=\psi(w)\wedge_{T}\psi(w^{\prime})=\phi_{t}\circ\eta^{j}(w)\wedge_{T}\phi_{t}\circ\eta^{j}(w^{\prime})=\phi_{t}\circ\eta^{j}(w\wedge_{S}w^{\prime})=\psi(w\wedge_{S}w^{\prime}).$
∎

######  Theorem 3.4.

Let $(T^{\prime},r)$ be $(T,r)$ curtailed via $(S,s)$. It follows that for any
$v\in V(T^{\prime})$, $(T^{\prime}_{v},v)\equiv^{\sharp}(T_{v},v)$. In
particular, $(T,r)\equiv^{\sharp}(T^{\prime},r)$.

###### Proof.

Clearly, $(T^{\prime}_{v},v)\leq^{\sharp}(T_{v},v)$. For the reverse
inequality, notice that if $(S,s)\equiv^{\sharp}(T_{v},v)$, then by Lemma 3.3
we are done. Otherwise, much in the same spirit as with Lemma 3.3, we
construct a minor embedding $\psi:V(T_{v})\to V(T_{v}^{\prime})$ by inducting
on the full subtrees $(T_{v}^{n},v)$ of $(T_{v},v)$ generated by
$L_{0}\cup\ldots\cup L_{n}$ where

$L_{n}=\\{w\in V(T_{v})\mid l_{T_{v}}(w)=n\\}.$

Base Case: since $L_{0}=\\{v\\}$ and, by assumption,
$(S,s)\not\equiv^{\sharp}(T_{v},v)$ then $v\in V(T^{\prime}_{v})$ and we let
$\psi(v)=v$. Take any $u\in L_{1}$. If $u\in V(T^{\prime}_{v})$ then we let
$\psi(u)=u$. Otherwise, if $u\not\in V(T^{\prime})$ then
$(S,s)\equiv^{\sharp}(T_{u},u)$ and $\operatorname{sp}(u)=1$. Let
$p_{u}:u=u_{1},\ldots,u_{n}$ be the maximal bare path starting at $u$ and put
$\operatorname{succ}(u_{n})=u^{*}$. Since $(S,s)\equiv^{\sharp}(T_{u},u)$ then
Lemma 3.2 yields $(T_{u},u)\equiv^{\sharp}(T_{u^{*}},u^{*})$ and by Lemma 3.3,
$(S,s)\equiv^{\sharp}(T^{\prime}_{u^{*}},u^{*})$. In turn,
$(T_{u},u)\equiv^{\sharp}(T^{\prime}_{u^{*}},u^{*})$ and we extend $\psi$ to
all $w\geq_{T}u$ by mapping $w\mapsto\phi_{u}(w)$ where $\phi_{u}:V(T_{u})\to
V(T^{\prime}_{u^{*}})$ witnesses
$(T_{u},u)\leq^{\sharp}(T^{\prime}_{u^{*}},u^{*})$. It is clear that $\psi$
restricted to $(T_{v}^{1},v)$ defines a meet semilattice homomorphism.

Inductive Case: set $k\in\mathbb{N}$ as the smallest for which we can find a
vertex $u\in V(T_{v}^{n_{k}})$ where $\psi$ is not defined. This implies that
$\psi(\operatorname{pred}(u))=\operatorname{pred}(u)$ and we apply the same
logic as with the base case; distinguish between whether or not
$(S,s)\equiv^{\sharp}(T_{u},u)$ and extend $\psi$ accordingly. The meet
semilattice homomorphism property of $\psi$ is verified by cases as with Lemma
3.3. ∎

###  3.2. Proof of Theorem 3.1

Let us begin by restating the theorem.

See 3.1

###### Proof.

(1) Start with a large tree $T$, a not eventually bare ray $v_{0}v_{1}\ldots$
witnessing this. Root $T$ at $r=v_{0}$ and put $v_{k}$ as the first vertex
along the ray with $(T_{v_{k}},v_{k})\leq^{\sharp}(T_{v_{j}},v_{j})$ for some
$j>k>0$ as rooted subtrees of $(T,r)$. Set $s=v_{k}$ and
$(S,s)=(T_{v_{k}},v_{k})$ and by Lemma 3.2 we can assume that both
$\operatorname{sp}(s)>1$ and $\operatorname{sp}(r)>1$. Since $(S,s)$ is self-
similar there exists an embedding $\phi:V(S)\to V(S)$ witnessing this. Set
$r_{1}=s$, $r_{n}=\phi^{n}(s)$ for $n>2$, and put $D:r_{1}\ldots$ as the
unique ray in $(T,r)$ containing all $r_{n}$’s. Observe that $\phi$
establishes that $(T_{r_{i}},r_{i})\equiv^{\sharp}(T_{r_{j}},r_{j})$ for all
$i,j\geq 1$ and that $\phi^{i}(V(D_{i}^{\uparrow}))\subseteq
V(D_{i}^{\uparrow})$. Let $(T^{\prime},r)$ denote the curtailed tree $(T,r)$
via $(S,s)$ and denote the ray $D^{\prime}$ in $(T^{\prime},r)$ as the
obtained from $D$ in $(T,r)$. A moments thought reveals that one can modify
$\phi$ into an embedding $\eta:V(T^{\prime}_{r_{1}})\to V(T^{\prime}_{r_{1}})$
with $\eta(r_{i})=r_{i+1}$ and $\eta(V(D^{{}^{\prime}\uparrow}))\subseteq
V(D^{{}^{\prime}\uparrow})$. This is since all vertices of degree at most $3$
in $D$ remain in $D^{\prime}$ (e.g., all roots $r_{i}$ remain in $D^{\prime}$)
and by Theorem 3.4 all subtrees of $(T_{r_{1}},r_{1})$ rooted on vertices off
the ray $D$ remain topologically equivalent to their curtailed versions. Set
$e_{n}=\\{\operatorname{pred}(r_{n}),r_{n}\\}$ in $(T^{\prime},r)$ for each
$n\in\mathbb{N}$, and for each $f\in\mathbb{N}^{\mathbb{N}}$ we let
$(T^{\prime}_{f},r)$ denote the tree generated from $(T^{\prime},r)$ by
replacing the edge $e_{n}$ with a bare path of length $f(n)$. Let
$D^{\prime}_{f}$ denote the subdivision of $D^{\prime}$ in
$(T^{\prime}_{f},r)$. Observe that by curtailing $(T,r)$ and subsequently
subdividing $D^{\prime}$ the only bare paths in $D^{\prime}_{f}$ above
$\operatorname{pred}(r_{1})$ are those bare paths ending in an $r_{n}$. Next
we prove that

  * •

$(T^{\prime}_{f},r)\equiv^{\sharp}(T,r)$ for all $f\in\mathbb{N}^{\mathbb{N}}$
(hence $T^{\prime}_{f}\equiv^{\sharp}T$), and

  * •

$\exists\mathcal{D}\subset\mathbb{N}^{\mathbb{N}}$ of size $\mathfrak{c}$
where $T^{\prime}_{f}\not\simeq T^{\prime}_{g}$ for all $f,g\in\mathcal{D}$.

######  Lemma 3.4.

Assume that $T^{\prime}_{f}\simeq T^{\prime}_{g}$ for some
$g,f\in\mathbb{N}^{\mathbb{N}}$ with bijection
$\psi^{f}_{g}:V(T^{\prime}_{f})\to V(T^{\prime}_{g})$ witnessing this. It
follows that $\psi^{f}_{g}$ must map a final segment of $D^{\prime}_{f}$
entirely within $D^{\prime}_{g}$.

###### Proof.

Assume that $\psi^{f}_{g}$ does not map a final segment of
$D^{\prime}_{f}=u_{1}\ldots$ entirely within $D^{\prime}_{g}$, put

$m=\min\\{n\in\mathbb{N}\mid\forall j\geq n,\psi^{f}_{g}(u_{j})\not\in
D_{g}^{\prime}\text{ and
}\psi^{f}_{g}(u_{j})\leq_{T^{\prime}_{g}}\psi^{f}_{g}(u_{j+1})\\}$

and $B=b_{1}\ldots$ the ray in $(T^{\prime}_{g},r)$ containing all vertices
$\psi^{f}_{g}(u_{n})$ with $n\geq m$ (i.e., the ray $B$ is disjoint from
$D_{g}^{\prime}$, as in Figure 2). Recall that all of the $r_{n}$ (i.e., those
used to define $D$ in $(T,r))$ are in $D^{\prime}_{f}$. Let $r_{l}\in
V(D^{\prime}_{f})$ so that $\phi^{f}_{g}(r_{l})=b_{j}$ for some $j>2$ (we can
choose an arbitrarily large value of $j$ since the $\psi^{f}_{g}(r_{i})$ are
cofinal in $B$ for $i\geq l$). It follows that the rooted tree
$([T^{\prime}_{g}]_{b_{j}},b_{j})$ (i.e., the full subtree of
$(T^{\prime}_{g},r)$ rooted at $b_{j}$) is isomorphic to $(T_{r_{l}},r_{l})$.
Indeed, $m$ was chosen so that $r_{l}>_{T_{g}^{\prime}}r_{l-1}$ and, thus,
$\psi^{f}_{g}$ preserves the order of $(T_{r_{l}},r_{l})$ in
$(T^{\prime}_{g},r)$. In turn, $([T^{\prime}_{g}]_{b_{j}},b_{j})$ and
$r_{l}>_{T_{g}^{\prime}}r_{l-1}$ are of the same topological type as $(S,s)$.
But since $\operatorname{sp}(\operatorname{pred}(b_{j}))>1$ in
$(T^{\prime}_{g},r)$ and $\operatorname{sp}(\operatorname{pred}(r_{l}))=1$ by
design of $(T^{\prime}_{f},r)$ this contradicts that $\psi^{f}_{g}$ is an
isomorphism. ∎

$b_{1}$$\psi_{g}^{f}(r)$$r$ Figure 2. The ray $B$ in green with initial vertex
$b_{1}$. Also the initial segment of the ray $\psi^{f}_{g}(D^{\prime}_{f})$
and the ray $D^{\prime}_{g}$ highlighted in red and blue, resp.

Next, we focus on proving that $T^{\prime}_{f}\not\simeq T^{\prime}_{g}$ for
continuum many $f,g\in\mathbb{N}^{\mathbb{N}}$. By Lemma 3.4 given
$f,g\in\mathbb{N}^{\mathbb{N}}$ with $\psi^{f}_{g}:V(T^{\prime}_{f})\to
V(T^{\prime}_{g})$ witnessing $T^{\prime}_{f}\simeq T^{\prime}_{g}$ it follows
that $\psi^{f}_{g}$ must map a final segment of $D^{\prime}_{f}$ entirely
within $D^{\prime}_{g}$. Given that $\psi^{f}_{g}$ maps bare paths to bare
paths, the final segment of $D^{\prime}_{f}$ that is mapped entirely within
$D_{g}^{\prime}$ must have its bare paths of the same length as those in
$D^{\prime}_{g}$. Let $\mathcal{D}$ be an a.d.f. of size $\mathfrak{c}$ and
for each $X=\\{x_{1},x_{2},\ldots\\}\in\mathcal{D}$ set $f_{X}(n)=x_{n}$. Then
$T^{\prime}_{f_{X}}\simeq T^{\prime}_{f_{Y}}$ precisely when $X=Y$. To finish
the proof of Theorem 3.1 (1) we establish the following result.

######  Lemma 3.5.

For all $f\in\mathbb{N}^{\mathbb{N}}$, $T_{f}^{\prime}\equiv^{\sharp}T$.

###### Proof.

Notice that curtailing $(T^{\prime}_{f},r)$ via $(S,s)$ yields
$(T^{\prime},r)$ and, thus, by Theorem 3.4 the result follows. ∎

(2) The construction of each $(T^{\prime}_{f},r)$, for
$f\in\mathbb{N}^{\mathbb{N}}$, based on an unrooted tree $T$ can be applied
directly to $(T,r)$ by selecting any ray $D$ in $(T,r)$ that is not eventually
bare. This means that there are continuum many unrooted trees $T_{f}^{\prime}$
with $T_{f}^{\prime}\not\simeq T$ (hence, $(T_{f}^{\prime},r)\not\simeq(T,r)$)
but with $(T_{f}^{\prime},r)\equiv^{\sharp}(T,r)$.

∎

##  4\. Small Trees

We now turn our attention to small trees. We first prove Theorem 1 for all
rooted small trees and extend the result to all small trees by adapting a
fixed-point result from Polat and Sabidussi to the topological minor relation.

###  4.1. Rooted small trees

Unlike the scenario encountered for large trees, given a pair small trees
$(T,r)\equiv^{\sharp}(S,s)$ (distinct from the ray) any rooted minor embedding
$\psi$ witnessing $(T,r)\leq^{\sharp}(S,s)$ must map $r$ to $s$. Indeed,
otherwise for any $\phi$ that witnesses $(T,r)\geq^{\sharp}(S,s)$ it follows
that the sequence $(\phi\circ\psi)^{n}(r)$ spans a ray in $T$ that is not
eventually bare. This observations then allows us to directly employ the
arguments developed in [15] and [4] to the rooted topological minor relation.
In the following lemmas for a rooted tree $(T,r)$, $S(T)=\\{(T_{u},u)\mid
u\in\operatorname{succ}(r)\\}$.

######  Lemma 4.1.

For any $(T_{u},u)\in S(T)$, $|[(T_{u},u)]|\leq|[(T,r)]|$.

###### Proof.

The proof of this lemma is exactly that of Lemma 2 in [15] but we reproduce it
here. Fix a $v\in\operatorname{succ}(r)$ and take any $(U,u)\in[(T_{v},v)]$
(i.e., a non-isomorphic tree of the same topological type as $(T_{v},v)$).
Construct the tree $(T_{U},r)$ as follows: for any
$w\in\operatorname{succ}(v)$

  * •

replace $(T_{w},w)$ with $(U,u)$ if $(T_{w},w)\equiv^{\sharp}(U,u)$, and

  * •

leave $(T_{w},w)$ as it is, otherwise.

By design, $(T_{U},r)\equiv^{\sharp}(T,r)$ and $(T_{U},r)\not\simeq(T,r)$.
Moreover, for any other $(W,w)\in[(T_{v},v)]$ non-isomorphic to $(U,u)$ it
follows that $(T_{U},r)\not\simeq(T_{W},r)$. ∎

In view of the above and given a small tree $(T,r)$ next we prove that if
$|[(T_{v},v)]|=1$ for all $v\in\operatorname{succ}(r)$, then $|[(T,r)]|=1$ or
$\geq\aleph_{0}$. This would then complete the proof of Theorem 1 for small
rooted trees. Indeed, by Lemma 4.2, that $\aleph_{0}>(T,r)>1$ implies the same
of at least one $(T_{v},v)$ for $v\in\operatorname{succ}(r)$, which in turn
implies the same of some $(T_{w},w)$ for $w\in\operatorname{succ}(v)$, and so
on. This would create a sequence
$\left(|[(T_{v_{n}},{v_{n}})]|\right)_{n\in\mathbb{N}}$ with
$v_{n}\in\operatorname{succ}(v_{n-1})$ and $|[(T_{v_{n}},{v_{n}})]|>1$. A
contradiction since any such sequence in a small tree ends on a terminal
vertex or spans an eventually bare ray (i.e., a tail of all $1$’s).

######  Lemma 4.2.

Let $(T,r)$ be small. If for all $(T_{u},u)\in S(T)$, $|[(T_{u},u)]|=1$ then
$|[(T,r)]|=1$ or $\geq\aleph_{0}$.

###### Proof.

Lemma 2 in [4] after exchanging rooted subgraph with rooted topological minor
but we also reproduce a simplified version here. Let $(T,r)$ and $(U,u)$ be
two non-isomorphic trees with $(U,u)\in[(T,r)]$, and rooted embeddings
$\phi_{T}:V(T)\to V(U)$ and $\phi_{U}:V(U)\to V(T)$ witnessing this. Further
assume that $|[(T_{v},v)]|=1$ for any $v\in\operatorname{succ}(r)$. We seek to
prove that $|[(T,r)]|\geq\aleph_{0}$.

Put $\mathcal{I}=\\{(X_{\alpha},\alpha)\mid\alpha\in\kappa\\}$ as the set of
all isomorphism types of all rooted trees $(T_{v},v)$ and $(U_{w},w)$ with
$v\in\operatorname{succ}(r)$ and $w\in\operatorname{succ}(u)$. Define
$f_{T}:\\{(T_{v},v)\mid v\in\operatorname{succ}(r)\\}\to\kappa$ so that
$f_{T}(T_{v},v)=\alpha$ provided $(T_{v},v)\simeq(X_{\alpha},\alpha)$ (resp.
$f_{U}:\\{(U_{w},w)\mid w\in\operatorname{succ}(u)\\}\to\kappa$ so that
$f_{U}(U_{w},w)=\alpha$ provided $(U_{w},w)\simeq(Y_{\alpha},\alpha)$). Since
$(T,r)\not\simeq(U,u)$ there must exist a $\beta\in\kappa$ for which, without
loss of generality, $|f_{T}^{-1}(\beta)|<|f_{U}^{-1}(\beta)|$. Put
$\eta=|f_{T}^{-1}(\beta)|$ and $\zeta=|f_{U}^{-1}(\beta)|$, and enumerate the
elements of $f_{T}^{-1}(\beta)$ and $f_{U}^{-1}(\beta)$ as $(T_{i},i)$ and
$(U_{j},j)$ with $i\in\eta$ and $j\in\zeta$. Before moving further with this
proof, let us establish some more notation: put $(T^{*},r)$ as $(T,r)$ but
with all full subtrees $(T_{i},i)$, $i\in\eta$, removed - of course, this also
includes the roots $i$ of each $(T_{i},i)$ and the edge joining $i$ with $r$.
In what follows we fix a $(T_{k},k)\in f_{T}^{-1}(\beta)$ and for each
cardinal $\lambda$ denote $(T^{*}_{\lambda},r)$ as $(T^{*},r)$ with $\lambda$
copies ($(T^{*}_{\lambda_{l}},{\lambda_{l}})$ with $l\in\lambda$) of
$(T_{k},k)$ attached to $r$. Clearly, $(T^{*}_{\eta},r)\simeq(T,r)$ and
$(T^{*}_{\nu},r)\not\simeq(T^{*}_{\xi},r)$ for any pair of cardinals
$\nu\not=\xi$.

Fix a $\gamma\in\zeta$ for which
$\phi_{U}(U_{\gamma},\gamma)\subseteq(T_{v},v)\not\in f_{T}^{-1}(\beta)$.
Next, for each $n\in\omega$, put $w_{n}\in\operatorname{succ}(s)$ so that
$(\phi_{T}\circ\phi_{U})^{n}[(U_{\gamma},\gamma)]\subseteq(U_{w_{n}},w_{n})$
and $v_{n}\in\operatorname{succ}(r)$ with
$\phi_{U}\circ(\phi_{T}\circ\phi_{U})^{n}[(U_{\gamma},\gamma)]\subseteq(T_{v_{n}},v_{n})$.
Clearly, $(U_{w_{0}},w_{0})=(U_{\gamma},\gamma)$ and
$(T_{v_{0}},v_{0})=(T_{v},v)$, and

$(U_{w_{0}},w_{0})\leq^{*}(T_{v_{0}},v_{0})\leq^{*}(U_{w_{1}},w_{1})\leq^{*}(T_{v_{1}},v_{1})\leq^{*}\ldots$

Observe that $w_{n}\not=w_{0}$ for any $n\in\omega$ since then
$(U_{\gamma},\gamma)\simeq(T_{v},v)$ from the assumption that
$|[(T_{u},u)]|=1$ for any $u\in\operatorname{succ}(r)$. The same logic
dictates $(U_{w_{n}},w_{n})\not\in f_{U}^{-1}(\beta)$ and
$(T_{v_{n}},v_{n})\not\in f_{T}^{-1}(\beta)$ for any $n>1$. Thus, we get two
sequences of non-repeating elements: $(w_{n})_{n\in\omega}$ and
$(v_{n})_{n\in\omega}$.

If $\eta$ is finite, then for each $\lambda\geq\eta$ it follows that
$(T^{*}_{\lambda},\lambda)\geq^{\sharp}(T,r)$. For any for each
$\lambda\in(\eta,\infty)$ put $\phi_{\lambda}:V(T^{*}_{\lambda})\to V(T)$ with

  * •

$\phi_{\lambda}((T^{*}_{\lambda_{l}},{\lambda_{l}}))=\phi_{U}\circ(\phi_{T}\circ\phi_{U})^{l}[(U_{\gamma},\gamma)]$
for each $l\leq\lambda$,

  * •

$\phi_{\lambda}((T_{v_{n}},{v_{n}}))=\phi_{U}\circ(\phi_{T}\circ\phi_{U})^{n+\lambda+1}[(U_{\gamma},\gamma)]$
for each $n\in\omega$, and

  * •

the identity everywhere else.

Since $(U_{\gamma},\gamma)\equiv^{\sharp}(T_{k},k)$, $\phi_{\lambda}$ is a
witness of $(T^{*}_{\lambda},\lambda)\leq^{\sharp}(T,r)$ for each finite
$\lambda\geq\eta$ but $(T^{*}_{\lambda},\lambda)\not\simeq(T,r)$ for any
$\lambda>\eta$. If $\eta\geq\aleph_{0}$, then since $\zeta>\eta$ one can
easily adapt the proof of finite $\eta$ to create $\zeta$ many non-isomorphic
$(T^{*}_{\lambda},\lambda)$, $\lambda\in\zeta$, of the same topological type
as $(T,r)$.

∎

###  4.2. Unrooted small trees

Given a rayless tree $T$ the authors of [13] showed that there exist a vertex
or edge of $T$ fixed by all automorphisms of $T$. By modifying the arguments
used in the above to minor embeddings one can easily obtain an analogous
result for all small trees different to the ray or double ray. Firstly, let us
remark that a minor self-embedding of a finite tree $T$ is necessarily an
automorphism and, as such, it must either fix a vertex or an edge of $T$ \-
[6], Lemma 2. In fact, more is true: given a set $S\subseteq V(T)$ let
$\overline{S}$ denote the subtree of $T$ generated by the union of all paths
joining any pair of vertices in $S$ , $F_{T}=\\{v\in
V(T_{\lambda})\mid\deg(v)>2\\}$ and $\text{TME}(T)$ as the collection of all
topological minor embeddings of $T$. Let $T$ be locally finite, small,
infinite and distinct from the ray or double ray. It follows that
$F_{T}\not=\emptyset$ is finite (since $T$ is small) and any minor embedding
$\psi$ induces a permutation of $F_{T}$ where $\deg(v)=\deg(\psi(v))$ for all
$v\in F_{T}$. It then follows that
$\psi\upharpoonright\overline{F_{T}}\in\text{TME}(\overline{F_{T}})$ and,
thus, that $\psi$ induces an automorphism of $\overline{F_{T}}$. Since
$\overline{F_{T}}$ is finite then $\psi$ restricted to $\overline{F_{T}}$ is
actually an automorphism and $\overline{F_{T}}$ has either a fixed edge or
vertex by all $\psi\in\text{TME}(T)$. It then follows that $T$ itself has a
fixed edge or vertex for all $\psi\in\text{TME}(T)$.

######  Lemma 4.3.

Any small and locally finite tree $T$, distinct from the ray or double ray,
has an edge or vertex fixed by all $\psi\in\text{TME}(T)$.

We extend the above to all small trees by means of employing the following
lemma from [13].

######  Lemma 4.4.

[Lemma 1.1, [13]] Let $G$ be a rayless graph,
$(A_{\alpha})_{\alpha\in\text{Ord}}$ a decreasing sequence of subsets of
$V(G)$ so that

  1. (1)

$A_{\alpha}=\bigcap_{\beta<\alpha}A_{\beta}$ for any limit $\alpha$ and

  2. (2)

each $A_{\alpha}$ induces a connected subgraph of $G$.

By $\kappa$ denote the smallest ordinal for which the sequence
$(A_{\alpha})_{\alpha\in\text{Ord}}$ becomes constant. If
$A_{\kappa}=\emptyset$, then $\kappa$ is a successor ordinal.

######  Lemma 4.5.

Given a small tree $T$, distinct from the ray or double ray, there exists a
vertex or edge of $T$ fixed by all $\psi\in\textrm{TME}(T)$.

###### Proof.

Given Lemma 4.3, let $T$ be small but not locally finite, and set
$T_{1}=\overline{\text{infinite}(T)}$, where $\text{infinite}(T)\subset V(T)$
is the set consisting all vertices of infinite degree. Clearly, $T_{1}$ is
connected, rayless, properly contained in $T$ \- since $T_{1}$ does not
contain any end-points or rays from $T$ \- and for any $\psi\in\text{TME}(T)$
we have $\psi\upharpoonright T_{1}\in\text{TME}(T_{1})$. For all
$\alpha\in\text{Ord}$ construct $T_{\alpha}$ as

$T_{\alpha}=\begin{cases}\overline{\text{inf}(T_{\beta})}&\text{ if
$\alpha=\beta+1$ and}\\\ \bigcap_{\beta<\alpha}T_{\beta},&\text{
otherwise}\end{cases}$

Let $\kappa$ denote the smallest ordinal for which $(T_{\alpha})$ is constant.
This happens when $T_{\kappa}=\emptyset$. Indeed, otherwise the sequence
$(T_{\alpha})$ would not be constant. Therefore, by Lemma 4.4, that
$\kappa=\lambda+1$ for some $\lambda\in\text{Ord}$. In turn, $T_{\lambda}$ is
locally finite and small and any such $\psi$ defines an automorphism on
$T_{\lambda}$. Hence, there must exist a vertex of edge of $T$ fixed by all
$\psi\in\text{TME}(T)$. ∎

Notice that Lemma 4.5 establishes a dichotomy in the following sense; if an
edge $e=\\{r,s\\}$ in a tree $T$ is fixed by all minor embeddings with
$\psi(r)=s$, for all $\psi\in\text{TME}(T)$, then $T$ does not have a fixed
vertex - the converse is also true. Indeed, all minor embeddings swap $s$ with
$r$ precisely when any vertex in $V$ is sent to the opposite connected
component of $T\smallsetminus\\{e\\}$. This observation is important for the
proof of the following theorem.

######  Theorem 4.1.

For any small tree $T$, $|[T]|=1$ or $|[T]|\geq\aleph_{0}$.

###### Proof.

The proof of Theorem 1 in [4] applies directly to this theorem and, for
completeness, we reproduce it below.

Let $T$ be small and assume that $|[T]|>1$. Next we show that
$|[T]|\geq\aleph_{0}$. Since $T$ is small then, by Lemma 4.5, $T$ has either a
fixed vertex $r$ or edge $\\{r,s\\}$. Let $T^{*}$ be a non-isomorphic
topologically equivalent tree to $T$, and $\psi_{T}$, $\psi_{T^{*}}$ denote
minor embedding witnessing $T\leq^{\sharp}T^{*}$ and $T\geq^{\sharp}T^{*}$,
respectively. Consider the tree $(T,r)$ and observe that $(T,r)$ and
$(T^{*},\psi_{T}(r))$ are not isomorphic. If $r$ is a fixed vertex of $T$ then
$(T,r)$ and $(T^{*},\psi_{T}(r))$ are of the same topological type as
witnessed by $\psi_{T}$, $\psi_{T^{*}}$. Hence, $|[(T,r)]|\geq\aleph_{0}$. If
$\\{r,s\\}$ is a fixed edge then the rooted embeddings $\psi_{T}$ and
$\psi_{T^{*}}\circ\psi_{T}\circ\psi_{T^{*}}$ witness
$(T,r)\equiv^{\sharp}(T^{*},\psi_{T}(r))$ and $|[(T,r)]|\geq\aleph_{0}$ as
well. In either case, $(T,r)$ is topologically equivalent to at least
infinitely many other non-isomorphic rooted trees.

By virtue of the previous paragraph, for a small tree $T$ with $|[T]|>1$ we
let $C=\\{(T_{i},r_{i})\mid i\in\omega\\}$ be an infinite collection of
mutually non-isomorphic rooted trees topologically equivalent to $(T,r)$. We
show that $\\{T_{i}\mid i\in\omega\\}$ contains an infinite collection of non-
isomorphic trees topologically equivalent to $T$. In order to achieve this
goal, we show that given any triplet $(T_{j},r_{j}),(T_{k},r_{k})$ and
$(T_{l},r_{l})$ from $C$ at least one pair from the set
$\\{T_{j},T_{k},T_{l}\\}$ are non-isomorphic. For the sake of contradiction
assume otherwise and let $h_{j}:V(T_{l})\to V(T_{j})$ and $h_{k}:V(T_{l})\to
V(T_{k})$ denote the respective isomorphisms - observe that
$h_{j}(r_{l})\not=r_{j}$ and $h_{k}(r_{l})\not=r_{k}$. Since $(T,r)$,
$(T_{j},r_{j}),(T_{k},r_{k})$ and $(T_{l},r_{l})$ are of the same topological
type there exist rooted minor embeddings $g_{j}:V(T_{j})\to V(T)$,
$g_{k}:V(T_{k})\to V(T)$ and $g_{l}:V(T)\to V(T_{l})$ witnessing this. Recall
that rooted minor embeddings between small trees must map roots to roots. It
follows that $g_{k}\circ h_{k}\circ g_{l}(r)\not=r\not=g_{j}\circ h_{j}\circ
g_{l}(r)$. In turn, the edge $\\{r,s\\}$ is a fixed edge of $T$ and
$g_{k}\circ h_{k}\circ g_{l}(r)=g_{j}\circ h_{j}\circ g_{l}(r)=s$ and
$g_{k}\circ h_{k}\circ g_{l}(s)=g_{j}\circ h_{j}\circ g_{l}(s)=r$. The
contradiction is then the following one: since
$h_{j}^{-1}(r_{j})=g_{l}(s)=h_{k}^{-1}(r_{k})$ then $h_{j}\circ h_{k}^{-1}$
establishes an isomorphism between $(T_{j},r_{j})$ and $(T_{k},r_{k})$ \-
impossible by our assumption. ∎

##  5\. Acknowledgments

We would like to extend our sincere gratitude to the reviewers for their many
helpful suggestions and comments.

## References

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# Higher-order generalized uncertainty principle corrections to the Jeans mass

Zhong-Wen Feng1 Email<EMAIL_ADDRESS>Guansheng He2 Xia Zhou1 Xueling
Mu3 Shi-Qi Zhou1 1\. Physics and Space Science College, China West Normal
University, Nanchong, 637009, China
2\. School of Mathematics and Physics, University of South China, Hengyang
421001, China
3\. School of Information Science and Engineering, Chengdu University, Chengdu
610106, China

###### Abstract

The Jeans instability is regarded as an important tool for analyzing the
dynamics of a self-gravitating system. However, this theory is challenging
since astronomical observation data show some Bok globules, whose masses are
less than the Jeans mass and still have stars or at least undergo the star
formation process. To explain this problem, we investigate the effects of the
higher-order generalized uncertainty principle on the Jeans mass of the
collapsing molecular cloud. The results in this paper show that the higher
order generalized uncertainty principle has a very significant effect on the
canonical energy and gravitational potential of idea gas, and finally leads to
a modified Jeans mass lower than the original case, which is conducive to the
generation of stars in small mass Bok globules. Furthermore, we estimate the
new generalized uncertainty principle parameter $\gamma_{0}$ by applying
various data of Bok globules, and find that the range of magnitude of
$\gamma_{0}$ is ${10^{11}}\sim{10^{12}}$.

Higher order generalized uncertainty principle; Jeans mass; GUP parameter; Bok
globules

##  I Introduction

Einstein’s theory of general relativity is regarded as the cornerstone of the
development of modern physics and astronomy. However, with the deepening of
research, it is found that general relativity has flaws, which lead to many
problems, such as the black hole information paradox and naked singularity of
spacetimes chb1 . One of the most promising candidates to solve those problems
is quantum gravity (QG). Now, based on these frameworks that explicate QG,
people hypothesize that exists a minimum measurable length of the order of the
Planck length in QG, which is supported by Gedanken experiments cha1 ; cha2 ;
cha3 .

According to the minimum measurable length, many such studies have converged
on the idea that the Heisenberg uncertainty principle (HUP) can be modified as
the generalized uncertainty principle (GUP). In 1995, Kempf, Mangano and Mann
proposed a quadratic form of the GUP, which we now call as the “KMM model”
with the expression, $\Delta x\Delta
p\geq\frac{\hbar}{2}\left[{1+\beta\Delta{p^{2}}}\right]$, where
$\beta={{{\beta_{0}}}\mathord{\left/{\vphantom{{{\beta_{0}}}{M_{p}^{2}{c^{2}}}}}\right.\kern-1.2pt}{M_{p}^{2}{c^{2}}}}={{{\beta_{0}}\ell_{p}^{2}{\rm{}}}\mathord{\left/{\vphantom{{{\beta_{0}}\ell_{p}^{2}{\rm{}}}{{\hbar^{2}}}}}\right.\kern-1.2pt}{{\hbar^{2}}}}$
and $\beta_{0}$ is the GUP parameter. It is easy to find that the KMM model
predicts a minimal length $\Delta
x_{\min}^{{\rm{KMM}}}\approx{\ell_{p}}\sqrt{{\beta_{0}}}$ cha4 . Subsequently,
incorporating the idea of maximal momentum, Ali, Das and Vagenas constructed
another GUP (ADV model) $\Delta x\Delta
p\geq\frac{\hbar}{2}\left[{1+{{2{\alpha_{0}}{\ell_{p}}\left\langle
p\right\rangle}\mathord{\left/{\vphantom{{2{\alpha_{0}}{\ell_{p}}\left\langle
p\right\rangle}\hbar}}\right.\kern-1.2pt}\hbar}}+{{{4\left\langle{{p^{2}}}\right\rangle\alpha_{0}^{2}\ell_{p}^{2}}\mathord{\left/{\vphantom{{4\left\langle{{p^{2}}}\right\rangle\alpha_{0}^{2}\ell_{p}^{2}}{{\hbar^{2}}}}}\right.\kern-1.2pt}{{\hbar^{2}}}}}\right]$,
where $\alpha_{0}$ is the GUP parameter. Moreover, this linear and quadratic
GUP model suggests the existence of a minimal length $\Delta
x_{\min}^{{\rm{ADV}}}\approx{\alpha_{0}}{\ell_{p}}$ and a maximal momentum
$\Delta
p_{\max}^{{\rm{ADV}}}\approx{{{\ell_{p}}}\mathord{\left/{\vphantom{{{\ell_{p}}}{{\alpha_{0}}}}}\right.\kern-1.2pt}{{\alpha_{0}}}}$cha5
. In recent years, the KMM model and ADV model have played important roles in
the research of many physics contexts. For instance, in Refs. cha5+ ; cha5++ ,
the Scardigli _et al._ , discussed the relation between GUP, general
relativity, and the Lorentz violation. Besides, by using the KMM model,
researchers derived the GUP corrected Hamilton-Jacobi equation and
investigated the modified tunneling rate of particles with arbitrary spins
from the event of curved spacetimes cha6 . Moreover, the KMM model and ADV
model can also be extended to the Proca equation, which leads to the modified
Hawking temperature of black holes cha7 ; cha8 ; cha8+ . In addition, Vagenas
_et al_. studied the validity of the no-cloning theorem within the framework
of GUP. They pointed out that the energy required to send information to a
black hole is affected by quantum gravity cha9 . In Refs. cha10 ; cha10+ , the
authors investigated how the KMM model affects the Casimir wormhole spacetime,
and obtained a class of asymptotically flat wormhole solutions.

Although the KMM model and ADV model are two of the most studied GUPs, they
still have some defects, e. g., the minimal length and the maximal momentum
are only valid for small GUP parameters, and do not imply noncommutative
geometry cha11 . To overcome these difficulties, Pedram introduced a
nonperturbative higher-order GUP, which agrees with various proposals of QG
cha12 . Subsequently, this higher order GUP was used to correct the blackbody
radiation spectrum and predict the cosmological constant. Based on this
heuristic work, many new forms of higher order GUP were presented (see, e. g.
Refs. cha13 ; cha14 ; cha15 ; cha15+  and references therein). Recently, in
Ref. cha16 , Chung and Hassanabadi proposed a generalized canonical
commutation relations as follows:

$\displaystyle[\hat{x},\hat{p}]=\frac{{i\hbar}}{{1-\gamma\left|{\hat{p}}\right|}},\quad\gamma>0$
(1)

which leads to a new higher order GUP (hereafter, we call this GUP as the CH
model):

$\displaystyle\Delta x\Delta p$
$\displaystyle\geq\frac{\hbar}{2}\left\langle{\frac{1}{{1-\gamma\left|p\right|}}}\right\rangle$
$\displaystyle=\frac{\hbar}{2}\left[{-\gamma\left({\Delta
p}\right)+\frac{1}{{1-\gamma\left({\Delta p}\right)}}}\right],$ (2)

where $\left|p\right|=\sqrt{\left|{{p^{2}}}\right|}$ and
$\gamma={{{\gamma_{0}}}\mathord{\left/{\vphantom{{{\gamma_{0}}}{{M_{p}}c}}}\right.\kern-1.2pt}{{M_{p}}c}}={{{\gamma_{0}}\ell_{p}{\rm{}}}\mathord{\left/{\vphantom{{{\gamma_{0}}\ell_{p}{\rm{}}}{{\hbar}}}}\right.\kern-1.2pt}{{\hbar}}}$
are associated with the dimensionless GUP parameter $\gamma_{0}$, the Planck
mass $M_{p}$, and the Planck length ${\ell_{p}}$. In addition, we set
$\left\langle p\right\rangle=0$,
$\left\langle{\left|p\right|}\right\rangle\geq 0$ and utilized the identities
$\left\langle{{{\left({{p^{2}}}\right)}^{n}}}\right\rangle\geq{\left({\left\langle{{p^{2}}}\right\rangle}\right)^{n}}$
and $\left|{\left\langle{\left({AB+BA}\right)}\right\rangle}\right|\geq
2\sqrt{\left\langle{{A^{2}}}\right\rangle}\sqrt{\left\langle{{B^{2}}}\right\rangle}$
cha16 . In addition, Eq. (I) guarantees the existence of an absolute smallest
uncertainty in position
$\Delta{x_{\min}}={{3\hbar\gamma}\mathord{\left/{\vphantom{{3\hbar\gamma}2}}\right.\kern-1.2pt}2}$
for $\Delta
p={1\mathord{\left/{\vphantom{1{2\gamma}}}\right.\kern-1.2pt}{2\gamma}}$,
which never appears in the framework of the HUP. In addition, the CH model has
merit since it involves no perturbation and overcomes some conceptual problems
raised in the previous forms of GUP (e.g. the divergence of the energy
spectrum of the eigenfunctions of the position operator), however, when
$\gamma\rightarrow 0$, it reduces to the HUP. Besides, for $\gamma\ll 1$, the
last term of Eq. (I) can be expanded as $\Delta x\Delta
p\geq\frac{\hbar}{2}\left[{1+{\gamma^{2}}\Delta{p^{2}}+\mathcal{O}\left({{\gamma^{3}}}\right)}\right]$,
which indicates that the CH model coincides with the KMM model to the second
order in $\gamma$; hence, one has the relation ${\gamma^{2}}\sim\beta$.

According to the CH model (I), people investigated the modified eigenfunctions
and eigenvalues for the particle in a box and one-dimensional hydrogen atom,
respectively. Moreover, one may also find that the higher order GUP corrected
the behavior of Bohr-Sommerfeld quantization, which can be used to calculate
the energy spectra of quantum harmonic oscillators and quantum bouncers cha15
. In light of previous works, it is believed that a higher-order GUP will not
only give new implications to quantum systems, but also to astrophysics. For
example, it is well known that the Jeans mass is an important theory to study
the collapse of molecular clouds. However, in recent years, this theory is has
been since some astronomical data have shown that there are some Bok globules,
whose masses are less than the Jeans mass that still have stars or at least
undergo the star formation process cha17 ; cha18 . For solving this problem,
Moradpour _et al_. corrected the limit of Jeans mass due to the KMM model.
Their results showed that the modifications were less than the original case
and were related to the properties of the GUP model cha19 . As we discussed
above, the higher order GUP has different properties from the KMM model and
the ADV model, and it is interesting to investigate how the higher order GUP
affects the Jeans mass. Therefore, in the present paper, we try to extend the
CH model (I) into the Jeans instability and calculate the GUP corrected limit
of Jeans mass of those especial Bok globules.

The paper is organized as follows: Section II is devoted to a review of the
Jeans instability limit in the classical case. In Section III, according to
the new higher order CH model, we compute the corrections to the potential
energy and canonical energy of a molecular cloud. Then, we discuss the
modified Jeans gravitational instability, and then derive the GUP corrected
Jeans mass to explain the problem of Bok globules. By using the GUP corrected
Jeans mass, the dimensionless parameter $\gamma_{0}$ of the GUP is constrained
in Section IV. Finally, conclusions can be found in Section V.

##  II The Jeans mass and the HUP

In this section, we briefly outline how to derive the Jeans mass in the
classical case. Based on the argument that relies upon the virial theorem in
Ref. cha19 , the collapse of the molecular cloud occurs if the gravitational
potential energy $E_{p}$ is larger than the canonical energy $U$, to wit

$\displaystyle
U<-{E_{p}\mathord{\left/{\vphantom{E_{p}2}}\right.\kern-1.2pt}2}.$ (3)

Obviously, to obtain the Jeans mass, it is necessary to derive the expressions
of gravitational potential energy and canonical energy.

It is well known that spacetime is flat in the classical case. Therefore, the
Newton’s law of gravitation is
${F_{0}}={{GMm}\mathord{\left/{\vphantom{{GMm}{{r^{2}}}}}\right.\kern-1.2pt}{{r^{2}}}}$
with the constant of universal gravitation $G$, and the gravitational
potential becomes
$V\left(r\right)=-{{GM}\mathord{\left/{\vphantom{{GM}r}}\right.\kern-1.2pt}r}$.
The corresponding potential energy is given by

$\displaystyle E_{p}$
$\displaystyle=\int_{0}^{M}{V\left(r\right)dr}=\int_{0}^{R}{4\pi\rho\left(r\right)rGMdr}$
$\displaystyle=-\frac{{3G{M^{2}}}}{{5R}},$ (4)

where the density of dark cloud $\rho\left(r\right)$ in Eq. (II) is assumed to
be a constant ${\rho_{0}}$ for simplicity.

Next, the classical fundamental commutation relation can be expressed as
$[\hat{x},\hat{p}]=i\hbar$, which implies HUP $\Delta x\Delta
p\geq{\hbar\mathord{\left/{\vphantom{\hbar 2}}\right.\kern-1.2pt}2}$. When
considering the Liouville theorem and HUP, the state density in momentum space
in the spherical coordinate system can be expressed as
$D\left(p\right)dp={{4\pi V{p^{2}}dp}\mathord{\left/{\vphantom{{4\pi
V{p^{2}}dp}{{h^{3}}}}}\right.\kern-1.2pt}{{h^{3}}}}$, which leads directly to
the original partition function of the ideal gas as follows

$\displaystyle Z$
$\displaystyle=\frac{{\mathcal{Z}^{N}}}{{N!}}=\frac{{{{\left({4\pi
V}\right)}^{N}}}}{{N!}}\int{{{\left[{\frac{{{p^{2}}}}{{{h^{3}}}}\exp\left({-\frac{{{p^{2}}}}{{2\mu{k_{B}}T}}}\right)}\right]}^{N}}dp}$
$\displaystyle=\frac{{{{\left({4\pi
V}\right)}^{N}}}}{{N!}}\frac{{{{\left({2\pi\mu{k_{B}}T}\right)}^{3N/2}}}}{{{h^{3N}}}},$
(5)

where $N$, $\mu$, and $T$ are the numbers, mass, and temperature of
noninteracting particles of the ideal gas, respectively che1 ; cha20 ; cha21 .
Besides, the total mass of the ideal gas satisfies the relationship $M=\mu N$.
According to Eq. (II), one can straightforward to derive the equation of state
of the ideal gas $PV=Nk_{B}T$, and the canonical energy can be expressed as

$\displaystyle U_{0}={k_{B}}{T^{2}}{\left({\frac{{\partial\ln Z}}{{\partial
T}}}\right)_{N,V}}=\frac{3}{2}N{k_{B}}T.$ (6)

Clearly, the original canonical energy is dependent only on the number of
noninteracting particles $N$ and their temperature $T$.

Now, substituting Eq. (II) and Eq. (6) into inequality (3), and supposing that
the distribution of ideal gas in space is spherical, the result is

$\displaystyle
M>{\left({\frac{{5{k_{B}}T}}{{G\mu}}}\right)^{\frac{3}{2}}}{\left({\frac{3}{{4\pi{\rho_{0}}}}}\right)^{\frac{1}{2}}},$
(7)

where the radius is
$R={\left({{{3M}\mathord{\left/{\vphantom{{3M}{4\pi{\rho_{0}}}}}\right.\kern-1.2pt}{4\pi{\rho_{0}}}}}\right)^{\frac{1}{3}}}$.
The above mentioned equation indicates a lower bound of the cloud mass to
collapse, which is known as the Jeans mass

$\displaystyle
M_{0}^{J}={\left({\frac{{5{k_{B}}T}}{{G\mu}}}\right)^{\frac{3}{2}}}{\left({\frac{3}{{4\pi{\rho_{0}}}}}\right)^{\frac{1}{2}}}.$
(8)

Although the formation of most stars can be examined by the original Jeans
instability and Jeans mass, people still observed some Bok globules, such as
CB 84 and CB 110, have masses less than ${M^{J}_{0}}$ but still have stars or
at least undergo the star formation process cha18 . To explain this problem,
we investigate the modified Jeans mass in the framework of the higher-order
GUP.

##  III The Jeans mass in the higher-order GUP

The GUP has various implications for a wide range of physical systems.
Moreover, they are regarded as a powerful tool to solve various difficult
problems in these research fields. Therefore, to obtain the modified Jeans
mass, we derive the GUP corrected entropy, gravitation potential energy, and
canonical energy of ideal gases. For the sake of simplicity, we shall takes
the units $\hbar=c=k_{B}=1$ in the next research.

###  III.1 GUP corrected gravitation potential energy

To derive the modified gravitation potential energy in the presence of the CH
model, we need to use the theory of entropic force, which is an intriguing
explanation for Newton’s law of gravity that is based on the holographic
principle and an equipartition rule. Now, by using Eq. (I), one can easily
obtain the following inequality

$\displaystyle\Delta p$
$\displaystyle\geq\frac{1}{{2\gamma}}\left({1-\sqrt{1-\frac{{4\gamma}}{{2\Delta
x+\gamma}}}}\right)$ $\displaystyle=\frac{1}{{2\Delta
x}}\left[{1+\frac{{{\gamma^{2}}}}{{4\Delta{x^{2}}}}+\mathcal{O}\left({{\gamma^{3}}}\right)}\right].$
(9)

Based on the arguments of Refs. cha23 ; chd1 ; chd2 ; chd3 ; chd3+ , a
massless quantum particle (e.g., a photon) can be used to determine the
position of quantum particles with the energy $\omega=pc$ (i.e.
$\Delta\omega=\Delta pc$), so that the HUP $\Delta
p\geq{1\mathord{\left/{\vphantom{1{2\Delta x}}}\right.\kern-1.2pt}{2\Delta
x}}$ can be translated to the lower bound
$\omega\geq{1\mathord{\left/{\vphantom{1{2\Delta
x}}}\right.\kern-1.2pt}{2\Delta x}}$. Accordingly, from Eq. (III.1), the GUP
version reads

$\displaystyle\omega\geq\frac{1}{{2\Delta
x}}\left[{1+\frac{{{\gamma^{2}}}}{{4\Delta{x^{2}}}}+\mathcal{O}\left({{\gamma^{3}}}\right)}\right].$
(10)

From Verlinde’s entropy force theory, to calculate the modified entropic
force, one should consider a spherically symmetric gravitational system (e.g.,
a black hole), which allows the quantum particles to enter or exit its horizon
cha28 . As Ref. chy1  pointed out, when a strong gravitational system absorbs
or emits a quantum particle with energy $\omega$ and size $\mathcal{R}$, the
minimal change in the area of the system becomes $\Delta{A_{\min}}\geq
8\pi\omega\mathcal{R}\ell_{p}^{2}$. Then, the arguments of Refs. cha24 ; chy2
implies that the size of a quantum particle cannot be smaller than its
uncertainty in the position (i.e. $\mathcal{R}\geq\Delta x$), which gives
$\Delta{A_{\min}}\geq 8\pi\omega\Delta x\ell_{p}^{2}$. Substituting this
relation into Eq. (10), one has

$\displaystyle\Delta{A_{\min}}\geq
4\pi\ell_{p}^{2}\left[{1+\frac{{{\gamma^{2}}}}{{4\Delta{x^{2}}}}+\mathcal{O}\left({{\gamma^{3}}}\right)}\right].$
(11)

Furthermore, considering that a spherical gravitational system with
Schwarzschild radius $r$, and the area of this system is $A=4\pi{r^{2}}$.
Moreover, near the event horizon, $\Delta x$ is approximately equal to the
orbital radius $r$; that is, $\Delta x\approx 2r$ cha25 ; cha26 ; chy3 ; chy4
; chy4+ . Hence, the relation between $\Delta x$ and $A$ is given by
$\Delta{x^{2}}=4{r^{2}}={A\mathord{\left/{\vphantom{A\pi}}\right.\kern-1.2pt}\pi}$.
Substituting this relation into Eq. (11), the minimal change in the horizon
area of the gravitational system can be recast as

$\displaystyle\Delta{A_{\min}}\simeq\lambda\ell_{p}^{2}\left[{1+\frac{{\pi{\gamma^{2}}}}{{4A}}+\mathcal{O}\left({{\gamma^{3}}}\right)}\right],$
(12)

where $\lambda$ is an undetermined coefficient. In Ref. cha24 , the authors
pointed out that the information of the gravitational system is reflected in
its area. On the other hand, based on the information theory, it is believed
that the area of one system can affect its smallest increase in entropy cha27
. Since the fundamental unit of entropy as one bit of information is
$\Delta{S_{\min}}=b=\ln 2$, one obtains

$\displaystyle\frac{{{\rm{d}}S}}{{{\rm{d}}A}}=\frac{{\Delta{S_{\min}}}}{{\Delta{A_{\min}}}}={\rm{}}\frac{b}{{\lambda\ell_{p}^{2}}}{\left[{1+\frac{{\pi{\gamma^{2}}}}{{4A}}+\mathcal{O}\left({{\gamma^{3}}}\right)}\right]^{-1}},$
(13)

Considering that $\gamma$ is a small parameter, one can obtain the GUP
corrected entropy according to Eq. (13),

$\displaystyle
S=\frac{{A}}{{4\ell_{p}^{2}}}\left[{1-\frac{{\pi{\gamma^{2}}}}{{4A}}\ln\left({4A}\right)+\mathcal{O}\left({{\gamma^{3}}}\right)}\right],$
(14)

where we fix
${b\mathord{\left/{\vphantom{b\lambda}}\right.\kern-1.2pt}\lambda}$ by
demanding matching the original entropy-area law in the limit $\gamma\to 0$.
Hence, one has
${b\mathord{\left/{\vphantom{b\lambda}}\right.\kern-1.2pt}\lambda}={{1}\mathord{\left/{\vphantom{{k_{B}}\pi}}\right.\kern-1.2pt}4}$
chy1 . Moreover, there is a logarithmic term $\ln\left({4A}\right)$ in the
bracketed term of Eq. (14), which is coincident with the requirements of QG
chb5 ; chb6 ; chb7 ; chf1 ; chf2 ; chb8 ; chg1 ; chg2 . According to the
holographic principle and entropy-area law, which is averagely distributed in
$\mathcal{N}$-bits information, the number of bits can be expressed as
follows:

$\displaystyle\mathcal{N}=4S=\frac{A}{{\ell_{p}^{2}}}\left[{1-\frac{{\pi{\gamma^{2}}}}{{4A}}\ln\left({4A}\right)+\mathcal{O}\left({{\gamma^{3}}}\right)}\right].$
(15)

Next, one can further denote the total energy of the gravitational system as
$E$, which is the average distribution in $\mathcal{N}$ bits (please see the
chg3  for details). Consequently, each bit contains
${{T}\mathord{\left/{\vphantom{{{k_{B}}T}2}}\right.\kern-1.2pt}2}$ energy,
then, following the equipartition rule, the total energy takes the form

$\displaystyle
E={{\mathcal{N}T}\mathord{\left/{\vphantom{{\mathcal{N}T}2}}\right.\kern-1.2pt}2}.$
(16)

On the other hand, Verlinde demonstrated that the entropic force is more
fundamental than gravity. For a gravitational system, the entropic force can
be expressed as follows cha28 :

$\displaystyle F\Delta x=T\Delta S,$ (17)

where $F$ is the entropy force, $T$ is the temperature, $\Delta S$ is the
change in entropy of the gravitational system, and $\Delta x$ represents the
displacement of the particle with the mass $m$ from the gravitational system,
which satisfies the relation $\Delta S=2\pi m\Delta x$ chb2 ; chb3 . Now,
substituting the expression of the change in entropy, Eq. (14)-Eq. (16) into
Eq. (17), then considering $E=M{c^{2}}$, $A=4\pi{r^{2}}$ and $\ell_{p}^{2}=G$,
Newton’s law of gravitation should be corrected as follows:

$\displaystyle{F_{{\rm{GUP}}}}=\frac{{GMm}}{{{r^{2}}}}\left[{1+\frac{{{\gamma^{2}}}}{{16{r^{2}}}}\ln\left({16\pi{r^{2}}}\right)+\mathcal{O}\left({{\gamma^{3}}}\right)}\right].$
(18)

Clearly, in the limit $\gamma=0$, Eq. (18) reduce to the original Newton’s
gravitation
${F_{0}}={{GMm}\mathord{\left/{\vphantom{{GMm}{{r^{2}}}}}\right.\kern-1.2pt}{{r^{2}}}}$.
Next, based on the modified Newton’s law of gravitation, the corresponding GUP
corrected gravitational potential is

$\displaystyle{V_{{\rm{GUP}}}}$
$\displaystyle=\int{\frac{{{F_{{\rm{GUP}}}}}}{m}dr}$
$\displaystyle=-\frac{{GM}}{r}\left[{1+\frac{{2+3\ln\left({16\pi{r^{2}}}\right)}}{{144{r^{2}}}}{\gamma^{2}}+\mathcal{O}\left({{\gamma^{3}}}\right)}\right].$
(19)

Applying Eq. (III.1) to a molecular cloud with radius $R$, mass $M$ and the
almost uniform density $\rho_{0}$, the following expression for the modified
potential energy is obtained

$\displaystyle E_{p}^{{\rm{GUP}}}$
$\displaystyle=\int_{0}^{R}{{V_{{\rm{GUP}}}}\left(r\right)dM}$
$\displaystyle=-\frac{{3G{M^{2}}}}{{5R}}\left[{1+\frac{{5\ln\left({16\pi{R^{2}}}\right)}}{{144{R^{2}}}}{\gamma^{2}}+\mathcal{O}\left({{\gamma^{3}}}\right)}\right].$
(20)

Obviously, the above equation is related to the mass $M$, radius $R$, and GUP
parameter $\gamma$. In addition, one can see that Eq. (III.1) lacks the first-
order correction term since the CH model (I) becomes the KMM model for small
$\gamma$. This indicates that the properties of the gravitational system are
affected by the higher-order term of $\gamma$. In the limit $\gamma=0$, the
result agrees with the original potential energy (II).

###  III.2 GUP corrected canonical energy of ideal gases

By virtue of Eq. (I) and the viewpoint in Refs. chb3+ ; chc11 , the partition
function of a dark cloud, which is approximated as the ideal gas containing
$N$ noninteracting particles with mass $m$ at temperature $T$, can be
expressed as follows:

$\displaystyle Z=\frac{{{\mathcal{Z}^{N}}}}{{N!}},$ (21)

where

$\displaystyle\mathcal{Z}=\frac{{4\pi
V}}{{N!}}\int_{0}^{\infty}{\frac{{{p^{2}}}}{{{h^{3}}}}{{\left({1-3\gamma\left|p\right|}\right)}^{3}}\exp\left({-\frac{{{p^{2}}}}{{2\mu
T}}}\right)dp}.$ (22)

Using the spherical coordinate and Gaussian integral, the GUP corrected
partition function for the dark cloud reads

$\displaystyle{Z_{{\rm{GUP}}}}=\frac{{{{\left({4\pi
V}\right)}^{N}}{{\left({2{k_{B}}\mu
T}\right)}^{3N/2}}}}{{N!{h^{3N}}}}{\Theta^{N}}\left(\gamma\right),$ (23)

where
$\Theta\left(\gamma\right)=\frac{{\sqrt{\pi}}}{4}\left[{1-6\sqrt{\frac{{2\mu
T}}{\pi}}\gamma+9\mu
T{\gamma^{2}}+\mathcal{O}\left({{\gamma^{3}}}\right)}\right]$. The GUP
corrected canonical energy is presented as

$\displaystyle U_{{\rm{GUP}}}=$
$\displaystyle{T^{2}}{\left({\frac{{\partial\ln{Z_{{\rm{GUP}}}}}}{{\partial
T}}}\right)_{N,V}}$ $\displaystyle=$
$\displaystyle{U_{0}}\left[{1-2\sqrt{\frac{2}{\pi}}{\cal
A}\gamma-\frac{6}{\pi}\left({4-\pi}\right){{\cal A}^{2}}{\gamma^{2}}+{\cal
O}\left({{\gamma^{3}}}\right)}\right],$ (24)

with $\mathcal{A}=\sqrt{\mu T}$ and the original canonical energy ${U}_{0}$ is
given in Eq. (6). Note that, the modified canonical energy is different from
the original case, it does only not relate to the original case $U_{0}$, but
also to parameter $\gamma$, the mass of the noninteracting particle $\mu$ and
its temperature $T$.

###  III.3 GUP corrected Jeans mass

With the above results in place, one can derive the modified Jeans mass and
further investigate how the GUP affects the Jeans mass. Using Eq. (III.1) and
Eq. (III.2) and recalling Eq. (3), one yields

$\displaystyle NT$ $\displaystyle\left[{1-2\sqrt{\frac{2}{\pi}}{\cal
A}\gamma-\frac{6}{\pi}\left({4-\pi}\right){{\cal A}^{2}}{\gamma^{2}}}{+{\cal
O}\left({{\gamma^{3}}}\right)}\right]<$
$\displaystyle\frac{{G{M^{2}}}}{{5R}}\left[{1+\frac{{5\ln\left({16\pi{R^{2}}}\right)}}{{144{R^{2}}}}{\gamma^{2}}+\mathcal{O}\left({{\gamma^{3}}}\right)}\right].$
(25)

Considering the radius
$R={\left({{{3M}\mathord{\left/{\vphantom{{3M}{4\pi{\rho_{0}}}}}\right.\kern-1.2pt}{4\pi{\rho_{0}}}}}\right)^{\frac{1}{3}}}$,
Eq. (III.3) can be rewritten as follows:

$\displaystyle\frac{{5T\left[{1-2\sqrt{\frac{2}{\pi}}{\cal
A}\gamma-\frac{6}{\pi}\left({4-\pi}\right){{\cal A}^{2}}{\gamma^{2}}}{+{\cal
O}\left({{\gamma^{3}}}\right)}\right]}}{{G\mu{{\left({{{4\pi{\rho_{0}}}\mathord{\left/{\vphantom{{4\pi{\rho_{0}}}3}}\right.\kern-1.2pt}3}}\right)}^{\frac{1}{3}}}}}<{M^{\frac{2}{3}}}\chi,$
(26)

where
$\chi={1+\frac{{5{\gamma^{2}}}}{{144{{\left({{{3M}\mathord{\left/{\vphantom{{3M}{4\pi{\rho_{0}}}}}\right.\kern-1.2pt}{4\pi{\rho_{0}}}}}\right)}^{2}}}}\ln\left[{16\pi{{\left({{{3M}\mathord{\left/{\vphantom{{3M}{4\pi{\rho_{0}}}}}\right.\kern-1.2pt}{4\pi{\rho_{0}}}}}\right)}^{2}}}\right]+\mathcal{O}\left({{\gamma^{3}}}\right)}$.
On the one hand, if $\gamma=0$, the upper bound of mass goes to the Jeans mass
$M_{\rm{0}}^{J}={\left({{{5T}\mathord{\left/{\vphantom{{5T}{G\mu}}}\right.\kern-1.2pt}{G\mu}}}\right)^{\frac{3}{2}}}{\left({{3\mathord{\left/{\vphantom{3{4\pi{\rho_{0}}}}}\right.\kern-1.2pt}{4\pi{\rho_{0}}}}}\right)^{\frac{1}{2}}}$.
On the other hand, because $M\gg 1$ and $\gamma$ is finite, the GUP corrected
Jeans mass is obtained by saturating Eq. (26), with the result

$\displaystyle M_{{\rm{GUP}}}^{J}=M$
$\displaystyle>M_{\rm{0}}^{J}{\left[{1-\sqrt{\frac{{8\mu
T}}{\pi}}\gamma-\left({4-\pi}\right)\frac{{6\mu
T}}{\pi}{\gamma^{2}}+\mathcal{O}\left({{\gamma^{3}}}\right)}\right]^{\frac{3}{2}}}.$
(27)

One may see that the GUP corrected Jeans mass $M_{{\rm{GUP}}}^{J}$ is related
to the original Jeans mass $M_{0}^{J}$, the mass of ideal gas $\mu$, the
temperature of gravity system $T$, and the GUP parameter $\gamma$.
Furthermore, it should be noted that the mass of ideal gas and the temperature
of the gravity system must be real numbers greater than zero. When considering
$\gamma>0$, $M_{{\rm{GUP}}}^{J}$ is positive when $1>\sqrt{{{8\mu
T}\mathord{\left/{\vphantom{{8\mu
T}\pi}}\right.\kern-1.2pt}\pi}}\gamma-\left({4-\pi}\right){{6\mu
T{\gamma^{2}}}\mathord{\left/{\vphantom{{6\mu
T{\gamma^{2}}}\pi}}\right.\kern-1.2pt}\pi}+{\cal
O}\left({{\gamma^{3}}}\right)$ is lower than $M_{\rm{0}}^{J}$. To date, the
astronomical observations have shown that some Bok globules’ masses are less
than their corresponding Jeans masses. The GUP corrections to the Jeans mass
can be a candidate to explain those observational facts. Our results show that
the effect of GUP is able to reduce the Jeans mass, which is conducive to the
generation of stars in small-mass Bok globules.

##  IV Constraints for GUP parameter $\gamma_{0}$

The GUP parameters are always assumed to be of order unity so that the
modified results are negligible unless energy approaches the Planck scale.
However, if the assumptions regarding the GUP parameters are not considered,
the bound of GUP parameters can be obtained by previous experimental and
observational data. In previous works, people mainly focused on the
constraints on the GUP parameters of the KMM model and the ADD model chc1 ;
chc2 ; chc3 ; chc4 ; chc5 ; chc5+ ; chy5 ; chc10 ; chc7 ; cha30 ; chc6 ; chc8
; cha31 ; chc9 ; chc10+ ; chy10 ; chy11 . Now, armed with the previous
results, we estimate the parameter of the CH model based on the data of Bok
globules. First, for simplicity, by keeping the leading order term of
$\gamma$, inequality (III.3) can be written as follows

$\displaystyle M_{{\rm{GUP}}}^{J}>M_{\rm{0}}^{J}{\left({1-\sqrt{\frac{{8\mu
T}}{\pi}}\frac{\gamma_{0}}{{{M_{p}}c}}}\right)^{\frac{3}{2}}}.$ (28)

By solving the above equation, one yields

$\displaystyle\gamma_{0}<\frac{{c{M_{p}}}}{2}\sqrt{\frac{\pi}{{2\mu
T}}}\left[{1-{{\left({\frac{{M_{{\rm{GUP}}}^{J}}}{{M_{\rm{0}}^{J}}}}\right)}^{2/3}}}\right].$
(29)

Now, based on the data of Bok globules in Ref. cha17 , the upper bounds on
$\gamma_{0}$ are shown in Table. 1.

Table 1:  The ID, temperature $T$, mass $M_{{\rm{GUP}}}^{J}$ and Jeans mass ${M_{0}^{J}}$ of Bok globules, $\gamma_{0}$ and $\gamma_{0}^{2}$ are the first power and second power of the upper bound on the GUP parameter of the CH model, $\beta_{0}^{\rm{KMM}}$ is the upper bound on the GUP parameter of the KMM model. ID |  $T({\rm{K}})$cha17  |  $M_{{\rm{GUP}}}^{J}({{{\rm{M}}_{\odot}}})$ cha17  |  ${M_{0}^{J}}({{{\rm{M}}_{\odot}}})$ cha17  | $\gamma_{0}$ | ${\gamma_{0}^{2}}$ |  $\beta_{0}^{\rm{KMM}}$ cha19
---|---|---|---|---|---|---
CB 87 | 11.4 | 2.73 | 9.6 | $5.31\times{10^{12}}$ | $2.82\times{10^{24}}$ | $6.33\times{10^{25}}$
CB 110 | 21.8 | 7.21 | 8.5 | $7.03\times{10^{11}}$ | $5.00\times{10^{23}}$ | $6.11\times{10^{24}}$
CB 131 | 25.1 | 7.83 | 8.1 | $1.41\times{10^{11}}$ | $1.99\times{10^{22}}$ | $1.02\times{10^{24}}$
CB 161 | 12.5 | 2.79 | 5.4 | $3.18\times{10^{12}}$ | $1.01\times{10^{25}}$ | $3.62\times{10^{26}}$
CB 188 | 19.0 | 7.19 | 7.7 | $3.24\times{10^{11}}$ | $1.05\times{10^{23}}$ | $3.05\times{10^{24}}$
FeSt 1-457 | 10.9 | 1.12 | 1.4 | $1.32\times{10^{12}}$ | $1.75\times{10^{24}}$ | $1.60\times{10^{25}}$
Lynds 495 | 12.6 | 2.95 | 6.6 | $1.70\times{10^{12}}$ | $1.37\times{10^{25}}$ | $4.20\times{10^{25}}$
Lynds 498 | 11.0 | 1.42 | 5.7 | $5.76\times{10^{12}}$ | $3.31\times{10^{25}}$ | $6.99\times{10^{25}}$

In Table. 1, we set $M_{p}=2.18\times{10^{-8}}{\rm kg}$,
$c=2.99\times{10^{8}}{\rm{m}}\cdot{{\rm{s}}^{-1}}$, and ${{\rm{M}}_{\odot}}$
denotes the Sun mass. The final results are determined by the mass of
noninteracting particles $\mu$. To obtain the exact value of the upper bounds
on $\gamma_{0}$, we further assume that the noninteracting particles are
composed of hydrogen atoms, which are the most abundant element in the
universe with the mass ${\mu_{\rm{H}}}=1.67\times{10^{-26}}{\rm kg}$. In this
case, one can see that the range of upper bounds of $\gamma_{0}$ is
${10^{11}}\sim{10^{12}}$. Furthermore, the GUP parameter in the CH model has a
special relationship with that of the KMM model, which can be inferred from
Eq. (I), as ${\gamma^{2}}\sim\beta$ (or $\gamma_{0}^{2}\sim{\beta_{0}}$).
Hence, it is easy to find that the upper bound of the GUP parameter in the CH
model $\gamma_{0}^{2}$ is 1-2 orders of magnitude more stringent than those in
the KMM model
$\beta_{0}^{\rm{KMM}}<{{\left[{1-{{\left({{{M_{GUP}^{J}}\mathord{\left/{\vphantom{{M_{GUP}^{J}}{M_{0}^{J}}}}\right.\kern-1.2pt}{M_{0}^{J}}}}\right)}^{{2\mathord{\left/{\vphantom{23}}\right.\kern-1.2pt}3}}}}\right]M_{p}^{2}c^{2}}\mathord{\left/{\vphantom{{\left[{1-{{\left({{{M_{GUP}^{J}}\mathord{\left/{\vphantom{{M_{GUP}^{J}}{M_{0}^{J}}}}\right.\kern-1.2pt}{M_{0}^{J}}}}\right)}^{{2\mathord{\left/{\vphantom{23}}\right.\kern-1.2pt}3}}}}\right]M_{p}^{2}c^{2}}{2\mu
T}}}\right.\kern-1.2pt}{2\mu T}}$ from Ref. cha19 .

##  V Discussion

In this paper, by incorporating a new higher-order GUP with virial equilibrium
and Verlinde’s entropy force theory, we investigated the modified canonical
energy $U_{{\rm{GUP}}}$ and the modified gravitational potential energy
$E_{p}^{{\rm{GUP}}}$ of a molecular cloud. Subsequently, according to those
modifications, the GUP corrected Jeans mass $M_{{\rm{GUP}}}^{J}$ is obtained.
It is found that the GUP can effectively increase the gravitational potential,
and reduce the canonical energy and the Jeans mass. This leads to the collapse
of Bok globules with masses less than the standard value $M_{\rm{0}}^{J}$,
which is consistent with astronomical observations. Finally, using the
different data of Bok globules, we constrain the upper bounds of the GUP
parameter $\gamma_{0}$, whose range turn out to be ${10^{11}}\sim{10^{12}}$.

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