Flip Graphs
Roy Harold Jennings
Department of Mathematics
Ph.D. Thesis
Submitted to the Senate of Bar-Ilan University

Ramat-Gan, Israel January 2020

  * This work was carried out under the guidance and supervision of Prof. Ron Adin and Prof. Yuval Roichman, Department of Mathematics, Bar-Ilan University.

### Acknowledgements

> “Perhaps we’ll have some answers, at least, before the end. I always dreamed
> of dying well-informed.”
>
> —Joe Abercrombie, Last Argument of Kings

I would like to express my gratitude to my advisors Prof. Ron M. Adin and
Prof. Yuval Roichman for their continuous support of my Ph.D. study, for their
patience and immense knowledge. Their encouragement and guidance helped me
build the confidence and courage needed for the fail and retry iterations I
went through during the research and writing of this thesis.

I would also like to thank my friends and fellow researchers Dr. Menachem
(Meny) Shlossberg and Dr. Arnon Netzer for their support and encouragement and
for many helpful discussions and suggestions.

Special thanks go to my parents Roger and Hava Ben-Ari for supporting me
throughout the writing of this thesis.

Last, but by no means least, I would like to thank Luie Jennings, my best
friend and partner for life, for being a source of energy, motivation and
inspiration; for being so supportive and understanding; but most of all, for
always putting things in proportions, for her lifesaving sense of humor and
her love.

###### Contents

  1. Abstract
  2. 1 Introduction
    1. 1.1 Main Results
    2. 1.2 The Structure of the Thesis
  3. 2 Mathematical Background
    1. 2.1 Graphs
      1. 2.1.1 Cayley and Schreier Graphs
      2. 2.1.2 Flip Graphs
    2. 2.2 Posets
      1. 2.2.1 Lattices
      2. 2.2.2 The Dominance Order on $\mathbb{Z}^{n}$
  4. 3 The Yoke Graphs $\mathrm{Y}_{n,m}$
    1. 3.1 Basic Definitions
    2. 3.2 Important Examples of Yoke Graphs
    3. 3.3 Group Actions on $\mathrm{Y}_{n,m}$
  5. 4 The Eccentricity of $0$ in $\mathrm{Y}_{n,m}$
    1. 4.1 The Shift Direction Lemma
    2. 4.2 Pivots and Walls
    3. 4.3 Computation of the Eccentricity
  6. 5 The Diameter of $\mathrm{Y}_{n,m}$
    1. 5.1 Proof for $m\leq n$ via the Dominance Order
    2. 5.2 The dYoke Graphs $\mathrm{Z}_{n,m}$
    3. 5.3 From Diameter to Eccentricity
    4. 5.4 The Eccentricity of $0$ in $\mathrm{Z}_{n,m}$
      1. 5.4.1 Pivots and Walls
      2. 5.4.2 Computation of the Eccentricity
  7. 6 Automorphisms of $\mathrm{Y}_{n,m}$
    1. 6.1 Fundamental Automorphisms
    2. 6.2 The Automorphism Group of $\mathrm{Y}_{n,m}$

### Abstract

Flip graphs are graphs on combinatorial objects in which the adjacency
relation reflects a local change in the underlying objects. In this thesis we
introduce Yoke graphs, a family of flip graphs that generalizes previously
studied families of flip graphs on colored triangle-free triangulations [2],
arc permutations [5] and geometric caterpillars [10]. Our main results are the
computation of the diameter of an arbitrary Yoke graph and a full
characterization of the automorphism group of this family of graphs. We also
show that Yoke graphs are Schreier graphs of the affine Weyl group of type
$\tilde{C}_{m}$.

The approach we take in the computation of the diameter is different from the
ones in [2] and [5]. We show that the approach in [5] (for arc permutation
graphs) does not extend to Yoke graphs. At the heart of our proof lies the
idea of transforming a diameter evaluation into an eccentricity problem. The
characterization of the automorphism group is a new result even for the above
mentioned three special families of Yoke graphs.

##  Chapter 1 Introduction

Over recent decades, there has been an increasing interest in flip graphs,
namely, graphs on combinatorial objects in which the adjacency relation
reflects a local change. For example, see [4, 6, 13, 14, 16]. In this work we
introduce Yoke graphs, a family of flip graphs that generalizes previously
studied families of flip graphs on colored triangle-free triangulations (CTFT)
[2], arc permutations [5] and geometric caterpillars [10].

Typical problems related to flip graphs include metric properties, such as
distance, diameter, finding antipodes and counting geodesics between them; and
algebraic properties, such as presentations using Cayley/Schreier graphs,
automorphism groups and eigenvalues. The generalization to Yoke graphs of the
three families of flip graphs mentioned above is motivated by their surprising
similarities in terms of algebraic, combinatorial and metric properties. In
particular, they carry similar group actions, are intimately related to posets
and have similar diameter formulas.

###  1.1 Main Results

In this thesis, as mentioned above, we introduce a family of flip graphs,
namely, Yoke graphs $\mathrm{Y}_{n,m}$. We compute the diameter of Yoke graphs
and prove (see Theorems 4.0.1, 5.3.5 and 5.4.33):

######  Theorem.

  1.   2. 1.

If $n=1$, then
$\operatorname{diam}(\mathrm{Y}_{n,m})=\binom{\lceil\frac{m}{2}\rceil+1}{2}+\binom{\lfloor\frac{m}{2}\rfloor+1}{2}$.

  3. 2.

If $0\leq m\leq n$, then
$\operatorname{diam}(\mathrm{Y}_{n,m})=\lfloor\frac{n(m+1)}{2}\rfloor$.

  4. 3.

If $2\leq n\leq m$, let
$d^{0}_{n,m}=\binom{\lfloor\frac{m+n}{2}\rfloor+1}{2}+\binom{\lceil\frac{m-n}{2}\rceil+1}{2}$.
Then

    1. (a)

if either $2\divides(m-n)$ or $n\leq\lceil\frac{m+1}{2}\rceil$, then
$\operatorname{diam}(\mathrm{Y}_{n,m})=d^{0}_{n,m}$;

    2. (b)

otherwise,
$\operatorname{diam}(\mathrm{Y}_{n,m})=d^{0}_{n,m}+n-\lceil\frac{m+1}{2}\rceil$.

We also give a full characterization of the automorphism group of this family
of graphs (for all values of $n$ and $m\neq 2$). For example (see Theorems
6.1.4 and 6.2.1 for the complete result):

######  Theorem.

Let $n\geq 1$ and $m\geq 3$ be two integers such that $(n,m)\neq(1,3)$.

  1. 1.

$\operatorname{Aut}(\mathrm{Y}_{1,m})\cong C_{2}\times C_{2}$ ($\forall m>3$).

  2. 2.

If $n>1$ and at least one of $n$ and $m$ is odd, then
$\operatorname{Aut}(\mathrm{Y}_{n,m})\cong D_{2n}$.

  3. 3.

If $n>1$ and both $n$ and $m$ are even, then
$\operatorname{Aut}(\mathrm{Y}_{n,m})\cong D_{n}\times C_{2}$.

Since Yoke graphs generalize the three families of flip graphs mentioned
above, these results provide a unified proof for known results regarding
diameter. Our approach in this proof is different from the ones in [2] and
[5]. At the heart of the proof lies the idea of transforming a diameter
evaluation into an eccentricity problem. The characterization of the
automorphism group is a new result for the mentioned three cases of Yoke
graphs.

###  1.2 The Structure of the Thesis

In Chapter 2, we recall and develop the theory that will be necessary for the
presentation of our main results. Chapter 3 introduces Yoke graphs, the main
object in this thesis. As noted in the introduction, the definition of Yoke
graphs was motivated by similarities between CTFT graphs, arc permutation
graphs and caterpillar graphs, in terms of their algebraic, combinatorial and
metric properties. In Section 3.2, we show that these graphs are indeed
special cases of Yoke graphs. In Section 3.3, the fact that CTFT graphs and
arc permutation graphs are Schreier graphs of the affine Weyl group of type
$\tilde{C}_{n}$ is extended to Yoke graphs.

In Chapter 4, we compute the eccentricity of the vertex $0$ in Yoke graphs.
Chapter 5 deals with the diameter of Yoke graphs. In Section 5.1, we develop
techniques and obtain partial results on the diameter, based on similarities
between Yoke graphs and the dominance order on $\mathbb{Z}^{n}$. In Sections
5.2, 5.3 and 5.4, we show that the problem of the diameter in Yoke graphs can
be converted into a problem of eccentricity in larger graphs, namely, dYoke
graphs. The terminology and techniques that were introduced in Chapter 4 are
extended and used in order to compute the eccentricity of $0$ in dYoke graphs.

Chapter 6 deals with the automorphism group of Yoke graphs. In Section 6.1,
three fundamental automorphisms of $\mathrm{Y}_{n,m}$ are introduced and we
compute the group they generate. In Section 6.2, we show that this group is in
fact the entire automorphism group of $\mathrm{Y}_{n,m}$ whenever $m\neq 2$
and $(n,m)\neq(1,3)$. We cover the case $(n,m)=(1,3)$ separately.

##  Chapter 2 Mathematical Background

In this chapter, we recall and develop the theory that will be necessary for
the presentation of our main results.

###  2.1 Graphs

A graph $G$ on a vertex set $V$ is an ordered pair $(V,E)$ where $E\subseteq
V\times V$ is a binary relation on $V$. $E$ is called the adjacency relation
of $G$ and its elements are called edges. Accordingly, if $(u,v)\in E$, then
$u$ and $v$ are said to be adjacent vertices or neighbors in $G$. An edge
$(u,u)\in E$ is called a loop. If the relation $E$ is symmetric, then we say
that $G$ is an undirected graph and denote adjacent vertices $u$ and $v$ by
$u\sim v$. Otherwise, we say that $G$ is a directed graph or a digraph. A
simple graph $G$ is an undirected graph without loops. In the rest of this
work, the term graph denotes a simple graph on a finite set of vertices
(directed and infinite graphs will be explicitly stated as such).

A graph $G^{\prime}=(V^{\prime},E^{\prime})$ is a subgraph of $G=(V,E)$,
denoted $G^{\prime}\leq G$, if $V^{\prime}\subseteq V$ and
$E^{\prime}\subseteq E$. A subgraph $G^{\prime}$ is said to be an induced
subgraph of $G$ if every edge $e$ in $G$ such that $e\in V^{\prime}\times
V^{\prime}$ is an edge in $G^{\prime}$. A complete graph is one in which all
of the vertices are adjacent. The complete graph on $n$ vertices is denoted by
$K_{n}$.

A path $P=(v^{0}\sim\dots\sim v^{d})$ between $v^{0}$ and $v^{d}$ in a
(simple) graph $G$ is a sequence $v^{0},\dots,v^{d}$ of vertices of $G$ such
that $v^{i-1}\sim v^{i}$ for every $1\leq i\leq d$. The path is a cycle if
$v^{0}=v^{d}$. Here, $d$ is called the length of $P$. $G$ is said to be
connected if every two vertices in $G$ are connected by a path. A geodesic (or
shortest path) between two vertices $u$ and $v$ in $G$ is a path with minimum
length between them. The distance $d_{G}(v,u)$ between $v$ and $u$ is the
length of a geodesic between them. We write $d(v,u)$ when $G$ is evident. The
diameter $\operatorname{diam}(G)$ of $G$ is the maximal distance between any
two vertices in $G$. Two vertices in $G$ are said to be antipodes if the
distance between them is equal to the diameter of the graph. The eccentricity
$\operatorname{ecc}_{G}(v)$ of a vertex $v$ in a graph $G$ is the maximal
distance of the vertex from any other vertex in $G$. We write
$\operatorname{ecc}(v)$ when $G$ is evident. A vertex $u$ is said to be
eccentric to the vertex $v$ if $d(v,u)=ecc(v)$. The valency $\delta_{G}(v)$ of
a vertex $v$ in a graph $G$ is the number of neighbors of $v$ in $G$.

An isomorphism of graphs $G$ and $H$ is a bijection $\varphi$ between the
vertex sets of $G$ and $H$ such that any two vertices $u$ and $v$ of $G$ are
adjacent in $G$ if and only if $\varphi(u)$ and $\varphi(v)$ are adjacent in
$H$. An isomorphism of a graph with itself is called an automorphism. The set
of automorphisms of a graph $G$, with respect to composition, forms a subgroup
$\operatorname{Aut}(G)$ of the symmetric group on $n$ elements, where $n$ is
the number of vertices in the graph. We call this group the automorphism group
of the graph. Throughout this work, composition of functions is computed
right-to-left: $(gf)(x)=g(f(x))$.

####  2.1.1 Cayley and Schreier Graphs

Let $G$ be a group and let $S$ be a symmetric ($S^{-1}=S$) generating set of
$G$. The (right) Cayley graph $X(G,S)$ is the graph on the set of elements of
$G$ in which $g_{1}\sim g_{2}$ if $g_{2}=g_{1}s$ for some $s$ in $S$. Let $H$
be a subgroup of $G$. The (right) Schreier coset graph (also Schreier graph or
coset graph) $X(G/H,S)$ is the graph on the set of (right) cosets $\\{Hg:g\in
G\\}$ of $H$, in which $Hg_{1}\sim Hg_{2}$ if $Hg_{2}=(Hg_{1})s$ for some $s$
in $S$. Note that the Cayley graph $X(G,S)$ is a special case of a Schreier
graph $X(G/H,S)$ for the trivial subgroup $H=\\{e\\}$.

######  Remark 2.1.1.

Assume that a group $G$ acts transitively on the vertices of a graph $Y$ such
that the edges of $Y$ correspond to the action of a symmetric generating set
$S$ of $G$. In this case, $Y$ is isomorphic to the Schreier graph
$X(G/\operatorname{St}(v),S)$ for every $v$ in $Y$ ($\operatorname{St}(v)$
denotes the stabilizer of a vertex $v$ in $Y$).

####  2.1.2 Flip Graphs

This thesis was motivated by three previously studied families of flip graphs
on colored triangle-free triangulations (CTFT) [2], arc permutations [5] and
geometric caterpillars [10]. In this subsection we introduce these three
families.

Colored Triangle-Free Triangulation Graphs

All definitions and claims in this subsection, related to CTFT graphs, can be
found in [2]. The flip graph of triangulations of a convex polygon [16]
inspired the definition of a few flip graphs on subsets of triangulations. One
such graph is the CTFT graph.

A triangulation of a convex polygon in the plane is its subdivision into
triangles using non-intersecting diagonals. An edge of a triangulation that
belongs to the polygon is called an external edge. Every other edge is called
a chord. Every triangulation of a convex $n$-gon ($n\geq 3$) has exactly $n-2$
triangles and $n-3$ chords. We denote by $P_{n}$ a convex polygon with $n\geq
5$ vertices labeled (counter clockwise) by the elements of the additive cyclic
group $\mathbb{Z}_{n}$.

A triangulation is called inner-triangle-free, or simply triangle-free, if it
contains no triangle with $3$ chords. A chord which forms a triangle with two
external edges is called a short chord and the triangle formed by these three
edges is called an outer triangle. A triangulation of $P_{n}$ is triangle-free
if and only if it contains exactly two outer triangles. Moreover, every
triangle-free triangulation $T$ of $P_{n}$ induces two opposite linear orders
on the chords of $T$ in which the two short chords are the minimum and maximum
elements of the order (see [2] for more details).

A coloring, or orientation of a triangle-free triangulation $T$ of $P_{n}$ is
a labeling of its chords by $\\{0,...,n-4\\}$ in one of the two linear orders
induced by $T$ on its chords. Denote the set of all colored triangle-free
triangulations of $P_{n}$ by $CTFT(n)$.

Each chord in a triangulation of $P_{n}$ is a diagonal of a unique
quadrilateral (the union of two adjacent triangles). Replacing a chord in a
triangulation by the other diagonal of its corresponding quadrilateral is
called a flip of the chord (see Figure 2.1).

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{}{} {}{} {}{} {}{} {}{}
{}{}{}\pgfsys@moveto{14.22638pt}{48.3696pt}\pgfsys@lineto{34.14322pt}{48.3696pt}\pgfsys@lineto{48.3696pt}{34.14322pt}\pgfsys@lineto{48.3696pt}{14.22638pt}\pgfsys@lineto{34.14322pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@lineto{0.0pt}{14.22638pt}\pgfsys@lineto{0.0pt}{34.14322pt}\pgfsys@lineto{14.22638pt}{48.3696pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{48.3696pt}{34.14322pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@lineto{34.14322pt}{48.3696pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{48.3696pt}{34.14322pt}\pgfsys@lineto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{34.14322pt}\pgfsys@lineto{34.14322pt}{48.3696pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\pgfsys@setdash{0.4pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{34.14322pt}{48.3696pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}{}\pgfsys@moveto{48.3696pt}{34.14322pt}\pgfsys@lineto{0.0pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}$ Figure
2.1: Flip of a chord

The flip graph of colored triangle-free triangulations $\Gamma_{n}$ is the
graph on $CTFT(n)$ where two triangulations are connected by an edge if one is
obtained from the other by a flip of a chord. For example, see $\Gamma_{6}$ in
Figure 2.2. This graph is closely related to a distinguished lower interval in
the weak order on the affine Weyl group $\widetilde{C}_{n}$. The diameter of
this flip graph was calculated using lattice properties of the order, see [2,
Theorem 5.1].

Figure 2.2: $\Gamma_{6}$

Arc Permutation Graphs

All definitions and claims relating to arc permutation graphs can be found in
[5]. A permutation $\pi$ in the symmetric group $S_{n}$ is called an arc
permutation if every prefix (and suffix) in $\pi$ forms an interval in
$\mathbb{Z}_{n}$, where the element $0$ in $Z_{n}$ is represented by $n$ for
convenience. For example, the permutation $3421576$ is an arc permutations in
$S_{7}$ (see Figure 2.3) but $5643127$ is not. Denote the set of arc
permutations in $S_{n}$ by $\mathcal{A}_{n}$.

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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.7}{0.0}{0.0}{0.7}{-16.51158pt}{11.34203pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\\\
\hline\cr\end{tabular}$ Figure 2.3: $\pi=3421576$ is an arc permutation in
$\mathcal{A}_{7}$

The flip graph of arc permutations $X_{n}$ is the graph on $\mathcal{A}_{n}$
in which two arc permutations $\pi$ and $\sigma$ in $\mathcal{A}_{n}$ are
connected by an edge if $\pi=\sigma\circ(i,i+1)$ for some $1\leq i\leq n-1$,
where $(i,i+1)$ is a transposition written in cycle notation. For example, see
$X_{4}$ in Figure 2.4. A transposition $(i,i+1)$ is also called a simple
reflection. Recall that the set of all simple reflections in $S_{n}$ is a
generating set of $S_{n}$. An equivalent definition of $X_{n}$ is as follows:
let $G$ be the Cayley graph associated with $(S_{n},S)$, where $S$ is the set
of simple reflections in $S_{n}$. Then $X_{n}$ is the subgraph of $G$ that is
induced by $\mathcal{A}_{n}$.

4321342132413214231421341234124314231432413243123412234121434123 Figure 2.4:
$X_{4}$

The diameter of the graph of arc permutations was computed using similarities
between the graph and the dominance order on $\mathbb{Z}^{n}$, see [5,
Subsection 6.2].

Geometric Caterpillar Graphs

To define a geometric caterpillar, start with the complete graph $K_{n}$, for
$n\geq 3$, whose vertices are labeled by $\mathbb{Z}_{n}$. Embed $K_{n}$ in
the plane such that its vertices are the vertices of a convex polygon (the
labels increasing clockwise) and its edges are straight line segments. Denote
this geometric graph by $\hat{K}_{n}$. A geometric caterpillar (or
caterpillar) of order $n$ is a non-crossing spanning tree of $\hat{K}_{n}$,
such that the vertices that are not leaves in the tree form an interval in
$\mathbb{Z}_{n}$. The spine of a caterpillar is its longest (simple) path
whose edges are on the boundary of the polygon. Caterpillars are also called
fishbones or combs and were studied by Keller, Khachatryan, Perles, Sagan,
Wachs and others in various contexts, see, e.g., [8, 9, 11, 10, 12].

The flip graph of caterpillars $\mathcal{G}_{n}$ is the graph on the set of
caterpillars of order $n$ in which two caterpillars are connected by an edge
if one is obtained from the other by shifting an endpoint of an edge along the
spine (for an example of two adjacent caterpillar in $\mathcal{G}_{8}$, see
Figure 2.5).

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2.5: Adjacent caterpillars in $\mathcal{G}_{8}$

###  2.2 Posets

A partial order $\leq$ on a set $P$ is a binary relation on $P$ which is
reflexive, transitive and antisymmetric. A set $P$ equipped with a partial
order $\leq$ is called a partially ordered set (or a poset) and is denoted by
$(P,\leq)$. If we refer to a set $P$ as a poset, then either the partial order
is evident, or we assume that the partial order on $P$ is denoted by
$\leq_{P}$.

$Q$ is called an induced subposet of $P$ if $Q\subseteq P$ and $x\leq_{Q}y$ if
and only if $x\leq_{P}y$ for every $x,y\in Q$. In this case, the order on $Q$
is called the induced order. In the rest of this work, a subposet means an
induced subposet.

A map $\varphi:P\rightarrow Q$ between two posets is called order preserving
if $x\leq_{P}y$ implies that $\varphi(x)\leq_{Q}\varphi(y)$. $\varphi$ is an
isomorphism if it is surjective and $\varphi(x)\leq_{Q}\varphi(y)$ if and only
if $x\leq_{P}y$ for every $x$ and $y$ in $P$. (Note that
$\varphi(x)=\varphi(y)$ implies both $x\leq y$ and $y\leq x$, therefore
$\varphi$ is injective.) In this case, $P$ and $Q$ are said to be isomorphic
and we denote it by $P\cong Q$.

Let $x$ and $y$ be two elements in $(P,\leq)$. If $x\leq y$ or $y\leq x$, then
we say that $x$ and $y$ are comparable. Otherwise, $x$ and $y$ are
incomparable. We denote by $x<y$ the case $x\leq y$ and $x\neq y$. A poset in
which every two elements are comparable is called a chain or a linearly
ordered set. A poset in which no two elements are comparable is called an
antichain. For convenience, we say that $C\subseteq P$ is a chain in $P$, if
$C$ is a chain as a subposet of $P$.

For every $x\leq y$ in a poset $P$, the interval $[x,y]$ is the subposet
$\\{z\in P:x\leq z\leq y\\}$. If every interval in $P$ is finite, then $P$ is
called a locally finite poset. In this thesis, every poset is locally finite.
A subposet $Q$ of $P$ is called convex if for every $x,y\in Q$ the interval
$[x,y]$ in $P$ is contained in $Q$.