# Models of Bihyperelliptic Curves

Omri Faraggi  Department of Mathematics, University College London, London
WC1H 0AY, UK<EMAIL_ADDRESS>
(Date: February 2021)

###### Abstract.

We give an explicit description of the minimal regular model of
bihyperelliptic curves with semistable reduction over a local field of odd
residue characteristic. We do this using a generalisation of the cluster
picture; a completely combinatorial object attached to a hyperelliptic curve
$y^{2}=f(x)$ over $K$ which contains the data of the $p$-adic distances
between the roots of $f$. We add some information, resulting in a chromatic
cluster picture, and show that this determines the minimal regular model of
$Y$ with the action of Frobenius.

###### Table of contents

  1. 1 Introduction
  2. 2 Cluster Pictures
  3. 3 Statement of Results
  4. 4 Proof

##  1\. Introduction

###  1.1. Summary

In this paper we give an explicit description of the minimal regular model of
a bihyperelliptic curve $Y$ with semistable reduction over a local field $K$
of residue characteristic $p>2$. A bihyperelliptic curve is a smooth curve
with maps to two distinct hyperelliptic curves111For us, a hyperelliptic curve
of genus $g$ is a curve given by equations $y^{2}=f(x)$ and
$v^{2}=u^{2g+2}f(1/u)$ glued along the maps $(x,y)\mapsto(1/x,y/x^{g+1})$.
$C_{1}$ and $C_{2}$. This is achieved using a generalisation of the cluster
picture, an invariant of hyperelliptic curves. Cluster pictures are a
relatively recent development; they are a completely combinatorial object
attached to a hyperelliptic curve $C/K$ from which much of the local
arithmetic of $C$ can be deduced. For example, cluster pictures have been used
to calculate the Galois representation, semistable model, conductor and
minimal discriminant of $C$ in [8], the minimal snc model if $C$ has tame
reduction in [9], the Tamagawa number in [3], the root number in [4] and
finally differentials in [10] and [14]. A survey article is available at [2].
Our generalisation, the chromatic cluster picture of $Y$, is an amalgamation
of the cluster pictures of $C_{1}$ and $C_{2}$, and is mostly sufficient to
determine the model of $Y$.

###  1.2. Bihyperelliptic Curves, Models and Clusters

Our main objects of study will be bihyperelliptic curves, a curve $Y$ with
maps to two distinct hyperelliptic curves $C_{1}$ and $C_{2}$.

######  Definition 1.1.

Let $C_{1}:y_{1}^{2}=f_{1}(x)$ and $C_{2}:y^{2}_{2}=f_{2}(x)$ be hyperelliptic
curves, given by their affine models, with $f_{1}$ and $f_{2}$ coprime. Then
$C_{h}:y_{h}^{2}=f_{1}(x)f_{2}(x)$ is their composite curve, also
hyperelliptic, and the curve $Y$, the smooth projective closure of

$\left\\{\begin{array}[]{c}y_{1}^{2}=f_{1}(x)\\\
y^{2}_{2}=f_{2}(x)\end{array}\right\\},$

is a bihyperelliptic curve. The curves fit into a tower:

${Y}$${C_{1}}$${C_{h}}$${C_{2}}$${\mathbb{P}^{1}}$

such that $Y/\mathbb{P}^{1}$ is a Galois cover with Galois group $C_{2}\times
C_{2}$.

We would like to find the minimal regular model $\mathscr{Y}^{\textrm{min}}$
of $Y$ in the case where $Y$ has semistable reduction. Recently there has been
a flurry of activity finding explicit descriptions for models of different
classes of curves, including [8], [9], [15], [14] (hyperelliptic curves), [6]
(superelliptic), [11], [5] (genus 3), [7] ($\Delta_{v}$-regular).

To a hyperelliptic curve such as $C_{1}$ we attach a cluster picture,
following [8].

######  Definition 1.2.

Let $C:y^{2}=f(x)$ be a hyperelliptic curve over $K$, with $\mathcal{R}$ the
set of roots of $f$. A cluster is a non-empty subset
$\mathfrak{s}\subseteq\mathcal{R}$ of the form $\mathfrak{s}=D\cap\mathcal{R}$
for some disc $D=z+\pi^{n}\mathcal{O}_{K}$, where $z\in\overline{K}$ and
$n\in\mathbb{Q}$. If $\mathfrak{s}$ is a cluster and $|\mathfrak{s}|>1$,
$\mathfrak{s}$ is a proper cluster and we define its depth

$d_{\mathfrak{s}}=\min_{r,r^{\prime}\in\mathfrak{s}}v_{K}(r-r^{\prime}).$

The cluster picture $\Sigma=\Sigma_{C/K}$ is the set of all clusters of the
roots of $f$.

The chromatic cluster picture $\Sigma_{\chi}$ of $Y$ is the cluster picture
$\Sigma_{h}$ of $C_{h}$, the composite curve of $C_{1}$ and $C_{2}$, along
with a colouring function $c:\mathcal{R}\rightarrow\\{\textrm{red, blue}\\}$,
such that the roots coming from $C_{1}$ (resp. $C_{2}$) are coloured red
(resp. blue). This information is sufficient to calculate the dual graph of
the special fibre $\mathscr{Y}^{\textrm{min}}_{k}$ of the minimal regular
model.

######  Theorem 1.3.

Let $K$ be a local field of odd residue characteristic and let
$C_{1}:y_{1}^{2}=f_{1}(x)$ and $C_{2}:y_{2}^{2}=f_{2}(x)$ be two hyperelliptic
curves over $K$. Let $C_{h}$ be their composite curve. Let $Y$ be the
bihyperelliptic curve arising from $C_{1}$ and $C_{2}$, such that $Y$ has
semistable reduction and all the depths in the chromatic cluster picture of
$Y$ are integers. Then the dual graph with genera of the special fibre
$\mathscr{Y}^{\mathrm{min}}$ of the minimal regular model of $Y$ is entirely
determined by the chromatic cluster picture of $Y$.

######  Example 1.4.

Let $Y$ be any bihyperelliptic curve with the chromatic cluster picture below,
where the number to the bottom right of a cluster $\mathfrak{s}$ indicates the
relative depth $\delta_{\mathfrak{s}}$ of the cluster: that is,
$d_{\mathfrak{s}}-d_{P(\mathfrak{s})}$ where $P(\mathfrak{s})$ is the cluster
immediately containing $\mathfrak{s}$. The dual graph of
$\mathscr{Y}^{\textrm{min}}_{k}$ for any such $Y$ is shown below.

$n$$\mathfrak{s}$$0$$\mathcal{R}$

$v_{\mathcal{R}}$3n

We prove this theorem by explicitly describing the dual graph. In order to do
this, we define a colouring on the remaining clusters of $\Sigma_{\chi}$. Each
cluster can be coloured red, blue, both or neither. In particular, clusters
with an odd number of red (resp. blue) children are coloured red (resp. blue).
Clusters which are coloured both red and blue are purple, and clusters which
are coloured neither are black.

From here, our description of $\mathscr{Y}^{\textrm{min}}_{k}$ is very much in
the spirit of [8, Theorem 8.5] — to a cluster $\mathfrak{s}$ we associate $1$,
$2$ or $4$ components of the special fibre, and the components of
$\mathfrak{s}$ and $\mathfrak{s}^{\prime}$ are linked by a chain of
$\mathbb{P}^{1}$s if $\mathfrak{s}^{\prime}$ is a maximal subcluster (child)
of $\mathfrak{s}$ (or vice versa). The length of this chain is determined by
$\delta_{\mathfrak{s}^{\prime}}=d_{\mathfrak{s}^{\prime}}-d_{\mathfrak{s}}$.
In Theorem 3.3, we also give the action of Frobenius on the the dual graph of
$\mathscr{Y}^{\textrm{min}}_{k}$. Before we state our theorem, let us define
some of the cluster terminology used.

######  Definition 1.5.

A cluster $\mathfrak{s}\in\Sigma_{\chi}$ is odd (resp. even) if
$|\mathfrak{s}|$ is odd (resp. even). A child
$\mathfrak{s}^{\prime}<\mathfrak{s}$ is a maximal subcluster of
$\mathfrak{s}$. The cluster $\mathfrak{s}$ is übereven if all its children are
even. It is proper if $|\mathfrak{s}|\geq 2$, principal if $|\mathfrak{s}|\geq
3$ (except in a few cases if $\mathfrak{s}=\mathcal{R}$, see Definition 2.10)
and a twin if $|\mathfrak{s}|=2$. It is chromatic if it is red, blue or
purple. It has polychromatic children if it has a purple child, or if it has
both a red and a blue child (it can have other children as well) and
monochromatic children if all of its chromatic children are red or all of its
chromatic children are blue. The relative depth of a cluster
$\mathfrak{s}^{\prime}<\mathfrak{s}$ is
$\delta_{\mathfrak{s}^{\prime}}=d_{\mathfrak{s}^{\prime}}-d_{\mathfrak{s}}$
and the relative depth between any two clusters $\mathfrak{s}_{1}$ and
$\mathfrak{s}_{2}$ is
$\delta_{\mathfrak{s}_{1},\mathfrak{s}_{2}}=d_{\mathfrak{s}_{1}}+d_{\mathfrak{s}_{2}}-2d_{\mathfrak{s}_{1}\wedge\mathfrak{s}_{2}}$
where $\mathfrak{s}_{1}\wedge\mathfrak{s}_{2}$ is the smallest cluster
containing both $\mathfrak{s}_{1}$ and $\mathfrak{s}_{2}$.

######  Definition 1.6.

Let $\mathfrak{s}$ be a cluster. We assign to $\mathfrak{s}$ a chromatic genus
$g_{\chi}(\mathfrak{s})$ as follows: let $\mathfrak{s}_{\chi}$ be the set of
chromatic children of $\mathfrak{s}$. If $\mathfrak{s}$ has polychromatic
children, then $g_{\chi}(\mathfrak{s})=|\mathfrak{s}_{\chi}|-3$ if
$\mathfrak{s}=\mathcal{R}$ and $f_{1}$ and $f_{2}$ both have even degree, or
$\mathcal{R}=\mathfrak{s}\sqcup\mathfrak{s}^{\prime}$, $\mathfrak{s}$ is even
and $f_{1}$ and $f_{2}$ both have even degree; and
$g_{\chi}(\mathfrak{s})=|\mathfrak{s}_{\chi}|-2$ otherwise. If $\mathfrak{s}$
has monochromatic children, then $g_{\chi}(\mathfrak{s})$ is such that
$2g_{\chi}(\mathfrak{s})+1=|\mathfrak{s}_{\chi}|$ or
$2g_{\chi}(\mathfrak{s})+2=|\mathfrak{s}_{\chi}|$. Otherwise
$g_{\chi}(\mathfrak{s})=0$.

If $\mathfrak{s}$ has monochromatic children or no chromatic children then
$g_{\chi}(\mathfrak{s})=g(\mathfrak{s})$, as in [8, Section 5.3]. Our main
result is the following. We have excluded the cases where $\mathcal{R}$ is not
principal, but these are dealt with in Theorem 3.1.

######  Theorem 1.7 (=Theorem 3.1).

Let $K$ be a local field of odd residue characteristic and let $C_{1}$ and
$C_{2}$ be two hyperelliptic curves over $K$. Let $C_{h}$ be their composite
curve. Let $Y$ be the bihyperelliptic curve arising from $C_{1}$ and $C_{2}$,
such that $Y$ has semistable reduction and all the depths in the chromatic
cluster picture of $Y$ are integers. Suppose further that $\mathcal{R}$ has at
least three children, none of which of size $2g(C_{h})$222This is a sufficient
condition for $\mathcal{R}$ to be principal.. Then the dual graph of
$\mathscr{Y}^{\mathrm{min}}_{k}$ has the following explicit description:

Each principal cluster $\mathfrak{s}$ contributes vertices of genus
$g_{\chi}(\mathfrak{s})$ to the dual graph. If $\mathfrak{s}$ is not übereven:
$1$ vertex $v_{\mathfrak{s}}$ if $\mathfrak{s}$ has polychromatic children and
$2$ vertices $v_{\mathfrak{s}}^{+},v_{\mathfrak{s}}^{-}$ if $\mathfrak{s}$ has
monochromatic children; and if übereven: $2$ vertices
$v_{\mathfrak{s}}^{+},v_{\mathfrak{s}}^{-}$ if $\mathfrak{s}$ has chromatic
children and $4$ vertices
$v_{\mathfrak{s}}^{+,+},v_{\mathfrak{s}}^{+,-},v_{\mathfrak{s}}^{-,+},v_{\mathfrak{s}}^{-,-}$
if $\mathfrak{s}$ has no chromatic children.

These are linked by edges as follows:

Name | From | To | Length | Condition
---|---|---|---|---
$L_{\mathfrak{s}^{\prime}}^{+}$ | $v_{\mathfrak{s}}^{+}$ | $v_{\mathfrak{s}^{\prime}}^{\sigma}$ | $\frac{1}{2}\delta_{\mathfrak{s}^{\prime}}$ | $\mathfrak{s}^{\prime}<\mathfrak{s}$ both principal, $\mathfrak{s}^{\prime}$ chromatic
$L_{\mathfrak{s}^{\prime}}^{-}$ | $v_{\mathfrak{s}}^{-}$ | $v_{\mathfrak{s}^{\prime}}^{-\sigma}$
$L_{\mathfrak{s}^{\prime}}^{+,+}$ | $v_{\mathfrak{s}}^{+,+}$ | $v_{\mathfrak{s}^{\prime}}^{+,+}$ | $\delta_{\mathfrak{s}^{\prime}}$ | $\mathfrak{s}^{\prime}<\mathfrak{s}$ both principal, $\mathfrak{s}^{\prime}$ black
$L_{\mathfrak{s}^{\prime}}^{+,-}$ | $v_{\mathfrak{s}}^{+,-}$ | $v_{\mathfrak{s}^{\prime}}^{+,-}$
$L_{\mathfrak{s}^{\prime}}^{-,+}$ | $v_{\mathfrak{s}}^{-,+}$ | $v_{\mathfrak{s}^{\prime}}^{-,+}$
$L_{\mathfrak{s}^{\prime}}^{-,-}$ | $v_{\mathfrak{s}}^{-,-}$ | $v_{\mathfrak{s}^{\prime}}^{-,-}$
$L_{\mathfrak{t}}$ | $v_{\mathfrak{s}}^{+}$ | $v_{\mathfrak{s}}^{-}$ | $\delta_{\mathfrak{t}}$ |  $\mathfrak{t}<\mathfrak{s}$, $\mathfrak{t}$ chromatic twin, $\mathfrak{s}$ principal
$L_{\mathfrak{t}}^{+}$ | $v_{\mathfrak{s}}^{+,+}$ | $v_{\mathfrak{s}}^{\sigma,-\sigma}$ | $2\delta_{\mathfrak{t}}$ | $\mathfrak{t}<\mathfrak{s}$, $\mathfrak{t}$ black twin, $\mathfrak{s}$ principal
$L_{\mathfrak{t}}^{-}$ | $v_{\mathfrak{s}}^{-,-}$ | $v_{\mathfrak{s}}^{-\sigma,\sigma}$

where $\sigma=\sigma(\mathfrak{s},\mathfrak{s}^{\prime})\in\\{-1\\}$ is
defined in Definition 2.13;
$v_{\mathfrak{s}}^{\pm,+}=v_{\mathfrak{s}}^{\pm,-}=v_{\mathfrak{s}}^{\pm}$ if
$\mathfrak{s}$ is non-übereven with monochromatic red children;
$v_{\mathfrak{s}}^{+,\pm}=v_{\mathfrak{s}}^{-,\pm}=v_{\mathfrak{s}}^{\pm}$ if
$\mathfrak{s}$ is non-übereven with has monochromatic blue children;
$v_{\mathfrak{s}}^{\pm,\pm}=v_{\mathfrak{s}}^{+}$,
$v_{\mathfrak{s}}^{\pm,\mp}=v_{\mathfrak{s}}^{-}$ if $\mathfrak{s}$ is
übereven with chromatic children; and
$v_{\mathfrak{s}}^{+}=v_{\mathfrak{s}}^{-}=v_{\mathfrak{s}}$ if $\mathfrak{s}$
is non-übereven with polychromatic children.

######  Remark 1.8.

The action of Frobenius can also be described explicitly in terms of the
chromatic cluster picture of $Y$. See Theorem 3.3 for details.

######  Remark 1.9.

A condition we require is that $Y$ have semistable reduction. This is hard to
check directly, but it is equivalent to the curves $C_{1}$, $C_{2}$ and
$C_{h}$ having semistable reduction, which can be checked using the
Semistability Criterion for hyperelliptic curves given in [8, Theorem 1.8].
This equivalence arises due to the existence of an isogeny between the
Jacobians of $Y$ and $C_{1}\times C_{2}\times C_{h}$. It cannot be found in
the literature (to our knowledge), but will appear in an upcoming paper of
Dokchitser and Morgan.

######  Example 1.10.

Let $C_{1}:y_{1}^{2}=(x^{2}-p^{2n})(x-1)$ and $C_{2}:y_{2}^{2}=x^{3}-2$ be two
elliptic curves with composite curve
$C_{h}:y_{h}^{2}=(x^{2}-p^{n})(x-1)(x^{3}-2)$ and associated bielliptic curve
$Y$. The chromatic cluster picture of $Y$ and the dual graph of
$\mathscr{Y}^{\textrm{min}}_{k}$ is below.

$n$$\mathfrak{s}$$0$$\mathcal{R}$

$v_{\mathcal{R}}$22n2n

The top cluster $\mathcal{R}$ is not übereven and has polychromatic children
so contributes one component of genus $2$. The twin is black and has depth $n$
so contributes two loops of length $2n$. For a description of the action of
Frobenius see Example 3.6.

######  Example 1.11.

Let $C_{1}:y^{2}_{1}=((x-1)^{4}-p^{12})((x-2)^{3}+p^{6})$ and
$C_{2}:y_{1}^{2}=(x^{3}-p^{18})((x-2)^{3}-p^{6})$ be hyperelliptic curves over
$K$ and let $Y$ be the associated bihyperelliptic curve. The chromatic cluster
picture and dual graph of $\mathscr{Y}^{\textrm{min}}_{k}$ are shown below.
The cluster picture $\Sigma_{\chi}$ consists of the top cluster $\mathcal{R}$,
a blue cluster $\mathfrak{s}_{1}$, a black cluster $\mathfrak{s}_{2}$ and a
purple cluster $\mathfrak{s}_{3}$. The top cluster has a purple child and
hence there is one component, $v_{\mathcal{R}}$, arising from $\mathcal{R}$.
The component is genus $0$ by Definition 1.6. The cluster $\mathfrak{s}_{1}$
has monochromatic (blue) children, so contributes two components,
$v_{\mathfrak{s}_{1}}^{+}$ and $v_{\mathfrak{s}_{1}}^{-}$ of genus $1$.
Furthermore, it is chromatic so there are two linking chains between the
components of $\mathcal{R}$ and $\mathfrak{s}_{1}$, each of length $3$. The
cluster $\mathfrak{s}_{2}$ also has monochromatic (red) children, but it is
black and hence there are four linking chains, each of length $3$, between
$v_{\mathcal{R}}$ and $v_{\mathfrak{s}_{2}}^{\pm}$. Finally $\mathfrak{s}_{3}$
is chromatic with polychromatic children so it contributes a single component
with two linking chains to $v_{\mathcal{R}}$.

$6$$\mathfrak{s}_{1}$$3$$\mathfrak{s}_{2}$$2$$\mathfrak{s}_{3}$$0$$\mathcal{R}$

$v_{\mathcal{R}}$$v_{\mathfrak{s}_{1}}^{+}$$v_{\mathfrak{s}_{1}}^{-}$$v_{\mathfrak{s}_{2}}^{+}$$v_{\mathfrak{s}_{2}}^{-}$$v_{\mathfrak{s}_{3}}$1111433331

######  Example 1.12.

Let $C_{1}:y_{1}^{2}=(x^{3}-p^{12})(x^{3}-2)$ and
$C_{2}:y_{2}^{2}=(x^{4}-p^{20})(x^{3}-1)$ be two hyperelliptic curves, $C_{h}$
their composite hyperelliptic curve and $Y$ the associated bihyperelliptic
curve. The chromatic cluster picture of $Y$ and the dual graph of
$\mathscr{Y}^{\textrm{min}}_{k}$ are below.

$1$$\mathfrak{s}_{1}$$4$$\mathfrak{s}_{2}$$0$$\mathcal{R}$

$v_{\mathcal{R}}$$v_{\mathfrak{s}_{1}}^{+}$$v_{\mathfrak{s}_{1}}^{-}$$v_{\mathfrak{s}_{2}}^{+}$$v_{\mathfrak{s}_{2}}^{-}$511112211

######  Remark 1.13.

A perhaps more suitable notion of a bihyperelliptic curve would be any curve
$Y$ which has a degree $2$ map to a hyperelliptic curve $C$, directly
mirroring the definition for bielliptic curves. However, the cover
$Y/\mathbb{P}^{1}$ is not necessarily Galois in this case. Since we require
this for our purposes, we shall restrict to the case given in Definition 1.1,
but the more general case is also of interest.

###  1.3. Structure

In Section 2 we define chromatic cluster pictures, the main object we will use
in our theorem to describe the minimal regular model of a bihyperelliptic
curve. Some familiarity with cluster pictures is helpful: for a comprehensive
definition we direct the reader to [8], and for many examples of cluster
pictures in use to [2]. In Section 3 we state our main Theorem 3.1 and
illustrate it with several examples. The proof of Theorem 3.1 is Section 4.
The strategy is to normalise a model $\mathscr{X}$ of $\mathbb{P}^{1}$
associated to $\Sigma_{\chi}$ in the function field of $Y$, resulting in a
model $\mathscr{Y}$ of $Y$ over $K^{\mathrm{ur}}$. Using the results of [8],
we deduce the normalisation of $\mathscr{X}$ in the function fields of
$C_{1}$, $C_{2}$ and $C_{h}$ and use these to give an explicit description of
$\mathscr{Y}$.

###  1.4. Notation

Throughout $K$ will be a local field with ring of integers $\mathcal{O}_{K}$
and residue field $k$ of characteristic $p>2$. The uniformiser of $K$ will be
$\pi$, and the valuation $v_{K}$. A fixed algebraic closure of $K$ will be
denoted $\overline{K}$ and $K^{\mathrm{ur}}$ will be the maximal unramified
extension of $K$. For us, a model $\mathcal{C}$ of a curve $C$ over $K$ is a
flat proper $\mathcal{O}_{K}$-scheme whose generic fibre is isomorphic to $C$.
Furthermore, this model is regular if $\mathcal{C}$ is regular as a scheme, it
is minimal if it is dominates no other regular model, and it is semistable if
the special fibre is reduced with at worst normal crossings such that any
component isomorphic to $\mathbb{P}^{1}$ intersects the rest of the special
fibre in at least two points.

In cluster pictures, red roots will be represented by spheres  and blue roots
by hexagons  . Red clusters will be denoted with dotted lines, blue clusters
by dashed lines, purple cluster by dot-dash lines and black clusters by solid
lines. Sometimes, we will need an empty cluster, which will look like this:  .
The relative depth of a cluster $\mathfrak{s}\neq\mathcal{R}$ will be written
to the lower right of $\mathfrak{s}$, and the depth of $\mathcal{R}$ will be
written to its lower right. The name of the cluster is above it.

Special fibres will be draw as their dual graphs. Components from principal
clusters will be the vertices, with their genus inside. If no genus is
written, the corresponding component has genus $0$. The edges between nodes
are chains of rational curves, and the number next to them indicates the
number of edges in the dual graph corresponding to the linking chain (which is
one more than the number of $\mathbb{P}^{1}s$ in the linking chain). An edge
of length $1$ in the dual graph indicates an intersection between the
components. When there are two edges between a pair of vertices, only one edge
will be labelled with its length, as the other edge will always have the same
length.

### Acknowledgements

I would like to thank my supervisor Vladimir Dokchitser for his continued
advice and support. I would also like to thank Holly Green for checking and
correcting many earlier versions of the theorem, and Adam Morgan for his
patience answering my sporadic questions. This work was supported by the
Engineering and Physical Sciences Research Council [EP/L015234/1], the EPSRC
Centre for Doctoral Training in Geometry and Number Theory (The London School
of Geometry and Number Theory), and University College London.

##  2\. Cluster Pictures

###  2.1. The Story so Far

A cluster picture is a set of subsets (“clusters”) of the roots of a
polynomial $f$, where each cluster contains roots that are $p$-adically close
together. There are many numerical invariants associated to cluster picture
which are used, and in this section we recall those that we will require.
Throughout this subsection, $C:y^{2}=f(x)=c_{f}\prod_{r\in\mathcal{R}}(x-r)$
will be a hyperelliptic curve of genus $g\geq 2$ with semistable reduction and
cluster picture $\Sigma$, as in Definition 1.2. Denote by $\mathcal{C}$ the
minimal regular model of $C$ as described in [8, Theorem 8.5].

The cluster picture $\Sigma$ is equivalent to an admissible collection of
disks, that is a non empty set $\mathcal{D}$ of disks
$D=D_{z,d}=\\{x\in\overline{K}\,|\,v(x-z_{D})\geq d_{D}\\}$ such that
$z_{D}\in K^{\mathrm{ur}}$ (called integral disks), $d_{D}\in\mathbb{Z}$,
where $\mathcal{D}$ has a maximal element with respect to inclusion and such
that if $D_{1}\subseteq D_{2}$ then any integral disk with $D_{1}\subseteq
D\subseteq D_{2}$ is in $\mathcal{D}$. This collection of disks in turn gives
rise to a model $\mathscr{X}$ of $\mathbb{P}^{1}$ which separates the branch
points of $C\rightarrow\mathbb{P}^{1}$. For the precise construction, see [8,
Definition 3.8]. To each disk in such a collection we define a valuation
extending $v_{K}$.

######  Definition 2.1.

Let $D$ be a $p$-adic disk. Then

$\nu_{D}(f)=v_{K}(c_{f})+\sum_{r\in\mathcal{R}}\min(d_{D},v_{K}(z_{D}-r)),$

and $\omega_{D}(f)\in\\{0,1\\}$ is such that $\omega_{D}(f)\equiv v_{D}(f)\mod
2$.

Each proper cluster $\mathfrak{s}$ is given a genus, which is the genus of the
component in $\mathcal{C}$ arising from $\mathfrak{s}$.

######  Definition 2.2.

Let $\mathfrak{s}\in\Sigma$ be a proper cluster. Then the genus of
$\mathfrak{s}$, $g(\mathfrak{s})$ is such that $2g(\mathfrak{s})+1$ or
$2g(\mathfrak{s})+2$ equals the number of odd children of $\mathfrak{s}$.

To each even cluster or cotwin, a cluster who has a child of size $2g$ whose
complement is not a twin, we assign a character of
$\textrm{Gal}(\overline{K}/K)$, which roughly tell us whether or not
$\sigma\in\textrm{Gal}(\overline{K}/K)$ swaps the points at infinity of the
component arising from $\mathfrak{s}$.

######  Definition 2.3.

Let $\mathfrak{s}$ be an even cluster or a cotwin. We define the character
$\epsilon_{\mathfrak{s}}:\textrm{Gal}(\overline{K}/K)\rightarrow\\{\pm 1\\}$
by
$\epsilon_{\mathfrak{s}}(\sigma)=\sigma(\theta_{\mathfrak{s}^{*}})/\theta_{\sigma(\mathfrak{s}^{*})}$,
where for any cluster $\mathfrak{t}$, $\theta_{\mathfrak{t}}$ is a choice of
square root $\sqrt{c_{f}\prod_{r\not\in\mathfrak{t}}(z_{\mathfrak{t}}-r)}$ and
$\mathfrak{t}^{*}$ is the smallest cluster containing $\mathfrak{t}$ whose
parent is not übereven (or $\mathcal{R}$ if no such cluster exists).

Note that the definition above extends to “empty clusters” — $p$-adic disks
which contain no roots. This will be useful when talking about red and blue
cluster pictures in the next section, which can contain empty clusters.

###  2.2. Chromatic Cluster Pictures

The combinatorial objects we will use to calculate the semistable model will
be chromatic cluster pictures. A chromatic cluster picture is similar to a
cluster picture, but instead of one polynomial we take a product of coprime
polynomials $f_{1}f_{2}$, and we colour the roots red and blue to indicate
whether they are roots of $f_{1}$ or $f_{2}$. In this way, the chromatic
cluster picture contains the information of the cluster pictures associated to
$f_{1}$, $f_{2}$ and $f_{1}f_{2}$.

######  Definition 2.4.

A chromatic cluster picture $\Sigma_{\chi}$ is a cluster picture $\Sigma$ on a
set $\mathcal{R}$ with a colouring function
$c:\mathcal{R}\longrightarrow\\{\textrm{red},\textrm{ blue}\\}$, assigning to
each root a colour.

This induces a colouring on the remaining clusters as follows:

  1. (i)

clusters with an odd number of blue children and an even number of red
children (resp. an odd number of red children and an even number of blue
children) are coloured blue (resp. red),

  2. (ii)

clusters with an odd number of blue children and an odd number of red children
are coloured purple,

  3. (iii)

all other clusters are coloured black.

where purple children are included in both the set of red and blue clusters.
Blue, red and purple clusters are called chromatic clusters.

Clusters with purple children, or clusters with both blue and red children
have polychromatic children, whereas clusters whose only chromatic children
are red or blue have monochromatic children.

We define the red (resp. blue) cluster picture $\Sigma_{1}$ (resp.
$\Sigma_{2}$) associated to $\Sigma_{\chi}$ to be the subset of $\Sigma$ where
the only clusters of size $1$ are the red (resp. blue) ones. We forget the
colouring on the rest of the clusters.

######  Lemma 2.5.

Let $\mathfrak{s}\in\Sigma$ be a cluster. If $\mathfrak{s}$ is odd then
$\mathfrak{s}$ is red or blue, and if $\mathfrak{s}$ is even then
$\mathfrak{s}$ is purple or black. Furthermore, purple clusters are odd in
$\Sigma_{1}$ and $\Sigma_{2}$ and even in $\Sigma$, red clusters are odd in
$\Sigma_{1}$ and $\Sigma$ and even in $\Sigma_{2}$, blue clusters are odd in
$\Sigma_{2}$ and $\Sigma$ and even in $\Sigma_{1}$, and finally black clusters
are even in all of $\Sigma_{1},\Sigma_{2}$ and $\Sigma$.

###### Proof.

This follows by induction on the size of the cluster, noting that it is
trivially true for singletons. ∎

######  Remark 2.6.

The red and blue cluster pictures aren’t cluster pictures (or chromatic
cluster pictures) in the conventional sense — they are cluster pictures with
some additional $p$-adic disks. They can have “empty” clusters which contain
no roots, or clusters which only contain another cluster and nothing else.
However, they give rise to the same admissible collection of discs, in the
sense of [8, Definition 3.4], as $\Sigma_{h}$. Therefore we can apply the
results of [8] to them.

We will be interested in the chromatic cluster pictures of bihyperelliptic
curves. In other words, if $Y:\\{y_{1}^{2}=f_{1},y_{2}^{2}=f_{2}\\}$ is a
bihyperelliptic curve, then its chromatic cluster picture arises from
colouring the roots of $f_{1}$ red, the roots of $f_{2}$ blue and the rest of
the clusters according to the rules (ii) - (iv).

######  Example 2.7.

Below (left) is the chromatic cluster picture of
$Y:\\{y_{1}^{2}=(x-p^{n})(x^{2}-1),y_{2}^{2}=(x+p^{n})(x^{2}-2)\\}$. On the
right is the red cluster picture, which is equivalent to the blue cluster
picture.

$n$$\mathfrak{s}$$0$$\mathcal{R}$

$n$$\mathfrak{s}$$0$$\mathcal{R}$

The cluster $\mathfrak{s}\in\Sigma_{\chi}$ contains both an odd number of red
children and an odd number of blue children (one of each), and hence is
purple. The cluster $\mathcal{R}\in\Sigma_{\chi}$ contains three red children
(since the purple child counts as a red and a blue child) and three blue
children and hence is also purple.

The red and blue clusters are equivalent, and in both the “cluster”
$\mathfrak{s}$ is a $p$-adic disk containing a single root. This wouldn’t be a
cluster in a conventional cluster picture.

######  Example 2.8.

From left to right, we have the chromatic, red and blue cluster pictures of
$Y:\\{y_{1}^{2}=(x^{2}-p^{n})(x-1),y_{2}^{2}=x^{3}-2\\}$. The cluster
$\mathfrak{s}\in\Sigma_{\chi}$ has an even number of both red and blue
children and hence is black. On the other hand, $\mathcal{R}$ has an odd
number of both red and blue children and hence is coloured purple. Note that
in the blue cluster picture $\mathfrak{s}$ is an empty cluster (i.e. it is a
$p$-adic disk which contains no roots).

$n/2$$\mathfrak{s}$$0$$\mathcal{R}$

$n/2$$\mathfrak{s}$$0$$\mathcal{R}$

$n/2$$\mathfrak{s}$$0$$\mathcal{R}$

######  Example 2.9.

From left to right, the chromatic, red, and blue cluster picture of the
bihyperelliptic curve
$Y:\\{y_{1}^{2}=(x^{3}-p^{9})(x^{3}-2),y_{2}^{2}=(x^{4}-p^{20})(x^{3}-1)\\}$.
In the red cluster picture $\mathfrak{s}_{1}$ is an empty cluster, and in the
blue cluster picture $\mathfrak{s}_{2}$ has a unique child. Neither of these
are traditionally clusters.

$2$$\mathfrak{s}_{1}$$3$$\mathfrak{s}_{2}$$0$$\mathcal{R}$

$2$$\mathfrak{s}_{1}$$3$$\mathfrak{s}_{2}$$0$$\mathcal{R}$

$2$$\mathfrak{s}_{1}$$3$$\mathfrak{s}_{2}$$0$$\mathcal{R}$

The following is an important definition, the clusters which will contribute
principal components to the semistable model.

######  Definition 2.10.

Let $\mathfrak{s}$ be a cluster. Then $\mathfrak{s}$ is chromatically
principal if $|\mathfrak{s}|\geq 3$, except in the following cases:

  1. (i)

$\mathfrak{s}=\mathcal{R}=\mathfrak{s}_{1}\sqcup\mathfrak{s}_{2}$, with
$\mathfrak{s}_{1}$ and $\mathfrak{s}_{2}$ of the same colour, or one of
$\mathfrak{s}_{1}$ or $\mathfrak{s}_{2}$ a singleton or a twin,

  2. (ii)

$\mathfrak{s}=\mathcal{R}$ has a unique proper child $\mathfrak{s}^{\prime}$
of size $2g(C_{h})$ such that either $\mathfrak{s}^{\prime}$ is purple or
$\mathfrak{s}$ and $\mathfrak{s}^{\prime}$ are both black.

We can characterise this as follows:

######  Lemma 2.11.

A cluster $\mathfrak{s}$ is chromatically principal if and only if it is
principal in at least one of $\Sigma_{1},\Sigma_{2},\Sigma_{h}$.

###### Proof.

If $\mathfrak{s}\neq\mathcal{R}$ this is clear as its size in $\Sigma_{\chi}$
is the same as its size in $\Sigma_{h}$. Therefore, suppose
$\mathfrak{s}=\mathcal{R}$. Suppose
$\mathfrak{s}=\mathfrak{s}_{1}\sqcup\mathfrak{s}_{2}$ with $\mathfrak{s}_{1}$
and $\mathfrak{s}_{2}$ the same colour. Then $\mathfrak{s}_{1}$ and
$\mathfrak{s}_{2}$ have the same parity in each of
$\Sigma_{1},\Sigma_{2},\Sigma_{h}$ by Lemma 2.5. If $\mathfrak{s}$ has a
unique proper child of size $2g(C_{h})$ then $\Sigma_{1}$ has a child of size
$2g(C_{1})$, and similarly for $\Sigma_{2}$ and $\Sigma_{h}$. The other
direction can be checked similarly (for example, if
$\mathfrak{s}=\mathfrak{s}_{1}\sqcup\mathfrak{s}_{2}$ with $\mathfrak{s}_{1}$,
$\mathfrak{s}_{2}$ principal of different colours, then $\mathfrak{s}_{1}$ and
$\mathfrak{s}_{2}$ will have opposite parities in at least one of
$\Sigma_{1}$, $\Sigma_{2}$ or $\Sigma_{h}$). ∎

We must update our definition of cotwin to exclude a case which cannot happen
for classic cluster pictures:

######  Definition 2.12.

A cluster $\mathfrak{s}$ is a cotwin if it has a child $\mathfrak{s}^{\prime}$
of size $2g(C_{h})$ whose complement isn’t a twin, unless $\mathfrak{s}$ is
purple and $\mathfrak{s}^{\prime}$ is black.

We finish the section with a definition which will be needed in stating our
main theorem.

######  Definition 2.13.

If $\mathfrak{s}$ is a cluster then $\mathfrak{s}_{\chi}$ is the set of
chromatic children of $\mathfrak{s}$. If $\mathfrak{s},\mathfrak{s}^{\prime}$
are two proper clusters then $\sigma(\mathfrak{s},\mathfrak{s}^{\prime})=-1$
in the following cases:

  1. (i)

$\mathfrak{s}$ and $\mathfrak{s}^{\prime}$ each have monochromatic children of
opposite colours,

  2. (ii)

$\mathfrak{s}$ is black übereven and $\mathfrak{s}^{\prime}$ is black with
monochromatic, blue children;

and $1$ otherwise.

##  3\. Statement of Results

Here we describe our main theorem: how to construct the minimal regular model
$\mathscr{Y}^{\min}$ of $Y$ given the chromatic cluster picture of $Y$. The
rough idea is that principal clusters give rise to components in the special
fibre, and components of $\mathfrak{s}$ are linked to the components of the
children of $\mathfrak{s}$. The number of components and linking chains, as
well as the lengths of the linking chains, are determined by the properties of
the clusters.

######  Theorem 3.1.

Let $K$ be a local field of odd residue characteristic and let $C_{1}$ and
$C_{2}$ be two hyperelliptic curves over $K$. Let $C_{h}$ be their composite
curve. Let $Y$ be the bihyperelliptic curve arising from $C_{1}$ and $C_{2}$,
such that $Y$ has semistable reduction and all the depths in the chromatic
cluster picture of $Y$ are integers. Then the dual graph of
$\mathscr{Y}^{\mathrm{min}}_{k}$ is entirely determined by the chromatic
cluster picture of $Y$.

In particular, each principal cluster $\mathfrak{s}$ contributes vertices of
genus $g_{\chi}(\mathfrak{s})$ to the dual graph of
$\mathscr{Y}^{\textrm{min}}_{k}$. If $\mathfrak{s}$ is not übereven: $1$
vertex $v_{\mathfrak{s}}$ if $\mathfrak{s}$ has polychromatic children and $2$
vertices $v_{\mathfrak{s}}^{+},v_{\mathfrak{s}}^{-}$ if $\mathfrak{s}$ has
monochromatic children; and if übereven: $2$ vertices
$v_{\mathfrak{s}}^{+},v_{\mathfrak{s}}^{-}$ if $\mathfrak{s}$ has chromatic
children and $4$ vertices
$v_{\mathfrak{s}}^{+,+},v_{\mathfrak{s}}^{+,-},v_{\mathfrak{s}}^{-,+},v_{\mathfrak{s}}^{-,-}$
if $\mathfrak{s}$ has no chromatic children.

These are linked by edges as follows:

Name | From | To | Length | Condition
---|---|---|---|---
$L_{\mathfrak{s}^{\prime}}^{+}$ | $v_{\mathfrak{s}}^{+}$ | $v_{\mathfrak{s}^{\prime}}^{\sigma}$ | $\frac{1}{2}\delta_{\mathfrak{s}^{\prime}}$ | $\mathfrak{s}^{\prime}<\mathfrak{s}$ both principal, $\mathfrak{s}^{\prime}$ chromatic
$L_{\mathfrak{s}^{\prime}}^{-}$ | $v_{\mathfrak{s}}^{-}$ | $v_{\mathfrak{s}^{\prime}}^{-\sigma}$
$L_{\mathfrak{s}^{\prime}}^{+,+}$ | $v_{\mathfrak{s}}^{+,+}$ | $v_{\mathfrak{s}^{\prime}}^{+,+}$ | $\delta_{\mathfrak{s}^{\prime}}$ | $\mathfrak{s}^{\prime}<\mathfrak{s}$ both principal, $\mathfrak{s}^{\prime}$ black
$L_{\mathfrak{s}^{\prime}}^{+,-}$ | $v_{\mathfrak{s}}^{+,-}$ | $v_{\mathfrak{s}^{\prime}}^{+,-}$
$L_{\mathfrak{s}^{\prime}}^{-,+}$ | $v_{\mathfrak{s}}^{-,+}$ | $v_{\mathfrak{s}^{\prime}}^{-,+}$
$L_{\mathfrak{s}^{\prime}}^{-,-}$ | $v_{\mathfrak{s}}^{-,-}$ | $v_{\mathfrak{s}^{\prime}}^{-,-}$
$L_{\mathfrak{t}}$ | $v_{\mathfrak{s}}^{+}$ | $v_{\mathfrak{s}}^{-}$ | $\delta_{\mathfrak{t}}$ |  $\mathfrak{t}<\mathfrak{s}$, $\mathfrak{t}$ chromatic twin, $\mathfrak{s}$ principal
$L_{\mathfrak{t}}^{+}$ | $v_{\mathfrak{s}}^{+,+}$ | $v_{\mathfrak{s}}^{\sigma,-\sigma}$ | $2\delta_{\mathfrak{t}}$ | $\mathfrak{t}<\mathfrak{s}$, $\mathfrak{t}$ black twin, $\mathfrak{s}$ principal
$L_{\mathfrak{t}}^{-}$ | $v_{\mathfrak{s}}^{-,-}$ | $v_{\mathfrak{s}}^{-\sigma,\sigma}$

where $\sigma=\sigma(\mathfrak{s},\mathfrak{s}^{\prime})$;
$v_{\mathfrak{s}}^{\pm,+}=v_{\mathfrak{s}}^{\pm,-}=v_{\mathfrak{s}}^{\pm}$ if
$\mathfrak{s}$ is non-übereven with monochromatic red children;
$v_{\mathfrak{s}}^{+,\pm}=v_{\mathfrak{s}}^{-,\pm}=v_{\mathfrak{s}}^{\pm}$ if
$\mathfrak{s}$ is non-übereven with has monochromatic blue children;
$v_{\mathfrak{s}}^{\pm,\pm}=v_{\mathfrak{s}}^{+}$,
$v_{\mathfrak{s}}^{\pm,\mp}=v_{\mathfrak{s}}^{-}$ if $\mathfrak{s}$ is
übereven with chromatic children; and
$v_{\mathfrak{s}}^{+}=v_{\mathfrak{s}}^{-}=v_{\mathfrak{s}}$ if $\mathfrak{s}$
is non-übereven with polychromatic children.

Moreover, if $\mathcal{R}$ is not principal333Recall that if $\mathcal{R}$ is
purple and has a unique proper child $\mathfrak{s}^{\prime}$ of size
$2g(C_{h})$ which is black, then $\mathcal{R}$ is principal and doesn’t fall
in this category., then there are the following additional edges:

Name | From | To | Length | Condition
---|---|---|---|---
$L_{\mathfrak{t}}$ | $v_{\mathfrak{s}}^{+}$ | $v_{\mathfrak{s}}^{-}$ | $\delta_{\mathfrak{s}}$ |  $\mathfrak{s}<\mathfrak{t}$, $\mathfrak{t}$ cotwin, $\mathfrak{s}$ purple
$L_{\mathfrak{t}}^{+}$ | $v_{\mathfrak{s}}^{+,+}$ | $v_{\mathfrak{s}}^{\sigma,-\sigma}$ | $2\delta_{\mathfrak{s}}$ | $\mathfrak{s}<\mathfrak{t}$, $\mathfrak{t}$ cotwin, $\mathfrak{s}$ black
$L_{\mathfrak{t}}^{-}$ | $v_{\mathfrak{s}}^{-,-}$ | $v_{\mathfrak{s}}^{-\sigma,\sigma}$
$L_{\mathfrak{s},\mathfrak{s}^{\prime}}^{+}$ | $v_{\mathfrak{s}}^{+}$ | $v_{\mathfrak{s}^{\prime}}^{+}$ | $\frac{1}{2}(\delta_{\mathfrak{s}}+\delta_{\mathfrak{s}^{\prime}})$ | $\mathcal{R}=\mathfrak{s}\sqcup\mathfrak{s}^{\prime}$, $\mathfrak{s},\mathfrak{s}^{\prime}$ both principal, same chromatic colour
$L_{\mathfrak{s},\mathfrak{s}^{\prime}}^{-}$ | $v_{\mathfrak{s}}^{-}$ | $v_{\mathfrak{s}^{\prime}}^{-}$
$L_{\mathfrak{s},\mathfrak{s}^{\prime}}^{+,+}$ | $v_{\mathfrak{s}}^{+,+}$ | $v_{\mathfrak{s}^{\prime}}^{+,+}$ | $\delta_{\mathfrak{s}}+\delta_{\mathfrak{s}^{\prime}}$ | $\mathcal{R}=\mathfrak{s}\sqcup\mathfrak{s}^{\prime}$, $\mathfrak{s},\mathfrak{s}^{\prime}$ both principal, black
$L_{\mathfrak{s},\mathfrak{s}^{\prime}}^{+,-}$ | $v_{\mathfrak{s}}^{+,-}$ | $v_{\mathfrak{s}^{\prime}}^{+,-}$
$L_{\mathfrak{s},\mathfrak{s}^{\prime}}^{-,+}$ | $v_{\mathfrak{s}}^{-,+}$ | $v_{\mathfrak{s}^{\prime}}^{-,+}$
$L_{\mathfrak{s},\mathfrak{s}^{\prime}}^{-,-}$ | $v_{\mathfrak{s}}^{-,-}$ | $v_{\mathfrak{s}^{\prime}}^{-,-}$
$L_{\mathfrak{t}}$ | $v_{\mathfrak{s}}^{+}$ | $v_{\mathfrak{s}}^{-}$ | $\delta_{\mathfrak{s}}+\delta_{\mathfrak{t}}$ |  $\mathcal{R}=\mathfrak{s}\sqcup\mathfrak{t}$, $\mathfrak{s}$ principal, $\mathfrak{t}$ twin, both purple
$L_{\mathfrak{t}}^{+}$ | $v_{\mathfrak{s}}^{+,+}$ | $v_{\mathfrak{s}}^{\sigma,-\sigma}$ | $2(\delta_{\mathfrak{s}}+\delta_{\mathfrak{t}})$ | $\mathcal{R}=\mathfrak{s}\sqcup\mathfrak{t}$, $\mathfrak{s}$ principal, $\mathfrak{t}$ twin, both black
$L_{\mathfrak{t}}^{-}$ | $v_{\mathfrak{s}}^{-,-}$ | $v_{\mathfrak{s}}^{-\sigma,\sigma}$

######  Remark 3.2.

Possibly after a totally ramified extension, we can still think of proper, not
principal clusters as contributing components to the special fibre. However,
these components may not be principal In other words, they contribute
components isomorphic to $\mathbb{P}^{1}$ which intersects the rest of the
special fibre in precisely two places. So for example, a loop
$L_{\mathfrak{t}}$ arising from a chromatic twin can be thought of as a
component $v_{\mathfrak{t}}$ with two linking chains
$L_{\mathfrak{t}}^{+},L_{\mathfrak{t}}^{-}$ to $v_{P(\mathfrak{t})}$ of length
$\frac{1}{2}\delta_{\mathfrak{t}}$ (i.e., the linking chains arising in the
first two rows of the first table). We have chosen not to state our theorems
in this way since the component $v_{\mathfrak{t}}$ sometimes only appears in
the minimal regular model after a totally ramified extension (and has
multiplicity $2$ otherwise), but for the purposes of our proof we will usually
take this point of view. That is, we will go to a totally ramified extension,
treat all proper clusters as if they give us components, and then use Lemma
4.3 to move between totally ramified extensions.

######  Theorem 3.3.

Denote the Frobenius automorphism by $\phi$. It acts on
$\mathscr{Y}^{\min}_{k}$ in the following way:

  1. (i)

$\phi(\Gamma_{\mathfrak{s}}^{\pm})=\Gamma_{\phi(\mathfrak{s})}^{\pm\epsilon_{\mathfrak{s},i}(\phi)}$
for $\mathfrak{s}$ with chromatic children, $i\in\\{1,2,h\\}$ with
$\mathfrak{s}\in\Sigma_{i}$ übereven,

  2. (ii)

$\phi(\Gamma_{\mathfrak{s}}^{\pm,\pm})=\Gamma_{\phi(\mathfrak{s})}^{\pm\epsilon_{\mathfrak{s},2}(\phi),\pm\epsilon_{\mathfrak{s},1}(\phi)}$
for $\mathfrak{s}$ übereven with no chromatic children,

  3. (iii)

$\phi(L_{\mathfrak{s}}^{\pm})=\Gamma_{\phi(\mathfrak{s})}^{\pm\epsilon_{\mathfrak{s},i}(\phi)}$
for $\mathfrak{s}$ chromatic, $i\in\\{1,2,h\\}$ with with
$\mathfrak{s}\in\Sigma_{i}$ even,

  4. (iv)

$\phi(L_{\mathfrak{s}}^{\pm,\pm})=L_{\phi(\mathfrak{s})}^{\pm\epsilon_{\mathfrak{s},2}(\phi),\pm\epsilon_{\mathfrak{s},1}(\phi)}$
for $\mathfrak{s}$ black,

  5. (v)

$\phi(L_{\mathfrak{t}})=\epsilon_{\mathfrak{t},h}(\phi)L_{\phi(\mathfrak{t})}$
for $\mathfrak{t}$ a chromatic twin, where $-L$ denotes $L$ with the opposite
orientation,

  6. (vi)

$\phi(L_{\mathfrak{t}}^{\pm})=\epsilon_{\mathfrak{t},j}(\phi)L_{\phi(\mathfrak{t})}^{\pm\epsilon_{\mathfrak{t},i}(\phi)}$
for $\mathfrak{t}$ a black twin, $i,j\in\\{1,2\\}$ such that $\mathfrak{t}$ is
empty in $\Sigma_{i}$ and $i\neq j$.

######  Remark 3.4.

For simplicity of proof, we have added the technical condition that the depth
of all clusters are integers, even though there exist semistable curves with
clusters of half integer depth (for example, any bihyperelliptic curve where
$C_{1}$ is the elliptic curve given by $y_{1}^{2}=(x^{2}-p^{3})(x-1))$.
However, this is only a very mild restriction, since this condition can always
be attained by going to the ramified extension of $K$ of degree $2$. The dual
graph of $\mathscr{Y}_{k}$ over $K$ is then the same as the dual graph of
$\mathscr{Y}_{k}$ over $K(\sqrt{\pi})$, except the lengths of all linking
chains are halved (see Lemma 4.3).

######  Example 3.5.

Recall example 1.4. Let $C_{1}:y_{1}^{2}=(x-p^{n})(x^{2}-1)$ and
$C_{2}:y_{2}^{2}=(x+p^{n})(x^{2}-2)$ be elliptic curves, with composite curve
$C_{h}:y_{h}^{2}=(x^{2}-p^{2n})(x^{2}-1)(x^{2}-2)$ and associated bielliptic
curve $Y$. The chromatic cluster picture of $Y$ and the dual graph of
$\mathscr{Y}^{\textrm{min}}_{k}$ are below.

$n$$\mathfrak{s}$$0$$\mathcal{R}$

$v_{\mathcal{R}}$3n

In order to understand the Frobenius automorphism $\phi$, we note that the
twin $\mathfrak{s}$ is only even in $\Sigma_{h}$, not $\Sigma_{1}$ or
$\Sigma_{2}$, so the action of Frobenius $\phi$ on the loop is
$\phi(L_{\mathfrak{s}})=\epsilon_{\mathfrak{s},h}(\phi)L_{\mathfrak{s}}$. We
can calculate $\epsilon_{\mathfrak{s},h}=\left(\frac{2}{p}\right)$ and so the
loop is inverted if and only if $2$ is not a quadratic residue mod $p$.

######  Example 3.6.

Recall example 1.10. Let $C_{1}:y_{1}^{2}=(x^{2}-p^{2n})(x-1)$ and
$C_{2}:y_{2}^{2}=x^{3}-2$ be two elliptic curves with composite curve
$C_{h}:y_{h}^{2}=(x^{2}-p^{n})(x-1)(x^{3}-2)$ and associated bielliptic curve
$Y$. The chromatic cluster picture of $Y$ and the dual graph of
$\mathscr{Y}^{\textrm{min}}_{k}$ are below.

$n$$\mathfrak{s}$$0$$\mathcal{R}$

$v_{\mathcal{R}}$22n2n

Frobenius, $\phi$ acts on the loops of $\mathfrak{s}$ as
$\phi(L_{\mathfrak{s}}^{\pm})=\epsilon_{\mathfrak{s},1}L_{\mathfrak{s}}^{\pm\epsilon_{\mathfrak{s},2}(\phi)}$,
since $\mathfrak{s}$ is empty in $\Sigma_{2}$. We can calculate
$\epsilon_{\mathfrak{s},1}(\phi)=\left(\frac{-1}{p}\right)$ and
$\epsilon_{\mathfrak{s},2}=\left(\frac{-2}{p}\right)$, and so the action of
Frobenius swaps the loops if $-2$ is not a quadratic residue mod $p$, and the
loops are inverted if $-1$ is not a quadratic residue mod $p$.

######  Example 3.7.

Let $C_{1}:y_{1}^{2}=((x+p^{2})^{2}-p^{14})(x-2+p^{3})$ and
$C_{2}:y_{2}^{2}=((x-p^{2})^{2}-p^{1}2)(x-2-p^{3})$ be hyperelliptic curves
over $K$ and let $Y$ be their associated bihyperelliptic curve. The chromatic
cluster picture of $Y$ consists of the übereven top cluster $\mathcal{R}$ with
chromatic children, the übereven cluster $\mathfrak{s}$ with no chromatic
children, and three twins $\mathfrak{t}_{1},\mathfrak{t}_{2}$ and
$\mathfrak{t}_{3}$, the first two black with monochromatic children and the
latter chromatic with polychromatic children. Note that $\mathcal{R}$ is
principal despite being the disjoint union of two clusters as its two children
are purple and black. The most subtle part of theorem is illustrated here:
that the components arising from $\mathcal{R}$, and the loops arising from
$\mathfrak{t}_{1}$ and $\mathfrak{t}_{2}$ link to different pairs of
$\Gamma_{\mathfrak{s}}^{+,+},\Gamma_{\mathfrak{s}}^{+,-},\Gamma_{\mathfrak{s}}^{-,+}$
and $\Gamma_{\mathfrak{s}}^{-,-}$; for example, $\Gamma_{\mathcal{R}}^{+}$
links to $\Gamma_{\mathfrak{s}}^{+,+}$ and $\Gamma_{\mathfrak{s}}^{-,-}$,
whereas $L_{\mathfrak{t}_{1}}^{+}$ links to $\Gamma_{\mathfrak{s}}^{+,+}$ and
$\Gamma_{\mathfrak{s}}^{+,-}$. This is because $\mathcal{R}$ is übereven with
chromatic children whereas $\mathfrak{t}_{1}$ is not übereven with
monochromatic (red) children.

$5$$\mathfrak{t}_{1}$$4$$\mathfrak{t}_{2}$$2$$\mathfrak{s}$$3$$\mathfrak{t}_{3}$$0$$\mathcal{R}$

$\Gamma_{\mathcal{R}}^{-}$$\Gamma_{\mathcal{R}}^{+}$$\Gamma_{\mathfrak{s}}^{+,+}$$\Gamma_{\mathfrak{s}}^{-,-}$$\Gamma_{\mathfrak{s}}^{+,-}$$\Gamma_{\mathfrak{s}}^{-,+}$$L_{\mathfrak{t}_{3}}$$L^{+}_{\mathfrak{t}_{1}}$$L^{+}_{\mathfrak{t}_{2}}$$L^{-}_{\mathfrak{t}_{1}}$$L^{-}_{\mathfrak{t}_{2}}$

######  Remark 3.8.

The above is an example of a curve whose minimal regular model has a special
fibre with a non-planar dual graph (its dual graph is a $K_{3,3}$). It is in
fact an example of minimal genus, since the special fibre is totally
degenerate (all of the components have genus $0$).

##  4\. Proof

The strategy of proof will be as follows: let $\mathscr{X}$ be the minimal
model of $\mathbb{P}^{1}$ which separates the branch points of the map
$Y\rightarrow\mathbb{P}^{1}$. We construct such a model from $\Sigma_{h}$
following the techniques of [8, Sections 3-4]. Normalising $\mathscr{X}$ in
the function field $K(Y)$ gives a regular model $\mathscr{Y}$ of $Y$, which
results in a semistable model $\mathscr{Y}^{\textrm{min}}$ after blowing down
components of multiplicity greater than $1$. We also obtain models
$\mathcal{C}_{1},\mathcal{C}_{h},\mathcal{C}_{2}$ of $C_{1},C_{h}$ and $C_{2}$
respectively by normalising in the appropriate function fields. The special
fibres of these intermediate models are computed using the result of [8,
Sections 5-6].

######  Notation 4.1.

There are many cases where we shall wish to refer to some object associated to
a cluster $\mathfrak{s}$ for each of the curves
$\mathbb{P}^{1},C_{1},C_{2},C_{h}$ and $Y$. For example, we may wish to refer
to the components arising from $\mathfrak{s}$. In this case, the component(s)
in $\mathscr{Y}_{k}$ will appear without subscript: $\Gamma_{\mathfrak{s}}$,
and those in $\mathscr{X}_{k}$ (resp.
$\mathcal{C}_{1,k},\mathcal{C}_{h,k},\mathcal{C}_{2,k}$) will be denoted
$\Gamma_{\mathfrak{s},\mathbb{P}^{1}}$ (resp.
$\Gamma_{\mathfrak{s},1},\Gamma_{\mathfrak{s},h},\Gamma_{\mathfrak{s},2}$).

######  Remark 4.2.

The models $\mathcal{C}_{1},\mathcal{C}_{2}$ and $\mathcal{C}_{h}$ are not
necessarily minimal models, as they come from the red (resp. blue resp.
chromatic) cluster picture of $Y$ \- the admissible collection of disks
arising from these is not in general the same as that arising from the cluster
picture of $C_{1}$ (resp. $C_{2}$, $C_{h}$).

######  Lemma 4.3.

Let $C$ be a curve over $K^{\mathrm{ur}}$ with semistable reduction and let
$\mathcal{C}$ be a semistable model of $C$. Let $L/K$ be a totally ramified
extension of degree $e$. Then
$\mathcal{C}\times_{\mathcal{O}_{K}}\mathcal{O}_{L}$ is a semistable model of
$C\times_{K}L$ with the same dual graph as $\mathcal{C}$, but with the lengths
of all edges multiplied by $e$.

###### Proof.

This follows from [12, Corollaries 10.3.36,10.3.25], noting that the thickness
([12, Definition 10.3.23]) of all double points multiplies by $e$ after
extending the field. ∎

This allows us to happily move between ramified extensions of $K$ when
describing the semistable model, which will prove invaluable.

######  Lemma 4.4.

Let $Y/K$ be a bihyperelliptic curve with semistable reduction. Then
$\mathscr{Y}$, the normalisation of $\mathscr{X}$ in $K(Y)$, is a proper
regular model of $Y$.

###### Proof.

The normalisation of $\mathscr{X}$ in $K(Y)$ is isomorphic to the
normalisation of $\mathcal{C}_{1}$ in $K(Y)$, so it is sufficient to prove
that the latter is a proper regular model of $Y$. Let
$\varphi:C_{1}\rightarrow\mathbb{P}^{1}$ be the canonical double cover and
write $\mathcal{D}=(\varphi_{1}^{-1}(f_{2}))=\sum m_{i}\Gamma_{i}$, the
divisor of (the pullback of) $f_{2}$ on $\mathcal{C}_{1}$. By [16, Lemma 2.1],
it is sufficient to prove that

  1. (i)

$\varphi_{1}^{-1}(f_{2})$ is square-free on $\mathcal{C}_{1}$,

  2. (ii)

any two $\Gamma_{i}$ for which $m_{i}$ is odd do not intersect and,

  3. (iii)

any $\Gamma_{i}$ for which $m_{i}$ is odd is regular.

Since $f_{1}$ and $f_{2}$ share no common roots, $\varphi_{1}^{-1}(f_{2})$ is
square-free on $\mathcal{C}_{1}$ and the horizontal components of
$\mathcal{D}$ do not intersect. We are left to consider the vertical
components of $\mathcal{D}$. Note that any vertical component of odd
multiplicity must arise as the preimage of some $E\in(f_{2})_{\mathrm{vert}}$
which appears with odd multiplicity. The component $E$ has either one or two
preimages in $\mathcal{C}_{1}$. In the first case, the preimage $\Gamma$ is
regular by [8, Theorem 5.2]. Since $E$ does not intersect any other component
of $(f_{2})$ of odd multiplicity, $\Gamma$ cannot intersect a component of
$\mathcal{D}$. In the second case, the two components are still regular and do
not intersect each other, and cannot intersect any other component
$\mathcal{D}$ as $E$ does not intersect any other component of $(f_{2})$ of
odd multiplicity. ∎

######  Proposition 4.5.

Let $Y$ be a semistable bihyperelliptic curve as in Theorem 3.1 and
$\mathscr{Y}$ the model obtained by normalising. Let $\mathscr{Y}^{\min}$ be
the minimal regular model of $\mathscr{Y}$. Then each principal cluster
$\mathfrak{s}$ contributes the following components to
$\mathscr{Y}_{k}^{\mathrm{min}}$: if übereven, $1$ component
$\Gamma_{\mathfrak{s}}$ if $\mathfrak{s}$ has polychromatic children and $2$
components $\Gamma_{\mathfrak{s}}^{+},\Gamma_{\mathfrak{s}}^{-}$ if
$\mathfrak{s}$ has monochromatic children; and if übereven: $2$ components
$\Gamma_{\mathfrak{s}}^{+},\Gamma_{\mathfrak{s}}^{-}$ if $\mathfrak{s}$ has
chromatic children and $4$ components
$\Gamma_{\mathfrak{s}}^{+,+},\Gamma_{\mathfrak{s}}^{+,-},\Gamma_{\mathfrak{s}}^{-,+},\Gamma_{\mathfrak{s}}^{-,-}$
if $\mathfrak{s}$ has no chromatic children.

###### Proof.

Consider a principal cluster $\mathfrak{s}$ and its corresponding component
$\Gamma_{\mathfrak{s},\mathbb{P}^{1}}$ in the model of $\mathbb{P}^{1}$. If
$\mathfrak{s}$ is not übereven, then $\Gamma_{\mathfrak{s},\mathbb{P}^{1}}$
lifts to one component $\Gamma_{\mathfrak{s},h}$ in $\mathcal{C}_{h}$ and so
lifts to either one or two components in $\mathscr{Y}_{k}$. Suppose
$\mathfrak{s}$ has polychromatic children. Then $\mathfrak{s}$ has one
corresponding component $\Gamma_{\mathfrak{s},1}\in\mathcal{C}_{1}$, and
$\Gamma_{\mathfrak{s},1}$ contains branch points of the morphism
$\mathscr{Y}\rightarrow\mathcal{C}_{1}$ (corresponding to the blue children of
$\mathfrak{s}$), and hence must lift to a single component in
$\mathscr{Y}_{k}$. If $\mathfrak{s}$ has monochromatic (e.g. red) children
then it has two components in either $\mathcal{C}_{1}$ or $\mathcal{C}_{2}$
(in this case $\mathcal{C}_{2}$), and hence must have two associated
components in $\mathscr{Y}$.

If $\mathfrak{s}$ is übereven, then it has two components in $\mathcal{C}_{h}$
so lifts to either two or four in $\mathscr{Y}_{k}$. If $\mathfrak{s}$ has
chromatic (e.g. red) children then it has a single component in either
$\mathcal{C}_{1}$ or $\mathcal{C}_{2}$ (in this case $\mathcal{C}_{1}$), so
can only lift to two components in $\mathscr{Y}_{k}$. If $\mathfrak{s}$ has no
chromatic children, then it has two corresponding components in each of
$\mathcal{C}_{1},\mathcal{C}_{2}$ and $\mathcal{C}_{h}$. Since $C_{2}\times
C_{2}$ acts on the components of $\mathscr{Y}_{k}$ arising from
$\mathfrak{s}$, and their images are the $\Gamma_{\mathfrak{s},i}^{\pm}$ under
the quotient of the three non trivial subgroups of $C_{2}\times C_{2}$, there
must be four components corresponding to $\mathfrak{s}$ in $\mathscr{Y}_{k}$.

Since $\mathfrak{s}$ is principal, it is principal in one of
$\Sigma_{1},\Sigma_{2}$ or $\Sigma_{h}$, say $\Sigma_{1}$. Then by [8, Theorem
8.5], any component $\Gamma_{\mathfrak{s},1}\in\mathcal{C}_{1}$ arising from
$\mathfrak{s}$ either has positive genus, or intersects at least three other
components. Therefore the same can be said for any $\Gamma\in\mathscr{Y}_{k}$
arising from $\mathfrak{s}$. We cannot blow such components down, and hence
the same components appear in $\mathscr{Y}^{\min}_{k}$. ∎

######  Proposition 4.6.

Let $Y,\mathscr{Y}^{\min}$ be as in Proposition 4.5, and let
$\mathfrak{s}^{\prime}<\mathfrak{s}$ be principal clusters of $Y$. Then the
components of $\mathfrak{s}$ and $\mathfrak{s}^{\prime}$ in $\mathscr{Y}_{k}$
are linked by two chains if $\mathfrak{s}^{\prime}$ is chromatic and four
otherwise, as described in the statement of Theorem 3.1. Furthermore, if
$\mathfrak{t}<\mathfrak{s}$ is a twin or $\mathfrak{s}<\mathfrak{t}$ a cotwin
then there is one loop if the child is chromatic and two loops if the child is
black, and if $\mathcal{R}=\mathfrak{s}\sqcup\mathfrak{s}^{\prime}$ is not
principal then the components of $\mathfrak{s}$ and $\mathfrak{s}^{\prime}$
are linked as in the statement of Theorem 3.1.

###### Proof.

Assume that we are in the case where $\mathfrak{s}^{\prime}<\mathfrak{s}$ are
both principal, since the other cases are checked similarly using Remark 3.2.
It is clear that components have linking chains to the components
corresponding to their parents, since this is the case for the model of
$\mathbb{P}^{1}$. Therefore, we have to calculate how many linking chains
there are ($2$ or $4$) and precisely which components are linked to which
others. The lengths of the linking chains is proved separately in Lemma 4.7.

Suppose that $\mathfrak{s}^{\prime}<\mathfrak{s}$ with $\mathfrak{s}^{\prime}$
chromatic. Then there are several cases for the different children
$\mathfrak{s}^{\prime}$ can have. If $\mathfrak{s}^{\prime}$ is red, then
$\Gamma_{\mathfrak{s},1}$ and $\Gamma_{\mathfrak{s}^{\prime},1}$ are linked by
one chain and $\Gamma_{\mathfrak{s},2}^{\pm}$ and
$\Gamma_{\mathfrak{s}^{\prime},2}^{\pm}$ are linked by two chains. Similarly
if $\mathfrak{s}^{\prime}$ is blue (swapping $1$ and $2$). In either case,
$\Gamma_{\mathfrak{s}}^{\pm}$ and $\Gamma_{\mathfrak{s}^{\prime}}^{\pm}$ are
linked by two chains. Similarly if $\mathfrak{s}^{\prime}$ is purple then
$\Gamma_{\mathfrak{s},h}$ and $\Gamma_{\mathfrak{s}^{\prime},h}$ are linked by
two chains but $\Gamma_{\mathfrak{s},1}$ and
$\Gamma_{\mathfrak{s}^{\prime},1}$ are linked by one chain so we get two
linking chains upstairs.

Now suppose that $\mathfrak{s}^{\prime}$ is black. Then
$\Gamma_{\mathfrak{s}^{\prime},1}^{\pm}$ has two linking chains up to the
components of its parent, as does $\Gamma_{\mathfrak{s}^{\prime},2}$, so by a
similar argument to Proposition 4.5 the components of $\mathfrak{s}^{\prime}$
have four linking chains up to the components of $\mathfrak{s}$. Therefore the
number of linking chains is correct. We must check that the correct components
are linked.

This is done on a case by case basis. First assume
$\mathfrak{s},\mathfrak{s}^{\prime}$ are not übereven. If $\mathfrak{s}$ has
polychromatic children then it only has one component and everything is
correct up to relabelling. Similarly if $\mathfrak{s}^{\prime}$ has
polychromatic children. So assume both $\mathfrak{s}$ and
$\mathfrak{s}^{\prime}$ have monochromatic children. If they have
monochromatic children of the same colour (say red), their components are
linked in the following way in the tower of models:

${\Gamma_{\mathfrak{s}}^{+}}$${\Gamma_{\mathfrak{s}^{\prime}}^{+}}$${\Gamma_{\mathfrak{s}}^{-}}$${\Gamma_{\mathfrak{s}^{\prime}}^{-}}$${\Gamma_{\mathfrak{s},1}}$${\Gamma_{\mathfrak{s}^{\prime},1}}$${\Gamma_{\mathfrak{s},h}}$${\Gamma_{\mathfrak{s}^{\prime},h}}$${\Gamma_{\mathfrak{s},2}^{+}}$${\Gamma_{\mathfrak{s}^{\prime},2}^{+}}$${\Gamma_{\mathfrak{s},2}^{-}}$${\Gamma_{\mathfrak{s}^{\prime},2}^{-}}$${\Gamma_{\mathfrak{s},\mathbb{P}^{1}}}$${\Gamma_{\mathfrak{s},\mathbb{P}^{1}}}$

If $\mathfrak{s}$ and $\mathfrak{s}^{\prime}$ have monochromatic children of
different colours (e.g. $\mathfrak{s}$ red and $\mathfrak{s}^{\prime}$ blue),
then their components are linked in the following way in the tower of models:

${\Gamma_{\mathfrak{s}}^{+}}$${\Gamma_{\mathfrak{s}^{\prime}}^{+}}$${\Gamma_{\mathfrak{s}}^{-}}$${\Gamma_{\mathfrak{s}^{\prime}}^{-}}$${\Gamma_{\mathfrak{s},1}}$${\Gamma_{\mathfrak{s}^{\prime},1}^{+}}$${\Gamma_{\mathfrak{s}^{\prime},1}^{-}}$${\Gamma_{\mathfrak{s},h}}$${\Gamma_{\mathfrak{s}^{\prime},h}}$${\Gamma_{\mathfrak{s},2}^{+}}$${\Gamma_{\mathfrak{s}^{\prime},2}}$${\Gamma_{\mathfrak{s},2}^{-}}$${\Gamma_{\mathfrak{s},\mathbb{P}^{1}}}$${\Gamma_{\mathfrak{s},\mathbb{P}^{1}}}$

From here the following notation arises: that
$\Gamma_{\mathfrak{s}}^{\pm,+}=\Gamma_{\mathfrak{s}}^{\pm,-}=\Gamma_{\mathfrak{s}}^{\pm}$
if $\mathfrak{s}$ has monochromatic red children and
$\Gamma_{\mathfrak{s}}^{+,\pm}=\Gamma_{\mathfrak{s}}^{-,\pm}=\Gamma_{\mathfrak{s}}^{\pm}$
if $\mathfrak{s}$ has monochromatic blue children. If $\mathfrak{s}$ or
$\mathfrak{s}^{\prime}$ is übereven, the different cases can be checked
similarly.

∎

######  Lemma 4.7.

Let $Y,\mathscr{Y}^{\min}$ be as in Proposition 4.5, and let
$\mathfrak{s}^{\prime}<\mathfrak{s}$ be two principal clusters. Then any
linking chain $L_{\mathfrak{s}^{\prime}}$ arising from this pair has length
$\frac{1}{2}\delta_{\mathfrak{s}^{\prime}}$ if $\mathfrak{s}^{\prime}$ is
chromatic and $\delta_{\mathfrak{s}^{\prime}}$ otherwise. If
$\mathfrak{t}<\mathfrak{s}$ is a twin or $\mathfrak{s}<\mathfrak{t}$ a cotwin
then any loop arising from $\mathfrak{t}$ has length $\delta_{\mathfrak{t}}$
if the child is chromatic, and $2\delta_{\mathfrak{t}}$ otherwise. If
$\mathcal{R}=\mathfrak{s}\sqcup\mathfrak{s}^{\prime}$ is not principal, then
any length of a linking chain arising from $\mathfrak{s}$ and
$\mathfrak{s}^{\prime}$ has length as described in Theorem 3.1.

###### Proof.

Possibly after a finite extension $L/K^{\mathrm{ur}}$, the map
$Y\rightarrow\mathbb{P}^{1}$ extends to a map of models
$\mathscr{Y}\rightarrow\mathscr{X}$ by [13, Theorem 2.3]. Furthermore, this
map induces a harmonic morphism of augmented $\mathbb{Z}$-graphs, in the sense
of [1, Section 2] (see also Sections 5,8) on the dual graphs of $\mathscr{Y}$
and $\mathscr{X}$. The length of an edge between two vertices in an augmented
$\mathbb{Z}$-graph is the thickness of the intersection point of the
components the vertices represent. Therefore the distance between two vertices
of degree $\geq 3$ is exactly the length of the linking chain between them.

If $\mathfrak{s}^{\prime}<\mathfrak{s}$ are principal, the lemma follows,
noting that a linking chain from a chromatic cluster to its parent has two
preimages in $\mathscr{Y}^{\prime}$ (so the length halves), but a linking
chain from a black cluster to its parent has four preimages so the length
stays the same.

If $\mathfrak{t}<\mathfrak{s}$ is a chromatic twin then (possibly after a
field extension), we can think of the loop $L_{\mathfrak{t}}$ as consisting of
a component $\Gamma_{\mathfrak{t}}$, the unique lift of
$\Gamma_{\mathfrak{t},\mathbb{P}^{1}}$ with two linking chains to
$\Gamma_{\mathfrak{s}}$. Since $\mathfrak{t}$ is chromatic, by the argument
above the linking chains will both have length
$\frac{1}{2}\delta_{\mathfrak{t}}$ and hence the total loop will have length
$\delta_{\mathfrak{t}}$. A similar argument is made if $\mathfrak{t}$ is a
black twin, or if $\mathcal{R}=\mathfrak{s}\sqcup\mathfrak{s}^{\prime}$ is not
principal. ∎

######  Proposition 4.8.

Let $Y,\mathscr{Y}^{\min}$ be as in Proposition 4.5 and let $\mathfrak{s}$ be
a principal cluster of $Y$. Then the components associated to $\mathfrak{s}$
have genus $g_{\chi}(\mathfrak{s})$.

###### Proof.

Let $\mathfrak{s}$ be a principal cluster. First suppose $\mathfrak{s}$ is not
übereven and has polychromatic children. In this case there is a unique
component $\Gamma_{\mathfrak{s}}$ arising from $\mathfrak{s}$. This is then a
direct application of Riemann-Hurwitz. The children of $\mathfrak{s}$
correspond to points on the component $\Gamma_{\mathfrak{s},\mathbb{P}^{1}}$
as in [8, Definition 3.7], and by Proposition 4.6 the points arising from
chromatic children are precisely the non-infinity branch points of
$\Gamma_{\mathfrak{s}}\rightarrow\Gamma_{\mathfrak{s},\mathbb{P}^{1}}$. In
addition, there is an extra branch point at infinity, unless $\mathfrak{s}$ is
the top cluster and $f_{1}$ and $f_{2}$ both have even degree, or the top
cluster $\mathcal{R}=\mathfrak{s}\bigsqcup\mathfrak{s}^{\prime}$ is not
principal, $\mathfrak{s}$ is even and $f_{1}$ and $f_{2}$ both have even
degree.

If $\mathfrak{s}$ is not übereven and has monochromatic children, then the
components $\Gamma_{\mathfrak{s}}^{+}$ and $\Gamma_{\mathfrak{s}}^{-}$ are
each isomorphic to $\Gamma_{\mathfrak{s},h}$ and so have genus
$g(\mathfrak{s})$. The same is true if $\mathfrak{s}$ is übereven and has
chromatic children, except with $\Gamma_{\mathfrak{s},1}$ if $\mathfrak{s}$
has red children and $\Gamma_{\mathfrak{s},2}$ if blue. If $\mathfrak{s}$ is
übereven with no chromatic children then its $4$ components must have genus
$0=g(\mathfrak{s})$ as well. ∎

######  Theorem 4.9.

Denote the Frobenius automorphism by $\phi$. It acts on the dual graph of
$\mathscr{Y}^{\min}_{k}$ in the following way:

  1. (i)

$\phi(\Gamma_{\mathfrak{s}}^{\pm})=\Gamma_{\phi(\mathfrak{s})}^{\pm\epsilon_{\mathfrak{s},i}(\phi)}$
for $\mathfrak{s}$ with chromatic children, $i\in\\{1,2,h\\}$ with
$\mathfrak{s}\in\Sigma_{i}$ übereven,

  2. (ii)

$\phi(\Gamma_{\mathfrak{s}}^{\pm,\pm})=\Gamma_{\phi(\mathfrak{s})}^{\pm\epsilon_{\mathfrak{s},2}(\phi),\pm\epsilon_{\mathfrak{s},1}(\phi)}$
for $\mathfrak{s}$ übereven with no chromatic children,

  3. (iii)

$\phi(L_{\mathfrak{s}}^{\pm})=\Gamma_{\phi(\mathfrak{s})}^{\pm\epsilon_{\mathfrak{s},i}(\phi)}$
for $\mathfrak{s}$ chromatic, $i\in\\{1,2,h\\}$ with with
$\mathfrak{s}\in\Sigma_{i}$ even,

  4. (iv)

$\phi(L_{\mathfrak{s}}^{\pm,\pm})=L_{\phi(\mathfrak{s})}^{\pm\epsilon_{\mathfrak{s},2}(\phi),\pm\epsilon_{\mathfrak{s},1}(\phi)}$
for $\mathfrak{s}$ black,

  5. (v)

$\phi(L_{\mathfrak{t}})=\epsilon_{\mathfrak{t},h}(\phi)L_{\phi(\mathfrak{t})}$
for $\mathfrak{t}$ a chromatic twin, where $-L$ denotes $L$ with the opposite
orientation,

  6. (vi)

$\phi(L_{\mathfrak{t}}^{\pm})=\epsilon_{\mathfrak{t},j}(\phi)L_{\phi(\mathfrak{t})}^{\pm\epsilon_{\mathfrak{t},i}(\phi)}$
for $\mathfrak{t}$ a black twin, $i,j\in\\{1,2\\}$ such that $\mathfrak{t}$ is
empty in $\Sigma_{i}$ and $i\neq j$.

###### Proof.

By Proposition 4.5, the components we must blow down to obtain
$\mathscr{Y}^{\min}_{k}$ from $\mathscr{Y}_{k}$ are all in linking chains, so
it is sufficient to calculate the action of Frobenius on $\mathscr{Y}_{k}$ as
the action on the shortened linking chains is the same as the originals. The
action of Frobenius commutes with the quotient maps, so we can deduce the
action of Frobenius on the components of $\mathscr{Y}_{k}$ from the
corresponding action of Frobenius on $\mathcal{C}_{1},\mathcal{C}_{2}$ and
$\mathcal{C}_{h}$, which is known by [8, Theorem 8.5]. First we focus on
clusters. For a principal cluster $\mathfrak{s}$, the set
$\mathcal{E}_{\mathfrak{s}}=\\{\Gamma_{\mathfrak{s}}^{\pm,\pm}\\}$ is mapped
to
$\mathcal{E}_{\phi(\mathfrak{s})}=\\{\Gamma_{\phi(\mathfrak{s})}^{\pm,\pm}\\}$
by Frobenius, since Frobenius maps the images of $\mathcal{E}_{\mathfrak{s}}$
to the images of $\mathcal{E}_{\phi(\mathfrak{s})}$ in
$\mathcal{C}_{1},\mathcal{C}_{2}$ and $\mathcal{C}_{h}$. It remains to show
which component of $\mathcal{E}_{\mathfrak{s}}$ is mapped to which of
$\mathcal{E}_{\phi(\mathfrak{s})}$.

If $\mathfrak{s}$ is a principal cluster with polychromatic children then
$\mathcal{E}_{\mathfrak{s}}$ consists of one component and hence there is
nothing to verify. If $\mathfrak{s}$ is a principal cluster with
monochromatic, red children (so therefore $\mathfrak{s}\in\Sigma_{2}$ is even)
then there are two components
$\Gamma_{\mathfrak{s}}^{+},\Gamma_{\mathfrak{s}}^{-}$ corresponding to
$\mathfrak{s}$, which are the lifts of two components
$\Gamma_{\mathfrak{s},2}^{+},\Gamma_{\mathfrak{s},2}^{-}\in\mathcal{C}_{2,k}$.
Therefore $\phi$ acts on $\Gamma_{\mathfrak{s}}^{\pm}$ as it does on
$\Gamma_{\mathfrak{s},2}^{\pm}$. But by [8, Theorem 8.5],
$\phi(\Gamma_{\mathfrak{s},2}^{\pm})=\Gamma_{\phi(\mathfrak{s}),2}^{\pm\epsilon_{2}(\phi)}$,
and so
$\phi(\Gamma_{\mathfrak{s}}^{\pm})=\Gamma_{\phi(\mathfrak{s})}^{\pm\epsilon_{2}(\phi)}$.
Similarly if $\mathfrak{s}$ has monochromatic, blue children or is übereven
with polychromatic children.

Now suppose that $\mathfrak{s}$ is an übereven cluster with no chromatic
children. In this case there are two components arising from $\mathfrak{s}$ in
$\mathcal{C}_{i,k}$ for $i=1,2$, $\Gamma_{\mathfrak{s},i}^{+}$ and
$\Gamma_{\mathfrak{s},i}^{-}$. Consider $i=1$. In $\mathcal{C}_{1}$, the
action of Frobenius is
$\phi(\Gamma_{\mathfrak{s},1}^{\pm})=\Gamma_{\phi(\mathfrak{s}),1}^{\pm\epsilon_{1}(\phi)}$.
The component $\Gamma_{\mathfrak{s},1}^{+}$ lifts to the components
$\Gamma_{\mathfrak{s}}^{+,+}$ and $\Gamma_{\mathfrak{s}}^{-,+}$ and so the set
$\\{\Gamma_{\mathfrak{s}}^{+,\pm},\Gamma_{\mathfrak{s}}^{-,\pm}\\}$ is mapped
to
$\\{\Gamma_{\mathfrak{s}}^{+,\pm\epsilon_{1}{\phi}},\Gamma_{\mathfrak{s}}^{-,\pm\epsilon_{1}{\phi}}\\}$.
The same argument for $i=2$ implies that the set
$\\{\Gamma_{\mathfrak{s}}^{\pm,+},\Gamma_{\mathfrak{s}}^{\pm,-}\\}$ is mapped
to the set
$\\{\Gamma_{\mathfrak{s}}^{\pm\epsilon_{2}(\phi),+},\Gamma_{\mathfrak{s}}^{\pm\epsilon_{2}(\phi),-}\\}$.
Combining these gives the action of Frobenius.

For linking chains between components of principal clusters
$\mathfrak{s}^{\prime}<\mathfrak{s}$, the action on the whole linking chain is
determined by the action on any component in the linking chain. Suppose $D$ is
a $p$-adic disk with $\mathfrak{s}^{\prime}<D<\mathfrak{s}$. If
$\mathfrak{s}^{\prime}$ is chromatic (i.e., case (iii)) then
$\Gamma_{D,\mathbb{P}^{1}}$ has two preimages in $\mathscr{Y}$ and these are
permuted like the principal components in (i). If $\mathfrak{s}^{\prime}$ is
black (case (iv)), then $\Gamma_{D,\mathbb{P}^{1}}$ has four preimages in
$\mathscr{Y}_{k}$ and these are permuted like the principal components in case
(ii).

Loops associated to twins and linking chains between $\mathfrak{s}$ and
$\mathfrak{s}^{\prime}$ when
$\mathcal{R}=\mathfrak{s}\sqcup\mathfrak{s}^{\prime}$ are not principal can be
dealt with using Remark 3.2, as in proofs in the rest of the section. ∎

## References

  * [1] Omid Amini, Matthew Baker, Erwan Brugallé, and Joseph Rabinoff.  Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta.  Research in the Mathematical Sciences, 2(1):1–67, 2015.
  * [2] Alex J. Best, L. Alexander Betts, Matthew Bisatt, Raymond van Bommel, Vladimir Dokchitser, Omri Faraggi, Sabrina Kunzweiler, Céline Maistret, Adam Morgan, Simone Muselli, and Sarah Nowell.  A user’s guide to the local arithmetic of hyperelliptic curves.  arXiv preprint arXiv:2007.01749, 2020.
  * [3] L. Alexander Betts.  On the computation of Tamagawa numbers and Néron component groups of Jacobians of semistable hyperelliptic curves.  Journal of Number Theory, to appear.
  * [4] Matthew Bisatt.  Clusters, inertia, and root numbers.  arXiv preprint arXiv:1902.08981, 2019.
  * [5] Irene Bouw, Nirvana Coppola, Pınar Kılıçer, Sabrina Kunzweiler, Elisa Lorenzo García, and Anna Somoza.  Reduction type of genus-3 curves in a special stratum of their moduli space.  Women in Numbers Europe III: Research Directions in Number Theory, to appear.
  * [6] Irene I Bouw and Stefan Wewers.  Computing L-functions and semistable reduction of superelliptic curves.  Glasgow Mathematical Journal, 59(1):77–108, 2017.
  * [7] Tim Dokchitser.  Models of curves over DVRs.  Duke Math. J, to appear.
  * [8] Tim Dokchitser, Vladimir Dokchitser, Céline Maistret, and Adam Morgan.  Arithmetic of hyperelliptic curves over local fields.  arXiv preprint arXiv:1808.02936, 2018.
  * [9] Omri Faraggi and Sarah Nowell.  Models of hyperelliptic curves with tame potentially semistable reduction.  Transactions of the London Mathematical Society, 7(1):49–95, 2020\.
  * [10] Sabrina Kunzweiler.  Differential forms on hyperelliptic curves with semistable reduction.  arXiv preprint arXiv:1902.07784, 2019.
  * [11] Reynald Lercier, Qing Liu, Elisa Lorenzo García, and Christophe Ritzenthaler.  Reduction type of smooth quartics.  Algebra and Number Theory, to appear.
  * [12] Qing Liu.  Algebraic Geometry and Arithmetic Curves.  Oxford Graduate Texts in Mathematics (0-19-961947-6). Oxford University Press, 2006.
  * [13] Qing Liu and Dino Lorenzini.  Models of curves and finite covers.  Compositio Mathematica, 118(1):61–102, 1999.
  * [14] Simone Muselli.  Models and integral differentials of hyperelliptic curves.  arXiv preprint arXiv:2003.01830, 2020.
  * [15] Julian Rüth.  Models of curves and valuations.  PhD thesis, Universität Ulm, 2015.
  * [16] Padmavathi Srinivasan.  Conductors and minimal discriminants of hyperelliptic curves with rational Weierstrass points.  arXiv preprint arXiv:1508.05172, 2015.


A Note on the Soft Group Category

Nazmiye Alemdar∗,1, Hasan Arslan∗,2

∗Department of Mathematics, Faculty of Science, Erciyes University, 38039,
Kayseri, Turkey
<EMAIL_ADDRESS>2<EMAIL_ADDRESS>

###### Abstract

The main purpose of this paper is to introduce the structure of soft group
category. In this category, we determine some special objects and morphisms
having a universal structure such as the final object and product. Therefore,
the category of soft groups is a symmetric monodial category.

Keywords: Soft groups, category, split monic, hyperoctahedral group

2020 Mathematics Subject Classification: 18A05, 18A10, 18A20, 19D23, 20F55

##  1 Introduction

The real world is too complex for us to understand and interpret it.
Therefore, simplified reality models of the real world are tried to be
obtained. But these mathematical models are also very complex and it is very
difficult to analyze them. Uncertainty in data makes classical methods
unsuccessful when trying to shape problems in engineering, physics, computer
science, economics, social sciences, health sciences, etc. Therefore, it is
not entirely appropriate to use classical set theory based on exact cases when
solving problems with such uncertainties. To deal with these problems, many
mathematical theories such as fuzzy set theory, intuitionistic fuzzy set
theory, indeterminate set theory, mathematical time theory and rough set
theory have been defined.These theories are used as tools against uncertain
situations. However, it has been seen that all these theories have their own
problems.

According to Molodtsov, these difficulties are most likely due to the
inadequacy of the theory’s parameterization tools. Molodtsov, freed from these
troubles, put forward the soft set theory as a mathematical theory in 1999 in
[1]. In his works, he successfully applied this new theory and its results to
other fields such as probability theory, Perron integration, Riemann
integration, research operation, game theory etc. In most of the games you
need to design human behaviors or human models. There are many approaches to
explain human behavior in game theory such as payment and selection functions.
A selection function is a transformation that relates a set of strategies to a
particular situation. Molodsov defined the soft function as a mathematical
tool that retains all the good sides of the selection function and eliminates
the drawbacks of the pay function and the selection function. Tripathy et al.
[7] also defined the basic definitions and concepts for soft sets in the
decision-making process and provided the use of soft sets in the decision-
making process based on game theory.

Recently, many authors have worked on the algebraic structures of soft sets.
Aktas and Cagman [2] introduced the soft group structure depending on
definition of soft sets given by Molodsov and showed a way to construct
algebraic structures based on concept of soft set.

As for category theory, the category theory was introduced by Samuel Elenberg
and Saunders Maclane in the 1940s. The main purpose of category theory is to
model and solve some problems in a simpler way by using objects and morphisms.

Category theory is a comprehensive area of study in mathematics that examines
in an abstract way the basic common language used to describe structures
occuring in different contexts. These developments in computer science plays
very important role in programming studies, logic and authentication in
computer science.

In category theory, all information about objects is encoded with morphisms
between them. In order to examine the internal structure of an object, not
only the object itself but also the relations of this object with other
objects in the category are considered. The characterization of the relations
between any particular type of object and the rest of the universe in which it
is located is called a universal structure and this situation is very common
in category theory. For this reason, it should be investigated whether the
category has special objects and morphisms, if any, in order to create the
universal structure associated with the category.

If any category has a finite product and a final object and also both the
product is a monoidal product and the final object is unit in this category,
then this category is called a cartesian monoidal category [3]. Any cartesian
monoidal category is a symmetric monoidal category [4]. Open games can be
considered as morphisms of a symmetric monoidal category with objects
consisting of pairs of sets. However, morphisms in the symmetric monoidal
category are also illustrations of Feynman diagrams in quantum field theory.

In this paper, we obtain the category of soft groups and investigate the
structures of both special objects (final object and product) and morphisms
(epimorphism, monomorphism) in the category of soft groups as an analogue to
Mac Lane’s study [3].

##  2 Preliminaries

######  Definition 2.1.

Let $U$, $P(U)$ and $E$ be a universal set, the power set of $U$ and a set of
parameters, respectively. If $F:A\rightarrow P(U)$ is a function, then the
pair $(F,A)$ is said to be a soft set over $U$, where $A\subseteq E$ [1].

######  Definition 2.2.

Let $(F,A)$ and $(G,B)$ be any two soft sets over the same universal set $U$.
If the following conditions are satisfied, then $(F,A)$ is called a soft
subset of $(G,B)$ and we denote it by $(F,A)\sqsubseteq(G,B)$ [5]:

  1. 1.

$A\subseteq B$

  2. 2.

$F(a)$ and $G(a)$ are two identical predictions for every $a\in A$.

See [1], [2], [5] for more detailed information about soft sets.

######  Definition 2.3.

Let $G$ be any group and let $(F,A)$ be soft set over $G$. If $F(a)$ is a
subgroup of $G$ for each $a\in A$, then the pair $(F,A)$ is called a soft
group over $G$ [2].

As a convention, throughout this paper, we denote by $(F,A)_{G}$ the soft
group $(F,A)$ over the group $G$.

######  Definition 2.4.

Let $(F,A)_{G}$ be a soft group. If $F(x)=\\{e\\}$ for all $x\in A$, then
$(F,A)_{G}$ is said to be a trivial or identity soft group, where $e$ stands
for the identity element of $G$ [2].

######  Definition 2.5.

Let $(F,A)_{G}$ be a soft group. If $F(x)=G$ for all $x\in A$, then
$(F,A)_{G}$ is called completely soft group [2].

######  Definition 2.6.

Let $(F,A)_{G}$ and $(H,B)_{K}$ be two soft groups. If there exists a group
homomorphism $f:G\rightarrow K$ and a function $p:A\rightarrow B$ such that

$\hat{f}\circ F=H\circ p$

then the pair $(f,p)$ is said to be a soft group homomorphism from $(F,A)_{G}$
to $(H,B)_{K}$, where $f$ and $\hat{f}$ are identical on the power set $P(G)$
[6].

In other words, the pair $(f,p)$ is a soft group homomorphism if and only if
the following diagram is commutative:

${A}$${P(G)}$${B}$${P(K)}$$\scriptstyle{F}$$\scriptstyle{p}$$\scriptstyle{\hat{f}}$$\scriptstyle{H}$
${G}$${K}$$\scriptstyle{f}$ (1)

######  Definition 2.7.

Let the pair $(f,p)$ be a soft group homomorphism from $(F,A)_{G}$ to
$(H,B)_{K}$. If $f$ is a group isomorphism and $p$ is a bijection then the
soft groups $(F,A)_{G}$ and $(H,B)_{K}$ are called isomorphic and written
$(F,A)_{G}\cong(H,B)_{K}$[6].

######  Theorem 2.8.

The composition of two soft group homomorphisms is a soft group homomorphism
[6].

######  Definition 2.9 ([2]).

Let $(F,A)_{G}$ and $(H,B)_{K}$ be two soft groups. Soft product of the soft
groups $(F,A)_{G}$ and $(H,B)_{K}$ is defined as

$U(x,y)=F(x)\times H(y)$

for all $(x,y)\in A\times B$, and it is represented by

$(F,A)_{G}\hat{\times}(H,B)_{K}=(U,A\times B)_{G\times K}.$

This concept can be generalized to three or finitely many soft groups in the
following way:

######  Definition 2.10.

Let $(F_{1},A_{1})_{G_{1}},(F_{2},A_{2})_{G_{2}},\cdots,(F_{n},A_{n})_{G_{n}}$
be soft groups. Soft product of these soft groups is defined as

$U(x_{1},x_{2},\cdots,x_{n})=F_{1}(x_{1})\times F_{2}(x_{2})\cdots\times
F_{n}(x_{n})$

for all $(x_{1},x_{2},\cdots,x_{n})\in A_{1}\times A_{2}\cdots\times A_{n}$,
and denoted by

$(F_{1},A_{1})_{G_{1}}\hat{\times}(F_{2},A_{2})_{G_{2}}\hat{\times}\cdots\hat{\times}(F_{n},A_{n})_{G_{n}}=(U,A_{1}\times
A_{2}\cdots A_{n})_{G_{1}\times G_{2}\cdots\times G_{n}}.$

##  3 Soft Group Category

The objects of the soft group category are soft groups and the morphisms
between these objects are soft group homomorphisms. The composition in this
category is defined as the composition of soft group homomorphisms.

######  Proposition 3.1.

Soft groups and soft group homomorphisms between them forms a category.

###### Proof.

The proof is clear from the equation (1) and Theorem 2.8. ∎

Note here that in this category for each object $(F,A)_{G}$, the unit morphism
is defined as the soft group homomorphism $(1_{G},1_{A})$, where
$1_{G}~{}:~{}G\mapsto G$ is the identity group homomorphism and
$1_{A}~{}:~{}A\mapsto A$ is an identity map. Throughout this paper, this
category is doneted by $SGp$.

We will give the definition of soft kernel which is not included in the
literature and will be needed to prove that a monic morphism is one-to-one in
the category of soft groups inspired by group theory.

######  Definition 3.2.

Let $(F,A)_{G}$ and $(H,B)_{K}$ be any two soft groups. Let
$(f,p):(F,A)_{G}\rightarrow(H,B)_{K}$ be a soft group homomorphism. Let
$A^{\prime}$ be the set consisting of $x\in A$ such that $F(x)=Kerf$. Having
fixed

$A^{\prime}=\\{x\in A:F(x)=Kerf\\},$ (2)

we define $F^{\prime}$ as the restriction function of $F$ to $A^{\prime}$.
Then the soft group $(F^{\prime},A^{\prime})_{G}$ is called the soft kernel of
$(f,p)$.

######  Remark 3.3.

The soft kernel of a soft group homomorphism
$(f,p):(F,A)_{G}\rightarrow(H,B)_{K}$ may not always be defined. One can
construct a soft group homomorphism with the soft kernel
$(f,p):(F,A)_{G}\rightarrow(H,B)_{K}$ by defining $F(x)=Kerf$ for some $x\in
A$.

######  Theorem 3.4.

Let $(F,A)_{G}$ and $(H,B)_{K}$ be any two soft groups. Assume that
$(f,p):(F,A)_{G}\rightarrow(H,B)_{K}$ a soft group homomorphism such that
$F(x)=Kerf$ for some $x\in A$. Then $f$ is injective if and only if
$(F^{\prime},A^{\prime})_{G}$, which is the soft kernel of $(f,p)$, is the
trivial soft subgroup.

###### Proof.

Let $f:G\rightarrow H$ be a monomorphism and let $F^{\prime}$ be the
restriction of the function $F$ to $A^{\prime}$, where $A^{\prime}$ is defined
as in the equation (2). Thus the pair $(F^{\prime},A^{\prime})_{G}$ is the
trivial soft subgroup of $(F,A)_{G}$. Conversely, let
$(F^{\prime},A^{\prime})_{G}$ be the trivial soft subgroup, where
$(F^{\prime},A^{\prime})_{G}$ is the soft kernel of $(f,p)$. We can write
$F^{\prime}(x)=Kerf=\\{e\\}$ for all $x\in A^{\prime}$. Therefore, $f$ is an
injective group homomorphism. ∎

######  Definition 3.5.

Let $(f,p)$ and $(g,q)$ be two soft group homomorphisms. Then $(f,p)$ is equal
to $(g,q)$ if and only if $f=g$ and $p=q$.

######  Theorem 3.6.

Let $(f,p):(F,A)_{G}\rightarrow(H,B)_{K}$ be a soft group homomorphism such
that $F(x)=Kerf$ for some $x\in A$. If $(f,p)$ is monic, then $f$ and $p$ are
injective.

###### Proof.

${A^{\prime}}$${P(Kerf)}$${A}$${P(G)}$${B}$${P(K)}$$\scriptstyle{F^{\prime}}$$\scriptstyle{i_{A^{\prime}}}$$\scriptstyle{\hat{i_{Kerf}}}$$\scriptstyle{F}$$\scriptstyle{p}$$\scriptstyle{\hat{f}}$$\scriptstyle{H}$
${Kerf}$${G}$${K}$$\scriptstyle{i_{Kerf}}$$\scriptstyle{f}$

Suppose that $f$ is not injective. Thus we have $Kerf\neq\\{e\\}$. When
$(f,p)\circ(i_{Kerf},i_{A^{\prime}})=(f,p)\circ(f^{\prime},i_{A^{\prime}})$,
we get $(i_{Kerf},i_{A^{\prime}})\neq(f^{\prime},i_{A^{\prime}})$, where
$f^{\prime}:Kerf\rightarrow G,f^{\prime}(x)=e$. Hence the map $(f,p)$ is not
monic.

Now $p$ is not injective. Then we have two different $x,y\in A$ such that
$p(x)=p(y)$.

${A}$${P(G)}$${A}$${P(G)}$${B}$${P(K)}$$\scriptstyle{F}$$\scriptstyle{p_{1},p_{2}}$$\scriptstyle{\hat{1_{G}}}$$\scriptstyle{F}$$\scriptstyle{p}$$\scriptstyle{\hat{f}}$$\scriptstyle{H}$
${G}$${G}$${K}$$\scriptstyle{1_{G}}$$\scriptstyle{f}$

If we define the maps $p_{1},p_{2}:A\rightarrow A$ as

$p_{1}(a)=\begin{cases}a,&a\in A-\\{x,y\\}\\\ x,&a\notin
A-\\{x,y\\}\end{cases}~{}~{}~{}\textrm{and}~{}~{}~{}~{}~{}p_{2}(a)=\begin{cases}a,&a\in
A-\\{x,y\\}\\\ y,&a\notin A-\\{x,y\\}\end{cases}$

then we have $p\circ p_{1}=p\circ p_{2}$. Now we want to show that the map
$(1_{G},p_{1}):(F,A)\rightarrow(F,A)$ is a soft group homomorphism. In what
follows, we need to prove the diagram

${A}$${P(G)}$${A}$${P(G)}$$\scriptstyle{F}$$\scriptstyle{p_{1},p_{2}}$$\scriptstyle{\hat{1_{G}}}$$\scriptstyle{F}$
${G}$${G}$$\scriptstyle{1_{G}}$

is commutative. Suppose $a\in A-\\{x,y\\}$. Then we get

$(F\circ p_{1})(a)=F(a)=\hat{1_{G}}(F(a))=(\hat{1_{G}}\circ F)(a).$

Due the fact that $\hat{f}\circ F=H\circ p$ and $f$ is injective, we conclude
$\hat{f}(F(x))=H(p(x))=H(p(y))=\hat{f}(F(y))$ and so

$F(x)=F(y).$ (3)

Now we first take $a=x$. For any $x\in A$ we write
$F(p_{1}(x))=F(x)=\hat{1_{G}}(F(x))$ and so $(F\circ
p_{1})(x)=(\hat{1_{G}}\circ F)(x)$. Secondly, if $a=y$, then we get
$F(p_{1}(y))=F(x)=F(y)=\hat{1_{G}}(F(y))$ and again $(F\circ
p_{1})(y)=(\hat{1_{G}}\circ F)(y)$. Therefore $(1_{G},p_{1})$ is a soft group
homomorphism.

In the similar way, we can prove that the map $(1_{G},p_{2})$ is a soft group
homomorphism. Eventually, we have

$\displaystyle(f,p)\circ(1_{G},p_{1})$ $\displaystyle=(f\circ 1_{G},p\circ
p_{1})$ $\displaystyle=(f\circ 1_{G},p\circ p_{2})$
$\displaystyle=(f,p)\circ(1_{G},p_{2}).$

However, $(f,p)$ is not monic since $(1_{G}\circ p_{1})\neq(1_{G}\circ
p_{2})$. This completes the proof. ∎

######  Theorem 3.7.

Let $(f,p):(F,A)_{G}\rightarrow(H,B)_{K}$ be a morphism in the soft group
category $SGp$. If both $f$ and $p$ are injective, then the morphism $(f,p)$
is monic.

###### Proof.

Assume that $f$ and $p$ are injective. Let
$(f_{1},p_{1}),~{}(f_{2},p_{2}):(L,C)_{M}\rightarrow(F,A)_{G}$ be any two
morphisms in $SGp$ such that
$(f,p)\circ(f_{1},p_{1})=(f,p)\circ(f_{2},p_{2})$. Thus it can been easily
seen that $(f\circ f_{1},p\circ p_{1})=(f\circ f_{2},p\circ p_{2})$. Then we
get $f\circ f_{1}=f\circ f_{2}$ and $p\circ p_{1}=p\circ p_{2}$. Since both
$f$ and $p$ are one-to-one, we get $f_{1}=f_{2}$ in $Gp$ and $p_{1}=p_{2}$ in
$Set$, where $Gp$ and $Set$ represent group category and set category,
respectively. Therefore the morphism $(f,p)$ is monic in $SGp$. ∎

######  Theorem 3.8.

If a soft group morphism $(f,p)$ in the $SGp$ is split monic, then both $f$
anf $p$ is injective.

###### Proof.

Let $(F,A)_{G}$ and $(H,B)_{K}$ be any two soft group and let
$(f,p):(F,A)_{G}\rightarrow(H,B)_{K}$ be a soft split monic morphism. Since
$(f,p)$ is a split monic morphism, then there exists
$(g,q):(H,B)_{K}\rightarrow(F,A)_{G}$ such that
$(g,q)\circ(f,p)=1_{(F,A)_{G}}=(1_{G},1_{A})$. Because of Definition 3.5, we
have $g\circ f=1_{G}$ and $q\circ p=1_{A}$. It follows that $f$ is a split
monic morphism in the group category $Gp$ and $p$ is a split monic morphism in
the set category $Set$. Thus, both $f$ anf $p$ is injective. ∎

######  Theorem 3.9.

Let $(f,p):(F,A)_{G}\rightarrow(H,B)_{K}$ be a morphism in the soft group
category $SGp$. If $f$ is an epimorphism in $Gp$ and $p$ is an onto function
in $Set$, then the morphism $(f,p)$ is epic.

###### Proof.

Assume that $(g_{1},q_{1}),~{}(g_{2},q_{2}):(H,B)_{K}\rightarrow(T,D)_{N}$ be
any two morphisms in $SGp$ such that
$(g_{1},q_{1})\circ(f,p)=(g_{2},q_{2})\circ(f,p)$. Thus we deduce that
$(g_{1}\circ f,q_{1}\circ p)=(g_{2}\circ f,q_{2}\circ p)$. From this, we get
$g_{1}\circ f=g_{2}\circ f$ and $q_{1}\circ p=q_{2}\circ p$. We obtain
$g_{1}=g_{2}$ in $Gp$ and $q_{1}=q_{2}$ in $Set$ due to the facts that $f$ is
an epimorphism and $q$ is onto. Hence, the morphism $(f,p)$ is epic in $SGp$.
∎

##  4 Properties of the Soft Group Category

In this section, we will show that the SGp category is a symmetric monoidal
category by proving the existence of the universal properties such as the
final object and the product.

######  Theorem 4.1.

Let $\\{e\\}$ be the trivial group and let $A=\\{a\\}$ be a singleton. Let
$(F,\\{a\\})_{\\{e\\}}$ be a soft group such that $F:\\{a\\}\rightarrow
P(\\{e\\}),~{}~{}F(a)=\\{e\\}$. Then $(F,\\{a\\})_{\\{e\\}}$ is a final object
in the soft group category $SGp$.

###### Proof.

Let $(H,B)_{K}$ be an object in the soft group category $SGp$. Then the
following diagram

${B}$${P(K)}$${\\{a\\}}$${P(\\{e\\})}$$\scriptstyle{H}$$\scriptstyle{p}$$\scriptstyle{\hat{f}}$$\scriptstyle{F}$

is commutative, where $p(b)=a$ for every $b\in B$ and

${K}$${\\{e\\}}$$\scriptstyle{f}$

states the group homomorphism defined by $f(k)=e$ for each $k\in K$. Thus
$(f,p)$ is the unique soft group homomorphism that can be defined from
$(H,B)_{K}$ to $(F,\\{a\\})_{\\{e\\}}$. As a result, any soft group
constructed by a parameter set with single-element and a group $\\{e\\}$ with
one element is a final object in the soft group category $SGp$. ∎

######  Proposition 4.2.

Let $(F_{1},A_{1})_{G_{1}}$ and $(F_{2},A_{2})_{G_{2}}$ be soft groups and let
$(U,A_{1}\times A_{2})_{G_{1}\times
G_{2}}=(F_{1},A_{1})_{G_{1}}\hat{\times}(F_{2},A_{2})_{G_{2}}$ be the soft
product of them. For $i=1,2$ let $\Pi_{i}:A_{1}\times A_{2}\rightarrow A_{i}$
be $i$-th projection function on the parameter sets and let $p_{i}:G_{1}\times
G_{2}\rightarrow G_{i}$ be $i$-th projection homomorphism on groups. Then the
map $(p_{i},\Pi_{i}):(U,A_{1}\times A_{2})_{G_{1}\times
G_{2}}\rightarrow(F_{i},G_{i})_{G_{i}}$ for each $i=1,2$ is a soft group
homomorphism.

###### Proof.

To prove that $(p_{i},\Pi_{i})$ is a soft group homomorphism for each $i=1,2$
we need to show that the following diagram is commutative:

${A_{1}\times A_{2}}$${P(G_{1}\times
G_{2})}$${A_{i}}$${P(G_{i})}$$\scriptstyle{U}$$\scriptstyle{\Pi_{i}}$$\scriptstyle{\hat{p_{i}}}$$\scriptstyle{F_{i}}$
${G_{1}\times G_{2}}$${G_{i}}$$\scriptstyle{p_{i}}$

For this purpose, it is sufficient to show that the equality $\hat{p_{i}}\circ
U=F_{i}\circ\Pi_{i}$ is satisfied. For any pair $(a_{1},a_{2})$, we get

$\displaystyle(\hat{p_{i}}\circ U)(a_{1},a_{2})$
$\displaystyle=\hat{p_{i}}(F_{1}(a_{1})\times F_{2}(a_{2}))$
$\displaystyle=F_{i}(a_{i})$ $\displaystyle=F_{i}(\Pi_{i}(a_{1},a_{2}))$
$\displaystyle=(F_{i}\circ\Pi_{i})(a_{1},a_{2}),$

and so the proof is completed. ∎

We can generalize the above proposition as follows:

######  Corollary 4.3.

Let $(F_{1},A_{1})_{G_{1}},(F_{2},A_{2})_{G_{2}},\cdots(F_{n},A_{n})_{G_{n}}$
be soft groups and let $(U,A_{1}\times A_{2}\times\cdots\times
A_{n})_{G_{1}\times G_{2}\cdots\times
G_{n}}=(F_{1},A_{1})_{G_{1}}\hat{\times}(F_{2},A_{2})_{G_{2}}\hat{\times}\cdots\hat{\times}(F_{n},A_{n})_{G_{n}}$
be the product of these soft groups in the sense of Definition 2.10. For each
$i=1,2,\cdots,n$, let $\Pi_{i}:A_{1}\times\cdots\times A_{n}\rightarrow A_{i}$
be $i$-th projection function on the parameter sets and let
$p_{i}:G_{1}\times\cdots\times G_{n}\rightarrow G_{i}$ be $i$-th projection
homomorphism on groups. Then the map
$(p_{i},\Pi_{i}):(U,A_{1}\times\cdots\times A_{n})_{G_{1}\times\cdots\times
G_{2}}\rightarrow(F_{i},A_{i})_{G_{i}}$ for each $i=1,2,\cdots,n$ is a soft
group homomorphism.

######  Theorem 4.4.

Let $(F_{1},A_{1})_{G_{1}}$ and $(F_{2},A_{2})_{G_{2}}$ be any two objects in
$SGp$ and let $(U,A_{1}\times A_{2})_{G_{1}\times
G_{2}}=(F_{1},A_{1})_{G_{1}}\hat{\times}F_{2},A_{2})_{G_{2}}$ be the soft
product of $(F_{1},A_{1})_{G_{1}}$ and $(F_{2},A_{2})_{G_{2}}$. Then the
product of $(F_{1},A_{1})_{G_{1}}$ and $(F_{2},A_{2})_{G_{2}}$ in $SGp$ is no
other than $((U,A_{1}\times A_{2})_{G_{1}\times G_{2}},(p_{i},\Pi_{i}))$,
where for each $i=1,2$, $\Pi_{i}:A_{1}\times A_{2}\rightarrow A_{i}$ is $i$-th
projection function on the parameter sets and $p_{i}:G_{1}\times
G_{2}\rightarrow G_{i}$ is $i$-th projection homomorphism on groups.

###### Proof.

Assume that ${(H,B)}_{K}$ is an object and each
$(g_{i},q_{i}):(H,B)_{K}\rightarrow(F_{i},A_{i})_{G_{i}}$,$i=1,2$ is a
morphism in $SGp$. To prove that uniqueness of the product of
$(F_{1},A_{1})_{G_{1}}$ and $(F_{2},A_{2})_{G_{2}}$, we need to show that
there exists a unique soft group homomorphism
$(\gamma,\theta):(H,B)_{K}\rightarrow(U,A_{1}\times A_{2})_{G_{1}\times
G_{2}}$, where we define $\theta=(q_{1},q_{2})$ and $\gamma=(g_{1},g_{2})$.
For this reason, we will verify that the diagram below is commutative:

${B}$${P(K)}$${A_{1}\times A_{2}}$${P(G_{1}\times
G_{2})}$$\scriptstyle{H}$$\scriptstyle{\theta=(q_{1},q_{2})}$$\scriptstyle{\hat{\gamma}=\hat{g_{1}}\times\hat{g_{2}}}$$\scriptstyle{U}$
${K}$${G_{1}\times G_{2}}$$\scriptstyle{\gamma}$

For any element $b$ of $B$, we have

$\displaystyle(\hat{\gamma}\circ H)(b)$ $\displaystyle=\hat{\gamma}(H(b))$
$\displaystyle=\hat{g_{1}}(H(b))\times\hat{g_{2}}(H(b))$
$\displaystyle=F_{1}(q_{1}(b))\times F_{2}(q_{2}(b))$
$\displaystyle=U(q_{1}(b),q_{2}(b))$ $\displaystyle=(U\circ(q_{1},q_{2}))(b)$
$\displaystyle=(U\circ\theta)(b),$

so $(\gamma,\theta)$ is a soft group homomorphism. We illustrated the
morphisms in the diagram below for the purpose of a better explanation of the
subject.

$(H,B)_{K}$$(U,A_{1}\times A_{2})_{G_{1}\times
G_{2}}$$(F_{1},A_{1})_{G_{1}}$$(F_{2},A_{2})_{G_{2}}$$(g_{2},q_{2})$$(g_{1},q_{1})$$(\gamma,\theta)$$(p_{1},\Pi_{1})$$(p_{2},\Pi_{2})$

Now we show that for each $i=1,2$ the realtion
$(p_{i},\Pi)\circ(\gamma,\theta)=(g_{i},q_{i})$. For any $k\in K$, we have

$(p_{i}\circ\gamma)(k)=p_{i}(\gamma(k))=p_{i}(g_{1}(k),g_{2}(k))=g_{i}(k),$

and so $p_{i}\circ\gamma=g_{i}$. Similarly, we can prove the relation
$\Pi_{i}\circ\theta=q_{i}$.

Finally, let
$(\gamma^{{}^{\prime}},\theta^{{}^{\prime}}):(H,B)_{K}\rightarrow(U,A_{1}\times
A_{2})_{G_{1}\times G_{2}}$ be another morphism satisfying the condition
$(p_{i},\Pi)\circ(\gamma^{{}^{\prime}},\theta^{{}^{\prime}})=(g_{i},q_{i})$.
Since

$(pi\circ\gamma^{{}^{\prime}})(k)=g_{i}(k)=(pi\circ\gamma)(k)$

for each $k\in K$, then we have $pi\circ\gamma^{{}^{\prime}}=pi\circ\gamma$.
We conclude that $\gamma^{{}^{\prime}}=\gamma$ due to the fact that each
$p_{i},~{}~{}i=1,2$ is a group monomorphism . In the similar manner, one can
see $\theta^{{}^{\prime}}=\theta$. Consequently, we obtain
$(\gamma^{{}^{\prime}},\theta^{{}^{\prime}})=(\gamma,\theta)$, which means
$(\gamma,\theta)$ is unique. Thus we complete the proof. ∎

##  5 EXAMPLE

In this section, we will give a soft group homomorphism with the soft kernel
based on the structure of the hyperoctahedral group, which is a finite real
reflection group.

We assume that $[m,n]:=\\{m,m+1,\cdots,n\\}$ for any $m,n\in\mathbb{Z}$ such
that $m\leq n$. Let $(W_{n},R_{n})$ be a Weyl group of type $B_{n}$, which is
also called a hyperoctahedral group, where
$R_{n}=\\{v_{1},r_{1},\cdots,r_{n-1}\\}$ is the canonical set of generators of
$W_{n}$. Any element $w\in W_{n}$ acts as a signed permutation on the set
$I_{n}=[-n,n]\backslash\\{0\\}$ such that $w(-i)=-w(i)$ for each $i\in I_{n}$.
The group $W_{n}$ has a Dynkin diagram with respect to the set of generators
$R_{n}=\\{v_{1},r_{1},\cdots,r_{n-1}\\}$ as follows:

$B_{n}$ :$v_{1}$$\frac{-1}{\sqrt{2}}$$r_{1}$$r_{2}$$r_{n-2}$$r_{n-1}$

The subgroup $W_{K}$ of $W_{n}$ generated by $K$ is said to be a standard
parabolic subgroup for any subset $K$ of $R_{n}$ and a subgroup of $W_{n}$
conjugate to $W_{K}$ for some $K\subset R_{n}$ is called a parabolic subgroup.
Let $v_{i}:=r_{i-1}v_{i-1}r_{i-1}$ for each $2\leq i\leq n$. It is well-known
that $W_{n}=S_{n}\rtimes\mathcal{V}_{n}$, where $S_{n}$ is the symmetric group
generated by $\\{r_{1},\cdots,r_{n-1}\\}$ and $\mathcal{V}_{n}$ is a normal
subgroup of $W_{n}$ generated by reflections in
$V_{n}=\\{v_{1},\cdots,v_{n}\\}$. That is, $W_{n}$ is a split group extension
of $\mathcal{V}_{n}$ by $S_{n}$ and clearly the cardinality of the group
$W_{n}$ is $2^{n}n!$. Note here that $r_{i}~{}~{}(i=1,\cdots,n-1)$ is
identified with

$r_{i}(j)=\begin{cases}i+1,&\textrm{if}~{}j=i;\\\ i,&\textrm{if}~{}j=i+1;\\\
j,&\textrm{otherwise.}\end{cases}$

and $v_{i}~{}~{}(i=1,\cdots,n)$ is defined by

$v_{i}(j)=\begin{cases}j,&\textrm{if}~{}j\neq i;\\\
-i,&\textrm{if}~{}j=i.\end{cases}$

The representation of $W_{n}$ is given by the following form (see [7]):

$\displaystyle W_{n}=$ $\displaystyle\langle
r_{1},\cdots,r_{n-1},v_{1},\cdots,v_{n}:r_{i}^{2}=(r_{i}r_{i+1})^{3}=(r_{i}r_{j})^{2}=e,|i-j|>1;$
$\displaystyle
v_{i}^{2}=(v_{1}r_{1})^{4}=e,v_{i}v_{j}=v_{j}v_{i},r_{i}v_{i}r_{i}=v_{i+1},r_{i}v_{j}=v_{j}r_{i},~{}j\neq
i,i+1\rangle$

where $e$ denotes the identity element of $W_{n}$. For more detailed
information about the Weyl group of type $B_{n}$, one can apply to [7].

A signed composition of $n$ can be considered of as an expression of $n$ as an
ordered sequence of nonzero integers. More precisely, a signed composition of
$n$ is a finite sequence $A=(a_{1},\cdots,a_{k})$ of nonzero integers
satisfying $\sum_{i=1}^{k}|a_{i}|=n$ [8]. Put $|A|=\sum_{i=1}^{k}|a_{i}|$. We
will denote the set of all signed compositions of $n$ by $\mathcal{SC}(n)$.

A bi-partition of $n$ is a pair $\mu=({\mu}^{+};{\mu}^{-})$ where ${\mu}^{+}$
and ${\mu}^{-}$ are partitions such that $|\mu|=|{\mu}^{+}|+|{\mu}^{-}|=n$. It
is well-known from [8] that
$\boldsymbol{\Lambda}:\mathcal{SC}(n)\rightarrow\mathcal{BP}(n),\boldsymbol{\Lambda}(A)=(\boldsymbol{\Lambda}^{+}(A);\boldsymbol{\Lambda}^{-}(A))$
is a surjective map, where $\boldsymbol{\Lambda}^{+}(A)$ (resp.
$\boldsymbol{\Lambda}^{-}(A)$) is rearrangement of the positive components
(resp. absolute value of negative components) of $A$ in decreasing order. For
$\mu=({\mu}^{+};{\mu}^{-})\in\mathcal{BP}(n)$,
$\hat{\mu}:=\mu^{+}\sqcup-\mu^{-}$ is a unique signed composition obtained by
concatenating the sequence of components of $\mu^{+}$ to that of $-\mu^{-}$
and $W_{\hat{\mu}}$ is a reflection subgroup of $W_{n}$ (see [8]).

We will denote by $\mathcal{BP}(n)$ the set of all bi-partitions of $n$. In
[8], Bonnafé and Hohlweg assigned each signed composition of $n$ to a
reflection subgroup of $W_{n}$ in the following way: The reflection subgroup
$W_{A}$ of $W_{n}$ with respect to $A=(a_{1},\cdots,a_{k})\in\mathcal{SC}(n)$
is generated by the reflection subset $R_{A}$, which is defined by

$\displaystyle R_{A}=$ $\displaystyle\\{r_{p}\in
S_{n}~{}:~{}|a_{1}|+\cdots+|a_{i-1}|+1\leq p<|a_{1}|+\cdots+|a_{i}|\\}$
$\displaystyle\cup\\{{v_{|a_{1}|+\cdots+|a_{j-1}|+1}}\in
V_{n}\\}~{}|~{}a_{j}>0\\}\subset R_{n}^{{}^{\prime}}$

where $R^{\prime}_{n}=\\{r_{1}\cdots r_{n-1},v_{1},v_{2},\cdots,v_{n}\\}.$

Let $\mathcal{P}(W_{n})$ denote the power set of $W_{n}$. If we pick the
parameter set as $\mathcal{BP}(n)$ and accordingly define the map
$G:\mathcal{BP}(n)\rightarrow\mathcal{P}(W_{n}),~{}~{}G(\mu)=W_{\hat{\mu}}$,
then the pair $(G,\mathcal{BP}(n))_{W_{n}}$ is a soft group.

Now we give another soft group example. If we take $\mathcal{SC}(n)$ as the
parameter set and define the map $F$ as $F(A)=W_{\hat{\mu}}$, where we can
write
$\hat{\mu}:=\boldsymbol{\Lambda}^{+}(A)\sqcup-\boldsymbol{\Lambda}^{-}(A)\in\mathcal{SC}(n)$
for every $A\in\mathcal{SC}(n)$, then the pair $(F,\mathcal{SC}(n))_{W_{n}}$
is a soft group due to the fact that $W_{\hat{\mu}}$ is a subgroup of $W_{n}$.

The pair
$(f,\boldsymbol{\Lambda}):(F,\mathcal{SC}(n))_{W_{n}}\rightarrow(G,\mathcal{BP}(n))_{W_{n}}$
is a soft group homomorphism with the soft kernel. Therefore, we say that the
following diagram is commutative:

${\mathcal{SC}(n)}$${\mathcal{P}(W_{n})}$${\mathcal{BP}(n)}$${\mathcal{P}(W_{n})}$$\scriptstyle{F}$$\scriptstyle{\boldsymbol{\Lambda}}$$\scriptstyle{\hat{f}}$$\scriptstyle{G}$

where $f~{}:~{}W_{n}\mapsto W_{n}$ is the trivial isomorphism and $\hat{f}$ is
the identity function on $\mathcal{P}(W_{n})$. In fact, let
$A\in\mathcal{SC}(n)$ and let $\boldsymbol{\Lambda}(A):=\mu$. Thus we can
write
$\hat{\mu}=\boldsymbol{\Lambda}^{+}(A)\sqcup-\boldsymbol{\Lambda}^{-}(A)$ and
so it is clear that $\boldsymbol{\Lambda}(\hat{\mu})=\mu$ for every
$\mu\in\mathcal{BP}(n)$. Therefore, for every $A\in\mathcal{SC}(n)$, we get

$\displaystyle(G\circ\boldsymbol{\Lambda})(A)$ $\displaystyle=W_{\hat{\mu}}$
$\displaystyle=\hat{f}(W_{\hat{\mu}})$ $\displaystyle=\hat{f}\circ F(A).$

Hence the above diagram is commutative. The soft kernel of this soft group
homomorphism equals to the set $\\{(-1,-1,\cdots,-1)\\}$, since
$F((-1,-1,\cdots,-1))=\\{e\\}=Kerf$ is the trivial subgroup of $W_{n}$.

## References

  * [1] Molodtsov, Dmitri A: Soft set theory-First results, Computers Mathematics With Applications, 37 (1999) 19-31.
  * [2] H. Aktas, N. Cagman: Soft sets and soft groups, Inf. Sci., 177 (2007) 2726-2735 .
  * [3] S. Mac Lane: Categories for the Working Mathematician, Graduate Text in Mathematics, Volume 5 Springer Verlag, Newyork, (1971).
  * [4] J. Baez and M. Stay: Physics, topology, logic and computation: a Rosetta Stone, in New Structures for Physics, ed. Bob Coecke, Lecture Notes in Physics vol. 813, Springer, Berlin, (2011) 95-174.
  * [5] P.K Maji, R. Bismas, A.R. Roy: Soft set theory, Comput. Math. Appl. 45(2003) 555-562.
  * [6] A. Sezgin , A.O. Atagun: Soft Groups and Normalistic Soft Groups, Comput. Math. Appl, 62 (2) (2011) 1457-1467.
  * [7] J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press Volume 29, 1990.
  * [8] C. Bonnafe, C. Hohlweg: Generalized descent algebra and construction of irreducible characters of hyperoctahedral groups, Ann. Inst. Fourier (Grenoble), 56(1) (2006) 131-181.