# Polyocollection ideals and primary decomposition of polyomino ideals

Carmelo Cisto  Dipartimento di Scienze Matematiche e Informatiche, Scienze
Fisiche e Scienze della Terra, Università di Messina, Viale Ferdinando Stagno
D’Alcontres 31, Messina, 98166, Italy<EMAIL_ADDRESS>,  Francesco
Navarra  Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e
Scienze della Terra, Università di Messina, Viale Ferdinando Stagno
D’Alcontres 31, Messina, 98166, Italy<EMAIL_ADDRESS>and  Dharm
Veer  Indian Institute of Technology Gandhinagar, Palaj, Gujarat 382355. India
Chennai Mathematical Institute, Siruseri, Tamilnadu 603103. India
<EMAIL_ADDRESS>
###### Abstract.

In this article, we study the primary decomposition of some binomial ideals.
In particular, we introduce the concept of polyocollection, a combinatorial
object that generalizes the definitions of collection of cells and polyomino,
that can be used to compute a primary decomposition of non-prime polyomino
ideals. Furthermore, we give a description of the minimal primary
decomposition of non-prime closed path polyominoes. In particular, for such a
class of polyominoes, we characterize the set of all zig-zag walks and show
that the minimal prime ideals have a very nice combinatorial description.

###### Key words and phrases:

Polyominoes, primary decomposition, zig-zag walk.

###### 2020 Mathematics Subject Classification:

05B50, 05E40

##  1\. Introduction

A polyomino is a finite collection of unit squares with vertices at lattice
points in the plane joined edge by edge. Qureshi [21] associated a finite type
$K$-algebra $K[{\mathcal{P}}]$ (over a field $K$) to a polyomino
${\mathcal{P}}$. An interesting question here is to relate the algebraic
properties of the $K$-algebra $K[{\mathcal{P}}]$ with the combinatorial
properties of the polyomino ${\mathcal{P}}$. For certain classes of
polyominoes, some algebraic properties of the associated algebra have been
studied. For example, the graded minimal free resolution of $K[{\mathcal{P}}]$
for convex polyominoes has been studied in [8, 10] and the Hilbert series of
$K[{\mathcal{P}}]$ has been studied in [24, 18, 22, 17, 6].
We say that a polyomino ${\mathcal{P}}$ is prime (resp. non-prime) if the
defining ideal of $K[{\mathcal{P}}]$, denoted by $I_{\mathcal{P}}$, is prime
(resp. non-prime). The ideal $I_{\mathcal{P}}$ is called the polyomino ideal.
It is known that simple polyominoes are prime [15, 23]. Hibi-Qureshi [16] and
Shikama [25] proved that the non-simple polyominoes obtained by removing a
convex polyomino from a rectangle are prime. Characterizing prime polyominoes
is a difficult task; to do that Mascia et. al. [19] defined zig-zag walks (see
Definition 4.2) for polyominoes. They proved [19, Corollary 3.6] that the
existence of zig-zag walks determines the non-primality of the polyomino and
conjectured that [19, Conjecture 4.6] this property characterizes non-prime
polyominoes. This conjecture has been verified for certain classes of
polyominoes, for example, for grid polyominoes [19], for closed path
polyominoes [3] and for weakly closed path polyominoes [4]. In [20], Mascia
et. al. have determined some prime polyominoes using Gröbner basis.
Concerning other algebraic properties, several mathematicians have studied the
combinatorial description of the height of polyomino ideals. For some classes
of polyominoes, it has been proved that the height of the polyomino ideal
$I_{{\mathcal{P}}}$ is the number of cells of ${\mathcal{P}}$. For instance,
Qureshi [21] proved this for convex polyominoes. Herzog-Madani [15] extended
this to simple polyominoes. Extending this further, Herzog et. al. [13]
recently proved this for the case when the polyomino ideal is unmixed. Dinu
and the second author of this article [7] proved this for closed path
polyominoes and conjectured that this holds for all polyomino ideals. In this
regard, we show that if removing a cell makes the polyomino simple, then the
above conjecture holds (Proposition 4.10).

In this article, we provide some properties about primary decomposition of
polyomino ideals and we study the minimal primary decomposition of non-prime
closed path polyominoes. The primary decomposition have been studied for some
classes of binomial ideals, for instance in [12] for binomial edge ideals and
in [11] for ideals generated by adjacent 2-minors. The first results explained
in this paper allows to consider something similar for polyomino ideals. In
particular, supported by some examples obtained by using the software
Macaulay2 ([9]) and inspired by the concept of _admissible set_ in [11], we
find that for studying the primary decomposition of polyomino ideals one can
consider a larger class of binomial ideals. This class is related to a
combinatorial object that generalizes the concept of collection of cells and
polyominoes. We call such a class of combinatorial objects _polyocollections_.
For non-prime closed path polyominoes we provide, in our main result, a
detailed description of the minimal primary decomposition. We show that the
polyomino ideal is the intersection of two minimal prime ideals (Theorem
4.19), and both minimal prime ideals have a very nice combinatorial
description. One of the minimal primes is the toric ideal appeared in [19,
Section 3] (see Theorem 4.16). We characterize zig-zag walks of non-prime
closed path polyominoes and use that to define the other minimal prime which
is generated by monomials and binomials. Finally, we also show that the height
of these two ideals is equal to the number of the cells of the polyomino, and
as a consequence the polyomino ideal is unmixed.
The article is organised as follows. In Section 2, we recall some basic
notions on combinatorics related to polyominoes. In Section 3, we define
polyocollections and we associate a binomial ideal to it, generalizing the
ideal associated to a collection of cells in [21]. Moreover, we provide a
characterization of the primality of that binomial ideal in terms of a lattice
ideal attached to the polyocollection and we give a primary decomposition of
the radical of the ideal associated to the polyocollection. Section 4 is
devoted to study the minimal primary decomposition of closed path polyominoes
having zig-zag walks, equivalently the polyomino is non-prime (see [3]). In
the last section, we highlight some possible future directions and open
questions about polyominoes and polyocollection ideals.

##  2\. Intervals, cells and polyominoes

Let $(i,j),(k,l)\in\mathbb{Z}^{2}$. We say that $(i,j)\leq(k,l)$ if $i\leq k$
and $j\leq l$. Consider $a=(i,j)$ and $b=(k,l)$ in $\mathbb{Z}^{2}$ with
$a\leq b$. The set $[a,b]=\\{(m,n)\in\mathbb{Z}^{2}:i\leq m\leq k,\ j\leq
n\leq l\\}$ is called an interval of $\mathbb{Z}^{2}$. In addition, if $i<k$
and $j<l$ then $[a,b]$ is a proper interval. If $j=l$ (or $i=k$) then $a$ and
$b$ are in horizontal (or vertical) position. We also denote
$]a,b[=\\{(m,n)\in\mathbb{Z}^{2}:i<m<k,\ j<n<l\\}$ and $]a,b]$ or $[a,b[$ with
the usual meaning. We also need to consider intervals in $\mathbb{R}^{2}$, for
which we use the notations
$\operatorname{c}([a,b])=\\{(m,n)\in\mathbb{R}^{2}:i\leq m\leq k,\ j\leq n\leq
l\\}$, that is the _closure_ of $[a,b]$ in $\mathbb{R}^{2}$, and
$\mathrm{int}([a,b])=\\{(r,s)\in\mathbb{R}^{2}:i<r<k,\ j<s<l\\}$, that is the
_interior_ of $[a,b]$ in $\mathbb{R}^{2}$.
Suppose that $[a,b]$ is a proper interval. In such a case we say $a,b$ the
diagonal corners of $[a,b]$ and $c=(i,l)$, $d=(k,j)$ the anti-diagonal corners
of $[a,b]$. Moreover, the elements $a$, $b$, $c$ and $d$ are called
respectively the lower left, upper right, upper left and lower right corners
of $[a,b]$. We also call $V([a,b])=\\{a,b,c,d\\}$ the set of _vertices_ of the
interval, while the sets $[a,c],[a,d],[b,d],[b,c]$ are the edges of the
interval and we denote $E([a,b])=\\{[a,c],[a,d],[b,d],[b,c]\\}$. If $[a,c]$ is
an edge of an interval, we call the set $\\{a,c\\}$ the _boundary of the
edge_.
Let $\mathcal{S}$ be a non-empty collection of intervals in $\mathbb{Z}^{2}$.
The set of the vertices and of the edges of $\mathcal{S}$ are respectively
$V(\mathcal{S})=\bigcup_{I\in\mathcal{S}}V(I)$ and
$E(\mathcal{S})=\bigcup_{I\in\mathcal{S}}E(I)$. We denote by $|\mathcal{S}|$
the number of intervals belonging to the collection $\mathcal{S}$, called also
the _rank_ of $\mathcal{S}$.
A proper interval $C=[a,b]$ with $b=a+(1,1)$ is called a cell of
${\mathbb{Z}}^{2}$. Consider now $\mathcal{S}$ be a collection of cells. If
$C$ and $D$ are two distinct cells of $\mathcal{S}$, then a walk from $C$ to
$D$ in $\mathcal{S}$ is a sequence $\mathcal{C}:C=C_{1},\dots,C_{m}=D$ of
cells of ${\mathbb{Z}}^{2}$ such that $C_{i}\cap C_{i+1}$ is an edge of
$C_{i}$ and $C_{i+1}$ for $i=1,\dots,m-1$. In addition, if $C_{i}\neq C_{j}$
for all $i\neq j$, then $\mathcal{C}$ is called a path from $C$ to $D$. We say
that $C$ and $D$ are connected in $\mathcal{S}$ if there exists a path of
cells in $\mathcal{S}$ from $C$ to $D$. If $\mathcal{P}$ is a non-empty and
finite collection of cells in $\mathbb{Z}^{2}$, then it is called a
_polyomino_ if any two cells of $\mathcal{P}$ are connected in $\mathcal{P}$.
For instance, see Figure 1.

Figure 1. A polyomino.

We say that a collection of cells $\mathcal{P}$ is simple if for any two cells
$C$ and $D$ not in $\mathcal{P}$ there exists a path of cells not in
$\mathcal{P}$ from $C$ to $D$. A finite collection of cells $\mathcal{H}$ not
in $\mathcal{P}$ is a hole of $\mathcal{P}$ if any two cells of $\mathcal{H}$
are connected in $\mathcal{H}$ and $\mathcal{H}$ is maximal with respect to
set inclusion. For example, the polyomino in Figure 1 is not simple with an
hole. Obviously, each hole of $\mathcal{P}$ is a simple polyomino and
$\mathcal{P}$ is simple if and only if it has not any hole. Consider two cells
$A$ and $B$ of $\mathbb{Z}^{2}$ with $a=(i,j)$ and $b=(k,l)$ as the lower left
corners of $A$ and $B$ and $a\leq b$. A cell interval $[A,B]$ is the set of
the cells of $\mathbb{Z}^{2}$ with lower left corner $(r,s)$ such that
$i\leqslant r\leqslant k$ and $j\leqslant s\leqslant l$. If $(i,j)$ and
$(k,l)$ are in horizontal (or vertical) position, we say that the cells $A$
and $B$ are in horizontal (or vertical) position. Let $\mathcal{P}$ be a
collection of cells. Consider two cells $A$ and $B$ of $\mathcal{P}$ in
vertical or horizontal position. The cell interval $[A,B]$, containing $n>1$
cells, is called a block of $\mathcal{P}$ of rank n if all cells of $[A,B]$
belong to $\mathcal{P}$. The cells $A$ and $B$ are called extremal cells of
$[A,B]$. Moreover, a block $\mathcal{B}$ of $\mathcal{P}$ is maximal if there
does not exist any block of $\mathcal{P}$ which contains properly
$\mathcal{B}$.

##  3\. Polyocollections and related ideals

Let $\mathcal{C}$ be a collection of intervals in $\mathbb{Z}^{2}$. We say
that $\mathcal{C}$ is a _polyocollection_ if for all $I,J\in\mathcal{C}$ with
$I\neq J$, we have $I\nsubseteq J$ and one of the following holds:

  1. (1)

$I\cap J$ is a common edge of $I$ and $J$.

  2. (2)

For all $F\in E(I)$ and for all $G\in E(J)$, $|F\cap G|\leq 1$.

######  Example 3.1.

$\mathcal{C}_{1}=\\{[(1,1),(2,2)],[(1,2),(2,3)],[(2,1),(5,2)],[(2,2),(5,3)],[(5,3),[7,4])]\\}$
is a polyocollection, it is pictured in Figure 2a.
The collection
$\mathcal{C}_{2}=\\{[(3,1),(4,2)],[(4,1),(5,2)],[(3,2),(4,6)],[(4,2),(5,6)],[(3,6),(4,7)],\newline
[(4,6),(5,7)],[(1,3),(2,4)],[(1,4),(2,5)],[(2,3),(6,4)],[(2,4),(6,5)],[(6,3),(7,4)],[(6,4),(7,5)]\\}$,
displayed in Figure 2b, is also a polyocollection.
The collection
$\mathcal{C}_{3}=\\{[(1,2),(3,4)],[(2,1),(4,3)],[(4,1),(5,2)],[(4,2),(5,3)]\\}$,
displayed in Figure 2c, is not a polyocollection. Because if we take
$I=[(2,1),(4,3)]$ and $J=[(4,1),(5,2)]$, then $I$ and $J$ does not satisfy any
of the two conditions for a polyocollection (the same hold also for
$I=[(2,1),(4,3)]$ and $J=[(4,2),(5,3)]$). Observe that, for $\mathcal{C}_{3}$,
the position of $[(1,2),(3,4)]$ with respect to $[(2,1),(4,3)]$ is not in
contradiction with the definition of polyocollection.
The collection
$\mathcal{C}_{4}=\\{[(1,1),(3,3)],[(1,3),(3,5)],[(3,1),(5,3)],[(3,3),(5,5)],[(2,2),(4,4)]\\}$,
displayed in Figure 2d, is another example of polyocollection.

(a)

(b)

(c)

(d)

Figure 2. (a), (b) and (d) are polyocollections, while (c) is not a
polyocollection.

An interval $I$ of $\mathbb{Z}^{2}$ is called an _inner interval of
$\mathcal{C}$_ if there exists $n\in\mathbb{N}$ and
$I_{1},I_{2},\ldots,I_{n}\in\mathcal{C}$ such that
$\operatorname{c}(I)=\bigcup_{i=1}^{n}\operatorname{c}(I_{i})$. We denote by
$G(\mathcal{C})$ the set of inner intervals of $\mathcal{C}$. By convention we
consider that the empty set is a polyocollection whose set of inner interval
is the empty set.
For instance, referring to Example 2, the intervals $[(1,1),(2,3)]$ and
$[(1,1),(5,3)]$ of $\mathcal{C}_{1}$ are examples of inner intervals. In
$\mathcal{C}_{2}$, $[(1,3),(7,5)]$ and $[(3,2),(5,7)]$ are examples of inner
intervals, while $[(3,3),(5,5)]$ is not an inner interval.

######  Proposition 3.2.

Let $\mathcal{C}$ be a polyocollection and $I$ be an interval of
${\mathbb{Z}}^{2}$. Suppose that $c(I)=\bigcup_{i=1}^{t}c(I_{i})$ with
$I_{i}\in\mathcal{C}$ for all $i\in\\{1,\ldots,t\\}$. Then there exists
$\\{i_{1},\ldots,i_{r}\\}\subseteq\\{1,\ldots,t\\}$ such that

$c(I)=\bigcup_{j=1}^{r}c(I_{i_{j}})$

with $\mathrm{int}(I_{i_{k}})\cap\mathrm{int}(I_{i_{l}})=\emptyset$ for all
$k\neq l$.

###### Proof.

Assume that $I=[(i,j),(m,n)]$. Since $(i,j)\in I\subset
c(I)=\bigcup_{i=1}^{t}c(I_{i})$ then there exists $i_{1}\in\\{1,\ldots,t\\}$
such that $(i,j)\in c(I_{i_{1}})$. By construction, $(i,j)$ is the lower left
corner of $I_{i_{1}}$. We denote by $(a_{1},b_{1})$ the right upper corner of
$I_{i_{1}}$, where $i<a_{1}\leq m$ and $j<b_{1}\leq n$. We distinguish two
cases.
Case I) Assume that $a_{1}=m$ or $b_{1}=n$. If $a_{1}=m$ and $b_{1}=n$, then
we are done. Without loss of generality, suppose that $a_{1}=m$ and
$j<b_{1}<n$ (the case $b_{1}=n$ and $i<a_{1}<m$ is similar). Since
$(i,b_{1})\in c(I)\cap{\mathbb{Z}}^{2}$, then there exists
$i_{2}\in\\{1,\ldots,t\\}\setminus\\{i_{1}\\}$ such that $(i,b_{1})\in
I_{i_{2}}$. Let us denote $I_{i_{2}}=[(i,c_{1}),(a_{2},b_{2})]$. Since
$\mathcal{C}$ is a polyocollection and $I_{i_{1}},I_{i_{2}}\in\mathcal{C}$,
the only possibility to avoid a contradiction is $c_{1}=b_{1}$, $a_{2}=m$ and
$b_{1}<b_{2}\leq n$. In Figure 3a, the only possible arrangement of
$I_{i_{1}}$ and $I_{i_{2}}$ described before is displayed. Moreover, observe
that $\mathrm{int}(I_{i_{1}})\cap\mathrm{int}(I_{i_{2}})=\emptyset$. If
$b_{2}=n$ we conclude, otherwise we can continue the argument in the same way
until we cover $c(I)$.

(a)

(b)

Figure 3. Arrangements of intervals.

Case II) Assume that $i<a_{1}<m$ and $j<b_{1}<n$. Since
$\operatorname{c}(I)=\bigcup_{i=1}^{t}\operatorname{c}(I_{i})$, considering
the vertices $(a_{1},j)$ and $(i,b_{1})$ there exists
$i_{2},i_{3}\in\\{1,\ldots,r\\}$ such that $(a_{1},j)\in I_{i_{2}}$ and
$(i,b_{1})\in I_{i_{3}}$. Arguing as done in the previous case, since
$I_{i_{1}},I_{i_{3}}\in\mathcal{C}$, the only possibility is
$I_{i_{2}}=[(a_{1},j),(a_{2},b_{1})]$ with $a_{2}\leq m$, and
$I_{i_{3}}=[(i,b_{1}),(a_{1},b_{2})]$ with $b_{2}\leq n$. In particular,
$I_{i_{1}}$, $I_{i_{2}}$ and $I_{i_{3}}$ are arranged as in Figure 3b.
Moreover,
$\mathrm{int}(I_{i_{1}})\cap\mathrm{int}(I_{i_{2}})=\mathrm{int}(I_{i_{1}})\cap\mathrm{int}(I_{i_{3}})=\mathrm{int}(I_{i_{2}})\cap\mathrm{int}(I_{i_{3}})=\emptyset$.
Also in this case, it is not difficult to see that we can continue with the
same argument until we cover $c(I)$. ∎

Observe that, considering the polyocollection $\mathcal{C}_{4}$ in Example 2,
the interval $[(1,1),(5,5)]$ is an inner interval, since
$\operatorname{c}([(1,1),(5,5)])=\operatorname{c}([(1,1),(3,3)])\cup\operatorname{c}([(1,3),(3,5)])\cup\operatorname{c}([(3,1),(5,3)])\cup\operatorname{c}([(3,3),(5,5)])\cup[(2,2),(4,4)]$.
We can also remove in the union the interval $[(2,2),(4,4)]$, in order to
cover $\operatorname{c}([(1,1),(5,5)])$ with intervals having disjoint common
interior.

Let $\mathcal{C}$ be a polyocollection. Define the polynomial ring
$S_{\mathcal{C}}=K[x_{a}\mid a\in V(\mathcal{C})]$, with $K$ a field. If
$I=[a,b]$ is an inner interval of $\mathcal{C}$, having anti-diagonal corners
$c,d$, then we call the binomial $f_{I}=x_{a}x_{b}-x_{c}x_{d}\in
S_{\mathcal{C}}$ an _inner 2-minor_ of $\mathcal{C}$. We define by
$I_{\mathcal{C}}$ the ideal generated by the binomials $f_{I}$ for all inner
intervals $I$ of $\mathcal{C}$ and we call it the polyocollection ideal of
$\mathcal{C}$. The quotient ring
$K[\mathcal{C}]=S_{\mathcal{C}}/I_{\mathcal{C}}$ is said the _coordinate ring_
of $\mathcal{C}$. For a monomial $u\in S_{\mathcal{C}}$, we define the support
of $u$, denoted by $\mathrm{supp}(u)$ to be the set of variables $x_{r}\in
S_{\mathcal{C}}$ such that $x_{r}$ divides $u$ in $S_{\mathcal{C}}$.

######  Remark 3.3.

Let $\mathcal{P}$ be a collection of cells. Observe that $\mathcal{P}$ is
trivially a polyocollection. In such a case the definition of inner intervals
reduces to the following: an interval $I\in\mathbb{Z}^{2}$ is an inner
interval of $\mathcal{P}$ if and only if each cell contained in $I$ belongs
$\mathcal{P}$. Moreover the ideal $I_{\mathcal{P}}$ as defined above for
polyocollections coincides with the ideal associated to a collection of cells
following the definition in [21].

If $\mathcal{C}$ is a polyocollection and $\mathcal{P}$ is a collection of
cells, we say that $\mathcal{C}$ and $\mathcal{P}$ are algebraically
isomorphic if $K[\mathcal{C}]$ is isomorphic to $K[\mathcal{P}]$ as
$K$-algebras. With reference to Example 2, note that $\mathcal{C}_{1}$,
$\mathcal{C}_{2}$ and $\mathcal{C}_{4}$ are algebraically isomorphic to the
collections of cells shown in Figures 4a, 4b and 4c respectively.

(a)

(b)

(c)

Figure 4. The collections of cells which are algebraically isomorphic to the
polyocollections of Figures 2a, 2b and 2d, respectively.

In the following discussion, we prove that there exists a polyocollection
$\mathcal{C}$ which is not algebraically isomorphic to any collection of
cells. Roughly speaking, that means the class of polyocollections is
algebraically strictly bigger than the class of collections of cells.

######  Discussion 3.4.

Let ${\mathcal{C}}$ be the polyocollection as shown in Figure 5a. Then,

$K[{\mathcal{C}}]=K[x_{21},x_{41},x_{12},x_{22},x_{42},x_{52},x_{13},x_{23},x_{33},x_{43},x_{53},x_{24},x_{34},x_{44}]$

and the polyocollection ideal

$\displaystyle I_{\mathcal{C}}=$
$\displaystyle(x_{42}x_{53}-x_{43}x_{52},\,x_{33}x_{44}-x_{34}x_{43},\,x_{23}x_{44}-x_{24}x_{43},\,x_{23}x_{34}-x_{24}x_{33},$
$\displaystyle x_{21}x_{42}-x_{22}x_{41},\,x_{12}x_{23}-x_{13}x_{22}).$

Macaulay2 computations show that $I_{\mathcal{C}}$ is a non-prime ideal of
height 5 and the minimal primary decomposition of the ideal $I_{\mathcal{C}}$
consists of two prime ideals $\mathfrak{p}_{1}$ and $\mathfrak{p}_{2}$, that
are the following:

$\displaystyle\mathfrak{p}_{1}=$ $\displaystyle
I_{\mathcal{C}}+(x_{12}x_{24}x_{41}x_{53}-x_{13}x_{21}x_{44}x_{52})$
$\displaystyle\mathfrak{p}_{2}=$
$\displaystyle(x_{43},x_{42},x_{33},x_{23},x_{22})$

The height of both ideals in the primary decomposition is 5. So,
$I_{\mathcal{C}}$ is a unmixed ideal.

Suppose that, if possible, there exists a collection of cells ${\mathcal{P}}$
such that $K[{\mathcal{C}}]$ is isomorphic to $K[{\mathcal{P}}]$. Then, by
[13, Theorem 3.1], ${\mathcal{P}}$ is a collection of 5 cells. Since
$I_{\mathcal{P}}$ is a non-prime ideal, ${\mathcal{P}}$ is a non-simple
collection of cells by [4, Theorem 3.3]. Therefore, up to rotations and
reflections, the only possible collections of cells ${\mathcal{P}}$ are those
in Figures 5b-5e. For ${\mathcal{P}}$ as in Figure 5c and 5d, Macaulay2
computations show that the dimension of the ring $K[{\mathcal{P}}]$ is $10$
and $11$ respectively. As the dimension of the ring $K[{\mathcal{C}}]$ is 9,
we get that $K[{\mathcal{C}}]$ is not isomorphic to $K[{\mathcal{P}}]$ when
${\mathcal{P}}$ is as in Figure 5c and 5d. For ${\mathcal{P}}$ as in Figure
5e, $K[\mathcal{P}]$ is a domain (see [4, Remark 3.4]), so it is not
isomorphic to $K[\mathcal{C}]$. When ${\mathcal{P}}$ is as in Figure 5b, the
minimal primary decomposition of the ideal $I_{\mathcal{P}}$ consists of two
prime ideals $\mathfrak{q}_{1}$ and $\mathfrak{q}_{2}$, that are the
following:

$\displaystyle\mathfrak{q}_{1}=$ $\displaystyle
I_{\mathcal{P}}+(x_{12}x_{25}x_{31}x_{43}-x_{13}x_{21}x_{35}x_{42},\,x_{12}x_{24}x_{31}x_{43}-x_{13}x_{21}x_{34}x_{42})$
$\displaystyle\mathfrak{q}_{2}=$
$\displaystyle(x_{33},\,x_{32},\,x_{23},\,x_{22},\,x_{24}x_{35}-x_{25}x_{34})$

One can immediately see that there is a monomial ideal generated by variables
in the primary decomposition of $I_{\mathcal{C}}$ but none of the ideals in
the primary decomposition of $I_{\mathcal{P}}$ is generated by monomials.
Hence $K[{\mathcal{C}}]$ is not isomorphic to $K[{\mathcal{P}}]$. ∎

(a)

(b)

(c)

(d)

(e)

Figure 5. The polyocollection and collections of cells examined in Discussion
3.4.

We want to show that some concepts and results introduced for collection of
cells in [21], can be extended in a natural way for polyocollections.

###  3.1. The lattice ideal related to a polyocollections

Let $\mathcal{C}$ be a polyocollection. For $a\in V(\mathcal{C})$ denote with
$\mathbf{v}_{a}$ the vector in $\mathbb{Z}^{|V(\mathcal{C})|}$ having $1$ in
the coordinate indexed by $a$ and $0$ in all other coordinates. If
$[a,b]\in\mathcal{C}$ is an interval having $a,b$ as diagonal corners, and
$c,d$ as anti-diagonal corners, we denote
$\mathbf{v}_{[a,b]}=\mathbf{v}_{a}+\mathbf{v}_{b}-\mathbf{v}_{c}-\mathbf{v}_{d}\in\mathbb{Z}^{|V(\mathcal{C})|}$.
We define $\Lambda_{\mathcal{C}}$ the sub-lattice of
$\mathbb{Z}^{|V(\mathcal{C})|}$ generated by the vectors $\mathbf{v}_{I}$ for
all $I\in\mathcal{C}$.
Let $n=|V(\mathcal{C})|$, we recall some known notations. If
$\mathbf{v}\in\mathbb{N}^{n}$ we denote as usual with $x^{\mathbf{v}}$ the
monomial in $S_{\mathcal{C}}$ having $\mathbf{v}$ as exponent vector. If
$\mathbf{e}\in\mathbb{Z}^{n}$ we denote with $\mathbf{e}^{+}$ the vector
obtained from $\mathbf{e}$ by replacing all negative components by zero, and
$\mathbf{e}^{-}=-(\mathbf{e}-\mathbf{e}^{+})$.
Let $L_{\mathcal{C}}$ be the lattice ideal of $\Lambda_{\mathcal{C}}$, that is
the following binomial ideal in $S_{\mathcal{C}}$:

$L_{\mathcal{C}}=(\\{x^{\mathbf{e}^{+}}-x^{\mathbf{e}^{-}}\mid\mathbf{e}\in\Lambda_{\mathcal{C}}\\})$

In [21, Theorem 3.5] it is shown that if $\mathcal{P}$ is a collection of
cells then $L_{\mathcal{P}}$ is a prime ideal. Using the same argument, we can
prove that the same holds considering also polyocollections. We provide the
proof of this fact for completeness.

Denote by $F(\mathcal{C})$ the set of elements in $V(\mathcal{C})$ that are
not lower left corner of an interval in $\mathcal{C}$.

######  Lemma 3.5.

Let $\mathcal{C}$ be a polyocollection and $\mathcal{B}=\\{\mathbf{v}_{I}\mid
I\in\mathcal{C}\\}\cup\\{\mathbf{v}_{a}\mid a\in F(\mathcal{C})\\}$. Then
$\mathcal{B}$ is a basis of the group $\mathbb{Z}^{|V(\mathcal{C})|}$.

###### Proof.

Let $n=|V(\mathcal{C})|$ and $M$ be the $n\times n$ matrix whose column
vectors are $\mathcal{B}=\\{\mathbf{v}_{I}\mid
I\in\mathcal{C}\\}\cup\\{\mathbf{v}_{a}\mid a\in F(\mathcal{C})\\}$. We refer
to $M$ as the matrix associated to $\mathcal{C}$. Observe that the columns of
$M$ can be labelled with the set $\\{I\in\mathcal{C}\\}\cup\\{a\in
F(\mathcal{C})\\}$ and the rows can be labelled with the set $V(\mathcal{C})$.
In order to prove our result we show that $\det(M)\neq 0$, by induction on
$|\mathcal{C}|$. If $|\mathcal{C}|=1$ it is easy to check that $\det(M)=\pm
1$. Suppose $|\mathcal{C}|>1$ and let $a\in V(\mathcal{C})$ be such that it is
a minimal element under the natural partial order on ${\mathbb{Z}}^{2}$.
Observe that in this case there exists an interval $J=[a,b]$ in $\mathcal{C}$
such that $a\notin I$ for all $I\in\mathcal{C}\setminus\\{J\\}$. Denote by
$c,d$ the anti-diagonal corner of $J$. We distinguish three cases.

  1. Case 1

Either $c\in F(\mathcal{C})$ or $d\in F(\mathcal{C})$. Without loss of
generality, we consider $c\in F(\mathcal{C})$ and $d\notin F(\mathcal{C})$.
Let $M^{\prime}$ be the matrix obtained by removing the row labelled with $a$
and the column labelled by $J$. Observe that the column removed contains all
$0$ except for $1$ in the position of $a$, so $\det(M)=\pm\det(M^{\prime})$.
Moreover, let $M^{\prime\prime}$ be the matrix obtained from $M^{\prime}$ by
removing the row labelled with $c$ and the column labelled by $c$. It is easy
to verify that $\det(M^{\prime})=\pm\det(M^{\prime\prime})$, so
$\det(M)=\pm\det(M^{\prime\prime})$. Furthermore, it is not difficult to
verify that $M^{\prime\prime}$ is the matrix associated to the polyocollection
$\mathcal{C}^{\prime}=\mathcal{C}\setminus\\{J\\}$, so by induction
$\det(M^{\prime\prime})\neq 0$, that is $\det(M)\neq 0$.

  2. Case 2

Both $c\in F(\mathcal{C})$ and $d\in F(\mathcal{C})$. In this case we can use
the same argument, considering $\det(M)=\pm\det(M^{\prime\prime\prime})$,
where $M^{\prime\prime\prime}$ is the matrix associated to
$\mathcal{C}^{\prime}=\mathcal{C}\setminus\\{J\\}$ and it is obtained from $M$
by removing the rows labelled with $a,c,d$ and the columns labelled by
$J,c,d$.

  3. Case 3

$c$ and $d$ are both the lower left corner of an interval in $\mathcal{C}$.
Also in this case we can use the same argument, considering
$\det(M)=\pm\det(M^{\prime})$, where $M^{\prime}$ is the matrix associated to
$\mathcal{C}^{\prime}=\mathcal{C}\setminus\\{J\\}$ and it is obtained from $M$
by removing the row labelled with $a$ and the column labelled by $J$.

∎

######  Theorem 3.6.

Let $\mathcal{C}$ be a polyocollection. Then $L_{\mathcal{C}}$ is a prime
ideal.

###### Proof.

Let $n=|V(\mathcal{C})|$. By Lemma 3.5, every $\mathbf{v}\in\mathbb{N}^{n}$
can be expressed in an unique way as $\mathbb{Z}$-linear combination of the
vectors in $\mathcal{B}=\\{\mathbf{v}_{I}\mid
I\in\mathcal{C}\\}\cup\\{\mathbf{v}_{a}\mid a\in F(\mathcal{C})\\}$. In
particular, for all $a\in V(\mathcal{C})$, we have
$\mathbf{v}_{a}=\sum_{I\in\mathcal{C}}\lambda_{I}^{(a)}\mathbf{v}_{I}+\sum_{b\in
F(\mathcal{C})}\mu_{b}^{(a)}\mathbf{v}_{b}$, with
$\lambda_{I}^{(a)},\mu_{b}^{(a)}\in\mathbb{Z}$. We define the following map:

$\psi:S_{\mathcal{C}}\longrightarrow K[\\{y_{b}^{\pm 1}\mid b\in
F(\mathcal{C})\\}]\ \ \mbox{with}\ \ \ x_{a}\mapsto\prod_{b\in
F(\mathcal{C})}y_{b}^{\mu_{b}^{(a)}}.$

It is not difficult to check that if $\mathbf{v}\in\mathbb{N}^{n}$ and
$\mathbf{v}=\sum_{I\in\mathcal{C}}\lambda_{I}^{(v)}\mathbf{v}_{I}+\sum_{b\in
F(\mathcal{C})}\mu_{b}^{(v)}\mathbf{v}_{b}$, with
$\lambda_{I}^{(v)},\mu_{b}^{(v)}\in\mathbb{Z}$, then
$\psi(x^{\mathbf{v}})=\prod_{b\in F(\mathcal{C})}y_{b}^{\mu_{b}^{(v)}}$.
Observe that $\ker\psi$ is a toric ideal, in particular it is prime. We show
that $L_{\mathcal{C}}=\ker\psi$.
$\subseteq)$ Let $x^{\mathbf{v}}-x^{\mathbf{w}}\in L_{\mathcal{C}}$, then
$\mathbf{v}-\mathbf{w}\in\Lambda_{\mathcal{C}}$. So
$\mathbf{v}=\mathbf{w}+\sum_{I\in\mathcal{C}}\alpha_{I}\mathbf{v}_{I}$, with
$\alpha_{I}\in\mathbb{Z}$, and by construction the coefficients $\alpha_{I}$
of $\mathbf{v}_{I}$, for $I\in\mathcal{C}$, does not contribute to the
computation of $\psi(x^{\mathbf{v}})$. Therefore
$\psi(x^{\mathbf{v}})=\psi(x^{\mathbf{w}})$.
$\supseteq)$ Let $x^{\mathbf{v}}-x^{\mathbf{w}}$ be a minimal generator of
$\ker\psi$. In particular,
$\mathrm{supp}(x^{\mathbf{v}})\cap\mathrm{supp}(x^{\mathbf{w}})=\emptyset$,
that is $\mathbf{e}=\mathbf{v}-\mathbf{w}\in\mathbb{Z}^{n}$ with
$\mathbf{e}^{+}=\mathbf{v}$ and $\mathbf{e}^{-}=\mathbf{w}$. Moreover
$\psi(x^{\mathbf{v}})=\psi(x^{\mathbf{w}})=\prod_{b\in
F(\mathcal{C})}y_{b}^{\mu_{b}}$ with $\mu_{b}\in\mathbb{Z}$. This means that
$\mathbf{v}=\sum_{I\in\mathcal{C}}\lambda_{I}^{(v)}\mathbf{v}_{I}+\sum_{b\in
F(\mathcal{C})}\mu_{b}^{(v)}\mathbf{v}_{b}$ and
$\mathbf{w}=\sum_{I\in\mathcal{C}}\lambda_{I}^{(w)}\mathbf{v}_{I}+\sum_{b\in
F(\mathcal{C})}\mu_{b}^{(w)}\mathbf{v}_{b}$ with
$\mu_{b}^{(v)}=\mu_{b}^{(w)}=\mu_{b}$ for each $b\in F(\mathcal{C})$. So
$\mathbf{v}-\mathbf{w}=\sum_{I\in\mathcal{C}}(\lambda_{I}^{(v)}-\lambda_{I}^{(w)})\mathbf{v}_{I}\in\Lambda_{\mathcal{C}}$,
in particular $x^{\mathbf{v}}-x^{\mathbf{w}}\in L_{\mathcal{C}}$. ∎

By the previous results, observe also that $L_{\mathcal{C}}$ does not contain
monomials. In fact, since it is a prime ideal, if it contains monomials then
it contains a variable $x_{a}$ for $a\in V(\mathcal{C})$. But all elements in
$L_{\mathcal{C}}$ have degree greater than 2.

######  Lemma 3.7.

Let $\mathcal{C}$ be a polyocollection. Then there exists a monomial $u\in
S_{\mathcal{C}}$ such that $L_{\mathcal{C}}=(I_{\mathcal{C}}:u)$. In
particular $I_{\mathcal{C}}\subseteq L_{\mathcal{C}}$.

###### Proof.

The proof is exactly the same as that provided by [20, Lemma 2.1] for a
collection of cells. ∎

######  Theorem 3.8.

Let $\mathcal{C}$ be a polyocollection. Then $I_{\mathcal{C}}$ is a prime
ideal if and only if $I_{\mathcal{C}}=L_{\mathcal{C}}$.

###### Proof.

If $I_{\mathcal{C}}=L_{\mathcal{C}}$, then trivially $I_{\mathcal{C}}$ is
prime by Theorem 3.6. Suppose that $I_{\mathcal{C}}$ is prime. By Lemma 3.7 we
have that $I_{\mathcal{C}}\subseteq L_{\mathcal{C}}$ and
$L_{\mathcal{C}}=(I_{\mathcal{C}}:u)$ for some monomial $u\in
S_{\mathcal{C}}$. So, if $f\in L_{\mathcal{C}}$ then $fu\in I_{\mathcal{C}}$
and observe that $u\notin I_{\mathcal{C}}$. Since $I_{\mathcal{C}}$ is prime
then $f\in I_{\mathcal{C}}$, so we proved $I_{\mathcal{C}}\supseteq
L_{\mathcal{C}}$. ∎

######  Remark 3.9.

Let $\mathcal{C}$ be a polyocollection and $I,J\in\mathcal{C}$. We say that
$I$ and $J$ are connected if there exists a sequence
$I_{1},\ldots,I_{m}\in\mathcal{C}$ with $I_{1}=I$ and $I_{m}=J$ such that
$I_{i}\cap I_{i+1}$ is a common vertex or a common edge for
$i\in\\{1,\ldots,m\\}$. We can consider in $\mathcal{C}$ the equivalence
relation $\simeq$, where $I\simeq J$ if and only if $I$ is connected with $J$.
Observe that all equivalence classes of $\simeq$ are polyocollection, that we
call the _connected components_ of $\mathcal{C}$. If
$\mathcal{C}_{1},\ldots,\mathcal{C}_{n}$ are the connected components of
$\mathcal{C}$, it is not difficult to see that
$L_{\mathcal{C}}=\sum_{i=1}^{n}L_{\mathcal{C}_{i}}$.

###  3.2. Primary decomposition in polyocollections

Let $\mathcal{C}$ be a polyocollection. A subset $X\subseteq V(\mathcal{C})$
is called an admissible set of $\mathcal{C}$ if for every inner interval $I$
of $\mathcal{C}$, $X\cap V(I)=\emptyset$ or $X\cap V(I)$ contains the boundary
of an edge of $I$. Observe that $\emptyset$ and $V(\mathcal{C})$ are
admissible sets.

For an admissible set $X$ of $\mathcal{C}$, define the following set:

$\mathcal{G}^{(X)}=\\{I\in G(\mathcal{C})\mid V(I)\cap X=\emptyset\\}.$

######  Proposition 3.10.

Let $\mathcal{C}$ be a polyocollection and $X$ be an admissible set. Then
there exists a polyocollection $\mathcal{C}^{(X)}$ such that
$\mathcal{G}^{(X)}$ is the set of inner intervals of $\mathcal{C}^{(X)}$.

###### Proof.

Let $\mathcal{C}^{(X)}$ be the set of minimal intervals in $\mathcal{G}^{(X)}$
with respect to set inclusion. We prove that $\mathcal{C}^{(X)}$ is a
polyocollection. Let $I,J\in\mathcal{C}^{(X)}$, by construction it is trivial
that $I\nsubseteq J$. Suppose $I\cap J\notin E(I)\cap E(J)$ and there exist
$F\in E(I)$ and $G\in E(J)$ such that $|F\cap G|>1$. Then
$I,J\notin\mathcal{C}$ and we can assume, without loss of generality, that
there exist $[a,b]\in E(I)$ and $[x,y]\in E(J)$, in particular $a,b\in V(I)$
and $x,y\in V(J)$, such that $[a,b]$ and $[x,y]$ belong to the same line,
$a\in[x,y]$ and $x\notin[a,b]$. Let $z\in V(J)$ such that $\\{x,z\\}$ is the
boundary of an edge of $J$ (see for instance Figure 6). Since
$I,J\in\mathcal{G}^{(X)}$ and $X$ is an admissible set, then $a,x,z\notin X$.

(a)  (b)

Figure 6. Two intervals $I,J$ such that $I\cap J\notin E(I)\cap E(J)$ and
$|F\cap G|>1$ for some $F\in E(I)$ and $G\in E(J)$.

Moreover $a,x,z$ are the vertices of an inner interval $K$ of $\mathcal{C}$,
otherwise there exist two intervals $I_{1},J_{1}\in\mathcal{C}$, with
$I_{1}\subseteq I$ and $J_{1}\subseteq J$, having no edges in common and two
edges intersecting in two or more points, that contradicts $\mathcal{C}$ is a
polyocollection. In particular $K\subseteq J$ and $|X\cap V(K)|\leq 1$.
Considering that $X$ is an admissible set, we have $X\cap V(K)=\emptyset$,
obtaining $K\in\mathcal{G}^{(X)}$ that contradicts the minimality of $J$. So
$\mathcal{C}^{(X)}$ is a polyocollection.

Finally, if $I\in\mathcal{G}^{(X)}$ we show that $I$ is union of intervals in
$\mathcal{C}^{(X)}$ having pairwise disjoint common interior, that is
$\mathcal{G}^{(X)}$ is the set of inner intervals of the polyocollection
$\mathcal{C}^{(X)}$. So, put $I=[a,b]$ with $V(I)=\\{a,b,c,d\\}$. If
$V(\mathcal{C})\cap(I\setminus V(I))\subseteq X$ then it is not difficult to
see that $I\in\mathcal{C}^{(X)}$, so suppose there exists $e\in
V(\mathcal{C})\cap(I\setminus V(I))$ such that $e\notin X$. Consider two
possibilities: either $e$ belongs to an edge of $I$ or $e\in\mathrm{int}(I)$.

  * •

Assume that $e$ belongs to an edge of $I$, say $[a,c]$ without loss of
generality. Then it is not difficult to see that in such a case there exists
$I_{1},I_{2}$ inner intervals of $\mathcal{C}$ with $\\{a,e,d\\}\subset
V(I_{1})$, $\\{e,c,b\\}\subset V(I_{2})$ such that
$\operatorname{c}(I)=\operatorname{c}(I_{1})\cup\operatorname{c}(I_{2})$.
Moreover, $|X\cap I_{1}|\leq 1$ and $|X\cap I_{2}|\leq 1$ and since $X$ is an
admissible set of $\mathcal{C}$ we obtain $X\cap I_{1}=\emptyset$ and $X\cap
I_{2}=\emptyset$, that is $I_{1},I_{2}\in\mathcal{G}^{(X)}$.

  * •

Assume that $e\in\mathrm{int}(I)$. In such a case is not difficult to see that
we can write
$\operatorname{c}(I)=\operatorname{c}(I_{1})\cup\operatorname{c}(I_{2})\cup\operatorname{c}(I_{3})\cup\operatorname{c}(I_{4})$,
with $\\{a,e\\}\subset V(I_{1})$, $\\{b,e\\}\subset V(I_{2})$,
$\\{c,e\\}\subset V(I_{3})$, $\\{d,e\\}\subset V(I_{4})$ and $I_{i}$ is an
inner interval of $\mathcal{C}$ for $i\in\\{1,2,3,4\\}$. For all
$i\in\\{1,2,3,4\\}$, observe that $X\cap V(I_{i})$ does not contain the
boundary of an edge of $I_{i}$, so $X\cap V(I_{i})=\emptyset$ considering that
$X$ is an admissible set of $\mathcal{C}$. Therefore
$I_{i}\in\mathcal{G}^{(X)}$ for all $i\in\\{1,2,3,4\\}$.

It follows that in every case $I$ is union of intervals in $\mathbb{R}^{2}$
belonging to $\mathcal{G}^{(X)}$ and having pairwise disjoint common interior.
If such intervals are not minimal, with respect to set inclusion, we can
repeat the previous arguments to these intervals obtaining the desired
conclusion. ∎

Denote by $\mathcal{C}^{(X)}$ the polyocollection having $\mathcal{G}^{(X)}$
as set of inner intervals. We define the following ideal

$J_{X}=(\\{x_{a}\mid a\in X\\})+L_{\mathcal{C}^{(X)}}\subseteq
S_{\mathcal{C}}.$

It is not difficult to see that $I_{\mathcal{C}}\subseteq J_{X}$ for all
admissible sets $X$. In fact, by the previous result, $(\\{x_{a}\mid a\in
X\\})+I_{\mathcal{C}}=(\\{x_{a}\mid a\in X\\})+I_{\mathcal{C}^{(X)}}\subseteq
J_{X}$. Moreover, since the set of variables involved in
$L_{\mathcal{C}^{(X)}}$ is disjoint to the set $\\{x_{a}\mid a\in X\\}$, it is
easy to see that $J_{X}$ is a prime ideal for all $X$ admissible sets of
$\mathcal{C}$.

######  Remark 3.11.

Let $\mathcal{C}$ be a polyocollection and suppose that $I_{\mathcal{C}}$ is
not a prime ideal. Then $J_{\emptyset}=L_{\mathcal{C}}$ is a minimal prime
ideal of $I_{\mathcal{C}}$; otherwise let $\mathfrak{p}$ be a prime ideal such
that $I_{\mathcal{C}}\subset\mathfrak{p}\subseteq L_{\mathcal{C}}$. If $f\in
L_{\mathcal{C}}$, then there exists a monomial $u\in S_{\mathcal{C}}$ such
that $fu\in I_{\mathcal{C}}\subset\mathfrak{p}$. If $u\in\mathfrak{p}$, since
$\mathfrak{p}$ is prime there exists $x_{a}\in\mathrm{supp}(u)$ such that
$x_{a}\in\mathfrak{p}$, but this is not possible since $L_{\mathcal{C}}$ does
not contain variables. So $u\notin\mathfrak{p}$, in particular
$f\in\mathfrak{p}$. Therefore $\mathfrak{p}=L_{\mathcal{C}}$.

######  Proposition 3.12.

Let $\mathcal{C}$ be a polyocollection and suppose that $I_{\mathcal{C}}$ is
not a prime ideal. Then for every minimal prime ideal $\mathfrak{p}$ of
$I_{\mathcal{C}}$, there exists an admissible set $X$ such that
$\mathfrak{p}=J_{X}$

###### Proof.

Let $\mathfrak{p}$ be a minimal prime ideal of $I_{\mathcal{C}}$ and let
$Y=\\{a\in V(\mathcal{C})\mid x_{a}\in\mathfrak{p}\\}$. Note that
$(\\{x_{a}\mid a\in Y\\})+I_{\mathcal{C}}\subset\mathfrak{p}$. We show that
$Y$ is an admissible set for $\mathcal{C}$. In fact, if $Y$ is not an
admissible set, then there exists an inner interval $I$ of $\mathcal{C}$ such
that $|Y\cap V(I)|=1$ or $Y\cap V(I)=\\{a,b\\}$ with $a,b$ both diagonal (or
anti-diagonal) corners of $I$. So, considering the binomial
$f_{I}=x_{a}x_{b}-x_{c}x_{d}$, we obtain that either
$x_{a}x_{b}\in\mathfrak{p}$ with $a,b\notin Y$ or $x_{c}x_{d}\in\mathfrak{p}$
with $c,d\notin Y$, that is a contradiction for the primality of
$\mathfrak{p}$.

Now, we can write $(\\{x_{a}\mid a\in Y\\})+I_{\mathcal{C}}=(\\{x_{a}\mid a\in
Y\\})+I_{\mathcal{C}^{(Y)}}$ and it is contained in $\mathfrak{p}$. If $f\in
L_{\mathcal{C}^{(Y)}}$ then by Lemma 3.7, there exists a monomial $u\in
S_{\mathcal{C}^{(Y)}}$ such that $fu\in
I_{\mathcal{C}^{(Y)}}\subseteq\mathfrak{p}$, and in particular
$\mathrm{supp}(u)\cap\\{x_{a}\mid a\in Y\\}=\emptyset$. As consequence
$u\notin\mathfrak{p}$, otherwise from the primality of $\mathfrak{p}$ there
exists a $x_{a}\in\mathrm{supp}(u)$ such that $x_{a}\in\mathfrak{p}$,
obtaining $a\in Y$, that is a contradiction. Therefore $f\in\mathfrak{p}$, so
$(\\{x_{a}\mid a\in Y\\})+L_{\mathcal{C}^{(Y)}}\subseteq\mathfrak{p}$ and by
the minimality of $\mathfrak{p}$ we obtain $(\\{x_{a}\mid a\in
Y\\})+L_{\mathcal{C}^{(Y)}}=\mathfrak{p}$. ∎

######  Theorem 3.13.

Let $\mathcal{C}$ be a polyocollection. Then
$\sqrt{I_{\mathcal{C}}}=\bigcap_{X}J_{X}$ where $X$ moves overall the
admissible sets of $\mathcal{C}$.

###### Proof.

We have $I_{\mathcal{C}}\subseteq\bigcap_{X}J_{X}$ where $X$ moves overall the
admissible sets of $\mathcal{C}$, so
$\sqrt{I_{\mathcal{C}}}\subseteq\bigcap_{X}J_{X}$. Moreover each minimal prime
of $I_{\mathcal{C}}$ is also a minimal prime of $\sqrt{I_{\mathcal{C}}}$.
Finally, since a radical ideal is the intersection of its minimal prime
ideals, the assertion of the theorem will follow. ∎

######  Remark 3.14.

If $\mathcal{C}$ is a polyocollection and $I_{\mathcal{C}}$ is radical, then
Theorem 3.13 provides a primary decomposition of it. Such decomposition is not
minimal since the converse of Proposition 3.12 does not hold in general. We
will provide an example of this fact in the last section.

##  4\. Primary decomposition of closed path polyominoes

In this section, we consider a particular class of polyominoes, called _closed
paths_ , for which we provide the minimal primary decomposition of the
polyomino ideal. We already observed that a collection of cells (or a
polyomino in particular) is also a polyocollection, but with respect to the
previous section we give a more detailed (and combinatorial) description of
the primary decomposition, taking advantage of the structure of closed path.

###  4.1. Closed paths and their zig-zag walks

In according to [3], we recall the definition of a closed path polyomino, and
the configuration of cells characterizing its primality. Closed paths are
polyominoes contained in the class of thin polyominoes, introduced in [20],
that are polyominoes not containing the square tetromino as subpolyomino.

######  Definition 4.1.

We say that a polyomino $\mathcal{P}$ is a closed path if it is a sequence of
cells $A_{1},\dots,A_{n},A_{n+1}$, $n>5$, such that:

  1. (1)

$A_{1}=A_{n+1}$;

  2. (2)

$A_{i}\cap A_{i+1}$ is a common edge, for all $i=1,\dots,n$;

  3. (3)

$A_{i}\neq A_{j}$, for all $i\neq j$ and $i,j\in\\{1,\dots,n\\}$;

  4. (4)

For all $i\in\\{1,\dots,n\\}$ and for all $j\notin\\{i-2,i-1,i,i+1,i+2\\}$
then $V(A_{i})\cap V(A_{j})=\emptyset$, where $A_{-1}=A_{n-1}$, $A_{0}=A_{n}$,
$A_{n+1}=A_{1}$ and $A_{n+2}=A_{2}$.

Figure 7. A closed path with an $L$-configuration and a ladder of three steps.

A path of five cells $C_{1},C_{2},C_{3},C_{4},C_{5}$ of $\mathcal{P}$ is
called an L-configuration if the two sequences $C_{1},C_{2},C_{3}$ and
$C_{3},C_{4},C_{5}$ go in two orthogonal directions. A set
$\mathcal{B}=\\{\mathcal{B}_{i}\\}_{i=1,\dots,n}$ of maximal horizontal (or
vertical) blocks of rank at least two, with $V(\mathcal{B}_{i})\cap
V(\mathcal{B}_{i+1})=\\{a_{i},b_{i}\\}$ and $a_{i}\neq b_{i}$ for all
$i=1,\dots,n-1$, is called a ladder of $n$ steps if $[a_{i},b_{i}]$ is not on
the same edge interval of $[a_{i+1},b_{i+1}]$ for all $i=1,\dots,n-2$. For
instance, in Figure 7 there is a closed path having an $L$-configuration and a
ladder of three steps. A closed path has no zig-zag walks if and only if it
contains an $L$-configuration or a ladder of at least three steps (see [3,
Section 6]).

######  Definition 4.2.

A zig-zag walk of a collection of cells ${\mathcal{P}}$ is a sequence
${\mathcal{W}}:I_{1},\ldots,I_{l}$ of distinct inner intervals of
${\mathcal{P}}$ where, for all $i=1,\ldots,l$, the interval $I_{i}$ has
diagonal corners $v_{i},z_{i}$ and anti-diagonal corners $u_{i},v_{i+1}$ or
anti-diagonal corners $v_{i},z_{i}$ and diagonal corners $u_{i},v_{i+1}$, such
that

  1. (1)

$I_{1}\cap I_{l}=\\{v_{1}=v_{l+1}\\}$ and $I_{i}\cap I_{i+1}=\\{v_{i+1}\\}$
for all $i=1,\ldots,l-1$;

  2. (2)

$v_{i}$ and $v_{i+1}$ are on the same edge interval of ${\mathcal{P}}$, for
all $i=1,\ldots,l$;

  3. (3)

for all $i,j\in\\{1,\ldots,l\\}$ with $i\neq j$, there exists no interval $J$
of ${\mathcal{P}}$ such that $z_{i}$ and $z_{j}$ belongs to $J$.

By using the concept of zig-zag walk, in [3, Theorem 6.2] the authors
characterize the primality of a closed path $\mathcal{P}$ proving that
$I_{\mathcal{P}}$ is prime if and only if $\mathcal{P}$ does not contain any
zig-zag walk.
Let $\mathcal{P}$ be a collection of cells. Following the notations in [19],
for a zig-zag walk ${\mathcal{W}}:I_{1},\ldots,I_{l}$ of ${\mathcal{P}}$, let
$f_{\mathcal{W}}$ be the corresponding binomial of ${\mathcal{W}}$, that is
$f_{\mathcal{W}}=\prod_{i=1}^{l}x_{z_{i}}-\prod_{i=1}^{l}x_{u_{i}}$. Moreover,
we denote

$N(\mathcal{W})=\\{v\in V({\mathcal{P}}):v\in[v_{i},v_{i+1}]\ \text{for}\
1\leq i\leq l\\}$

and we call it the _necklace_ of $\mathcal{W}$. Furthermore, it is shown in
[19, Proposition 3.5] that if a polyomino $\mathcal{P}$ has a zig-zag walk
$\mathcal{W}$, then:

$\pm
x_{v_{1}}f_{\mathcal{W}}=\prod_{k>1}x_{z_{k}}f_{I_{1}}+\ldots+(-1)^{i+1}\prod_{j<i}x_{u_{j}}\prod_{k>i}x_{z_{k}}f_{I_{i}}+\ldots+(-1)^{l+1}\prod_{j<l}x_{u_{j}}f_{I_{l}}$

where $f_{I_{i}}$ is the binomial related to the inner interval $I_{i}$, for
$i\in\\{1,\ldots,l\\}$ and the sign depends if $v_{1}$ is a diagonal or an
anti-diagonal corner of $I_{1}$. For $i\in\\{2,\ldots,l\\}$, it is easy to
obtain a similar expression also for $x_{v_{i}}f_{\mathcal{W}}$, by
relabelling the indices $1,2,\ldots,l$. In particular, for all zig-zag walk
$\mathcal{W}$ then $x_{v_{i}}f_{\mathcal{W}}\in I_{\mathcal{P}}$ for all
$1\leq i\leq l$. Observing the previous formula, we can also extend the same
claim also in the case $\mathcal{P}$ is a collection of cells.

######  Lemma 4.3.

Let $\mathcal{P}$ be a collection of cells with a zig-zag walk ${\mathcal{W}}$
and ${\mathfrak{p}}$ be a minimal prime ideal of $I_{\mathcal{P}}$. Under the
notations as above, if $f_{\mathcal{W}}\notin{\mathfrak{p}}$, then
$x_{v}\in{\mathfrak{p}}$ for all $v\in N({\mathcal{W}})$.

###### Proof.

Since $f_{\mathcal{W}}\notin{\mathfrak{p}}$, by above discussion, we get that
$x_{v_{i}}\in{\mathfrak{p}}$ for all $1\leq i\leq l$. Fix an $i$. We have
$x_{u_{i}}x_{v}-x_{v_{i}}x_{b_{v}}\in I_{{\mathcal{P}}}$ for all
$v\in]v_{i},v_{i+1}[$, where $b_{v}\in V({\mathcal{P}})$ such that the
rectangle with the vertices $u_{i},v,v_{i}$ and $b_{v}$ is in ${\mathcal{P}}$.
Since $x_{v_{i}}\in{\mathfrak{p}}$, $x_{v}x_{u_{i}}\in{\mathfrak{p}}$.
Therefore, either $x_{v}\in{\mathfrak{p}}$ for all $v\in]v_{i},v_{i+1}[$ or
$x_{u_{i}}\in{\mathfrak{p}}$.
Similarly, by repeating the above argument for
$x_{b_{v}}x_{v_{i+1}}-x_{v}x_{z_{i}}\in I_{{\mathcal{P}}}$ for all
$v\in]v_{i},v_{i+1}[$, where $b_{v}\in V({\mathcal{P}})$ such that the
rectangle with the vertices $z_{i},v,v_{i+1}$ and $b_{v}$ is in
${\mathcal{P}}$. We get that either $x_{v}\in{\mathfrak{p}}$ for all
$v\in]v_{i},v_{i+1}[$ or $x_{z_{i}}\in{\mathfrak{p}}$. If
$x_{u_{i}},x_{z_{i}}\in{\mathfrak{p}}$, then
$f_{\mathcal{W}}\in{\mathfrak{p}}$ which contradicts the assumption. Hence,
$x_{v}\in{\mathfrak{p}}$ for all $v\in]v_{i},v_{i+1}[$. ∎

$m$$e$$e_{1}$$d$$c$$b$$d_{1}$$a$$Z$$X$$Y$$W$ (a)

$d$$c$$b$$a$$e$$\ell$$W$$X$$Y$$Z$ (b)

Figure 8. Possible changes of direction in a closed path having a zig-zag
walk.

From now onwards, we assume that ${\mathcal{P}}$ is a non-prime closed path
polyomino. By [3, Theorem 6.2], ${\mathcal{P}}$ contains a zig-zag walk. As a
consequence, by [3, Proposition 6.1] $\mathcal{P}$ does not contain any
$L$-configuration, so any two maximal inner intervals of ${\mathcal{P}}$
intersect themselves in the cells displayed in Figure 8a or 8b (up to
reflections or rotations). We wish to determine all zig-zag walks of
${\mathcal{P}}$.

######  Discussion 4.4.

Let ${\mathcal{W}}:I_{1},\ldots,I_{l}$ be a zig-zag walk of ${\mathcal{P}}$.
We show that from the labelled vertices of Figure 8a, only
$d\in\\{v_{0},\ldots,v_{l}\\}$. By (3) of Definition 4.2, $I_{i}$ and $I_{j}$
are not contained in a maximal inner interval if $i\neq j$. By (1) of
Definition 4.2, $m,a,c,e$ and $e_{1}$ as in Figure 8a do not belong to
$\\{v_{0},\ldots,v_{l}\\}$. Since $[v_{1},v_{2}]\cup\cdots\cup[v_{l},v_{1}]$
is a closed path of edge intervals, we get that
$d\in\\{v_{0},\ldots,v_{l}\\}$. Now it suffice to show that
$d_{1},b\notin\\{v_{0},\ldots,v_{l}\\}$. Assume that
$d_{1}\in\\{v_{0},\ldots,v_{l}\\}$. Then, there exists a $j$ such that
$d,d_{1}\in V(I_{j})$. By Definition 4.2, $I_{j}=\\{X\\}$ or $I_{j}=\\{Z\\}$.
We may assume that $I_{j-1}$ is the inner interval such that $d_{1}\in
V(I_{j-1})$ and $W\in I_{j-1}$, and $I_{j+1}$ is the inner interval such that
$d\in V(I_{j+1})$ and $Y\in I_{j+1}$. If $I_{j}=\\{X\\}$, then $b,d\in
I_{j}\cap I_{j+1}$ and if $I_{j}=\\{Z\\}$, then $d_{1},e_{1}\in I_{j}\cap
I_{j-1}$. Both of them contradict (1) of Definition 4.2; thus,
$d_{1}\notin\\{v_{0},\ldots,v_{l}\\}$. Similarly,
$b\notin\\{v_{0},\ldots,v_{l}\\}$. Hence, from the labelled vertices of Figure
8a, only $d\in\\{v_{0},\ldots,v_{l}\\}$.
We now show that from the labelled vertices of Figure 8b, exactly one of $e$
and $\ell$ is in $\\{v_{0},\ldots,v_{l}\\}$. By (1) of Definition 4.2, $a,b,c$
and $d$ as in Figure 8b do not belong to $\\{v_{0},\ldots,v_{l}\\}$. Since
$v_{i}$ is the vertex where the inner intervals $I_{i-1}$ and $I_{i}$
intersects and, $e$ and $\ell$ are the vertices belongs to two maximal inner
intervals, either $e\in\\{v_{0},\ldots,v_{l}\\}$ or
$\ell\in\\{v_{0},\ldots,v_{l}\\}$. Now, we show that exactly one of $e$ and
$\ell$ is in $\\{v_{0},\ldots,v_{l}\\}$. Suppose both $e$ and $\ell$ are in
$\\{v_{0},\ldots,v_{l}\\}$. Then there exists a $j\in\\{1,\ldots,l\\}$ such
that $I_{j}=\\{X\\}$ or $I_{j}=\\{Y\\}$. Let $I_{j-1}$ (respectively
$I_{j+1}$) be the inner interval such that $e\in V(I_{j-1})$ (respectively
$\ell\in V(I_{j+1})$) and $W\in I_{j-1}$ (respectively $Z\in I_{j+1}$). If
$I_{j}=\\{X\\}$, then $a,e\in I_{j-1}\cap I_{j}$ which is a contradiction.
Similarly, if $I_{j}=\\{Y\\}$, then $b,\ell\in I_{j}\cap I_{j+1}$ which is a
contradiction. Hence, exactly one of $e$ and $\ell$ is in
$\\{v_{0},\ldots,v_{l}\\}$. ∎

$d_{1}$$d_{2}$$d_{4}$$d_{3}$$b_{1}$$b_{2}$$c_{1}$$c_{2}$ Figure 9. An example
of a closed path having zig-zag walks.

######  Example 4.5.

We illustrate the above discussion now. Let ${\mathcal{P}}$ be the polyomino
as shown in the Figure 9. Note that ${\mathcal{P}}$ is a non-prime polyomino.
Let ${\mathcal{W}}:I_{1},\ldots,I_{l}$ be a zig-zag walk of ${\mathcal{P}}$.
By Discussion 4.4,
$\\{d_{1},d_{2},d_{3},d_{4}\\}\subset\\{v_{1},\ldots,v_{l}\\}$. Also, by
Discussion 4.4, exactly one of $b_{1}$ and $b_{2}$ is in
$\\{v_{1},\ldots,v_{l}\\}$ and exactly one of $c_{1}$ and $c_{2}$ is in
$\\{v_{1},\ldots,v_{l}\\}$. So in conclusion, we get that $l=6$ and
$\\{v_{1},\ldots,v_{6}\\}$ is either
$\\{d_{1},d_{2},d_{3},d_{4},b_{1},c_{1}\\}$ or
$\\{d_{1},d_{2},d_{3},d_{4},b_{1},c_{2}\\}$ or
$\\{d_{1},d_{2},d_{3},d_{4},b_{2},c_{1}\\}$ or
$\\{d_{1},d_{2},d_{3},d_{4},b_{2},c_{2}\\}$.

######  Remark 4.6.

Based on the Discussion 4.4, we make few remarks on the zig-zag walks of a
non-prime closed path polyomino.

  1. (1)

The number of inner intervals in a zig-zag is equal to the number of maximal
inner intervals in ${\mathcal{P}}$ with at least three cells.

  2. (2)

Let ${\mathcal{W}}:I_{1},\ldots,I_{l}$ and
${\mathcal{W}}^{\prime}:I^{\prime}_{1},\ldots,I^{\prime}_{l}$ be two zig-zag
walks of ${\mathcal{P}}$. Then $N(\mathcal{W})=N(\mathcal{W}^{\prime})$.

  3. (3)

Under the labelling of Figure 8a $x_{b},x_{d_{1}}\in N({\mathcal{P}})$ and if
$f_{\mathcal{W}}=\prod_{i=1}^{l}x_{z_{i}}-\prod_{i=1}^{l}x_{u_{i}}$ is any
zig-zag walk then either $x_{c}\in\mathrm{supp}(\prod_{i=1}^{l}x_{z_{i}})$ and
$x_{e}\in\mathrm{supp}(\prod_{i=1}^{l}x_{u_{i}})$ or
$x_{c}\in\mathrm{supp}(\prod_{i=1}^{l}x_{u_{i}})$ and
$x_{e}\in\mathrm{supp}(\prod_{i=1}^{l}x_{z_{i}})$.

######  Notation 4.7.

Consider the following notation:

  * •

$N({\mathcal{P}})\mathrel{\mathop{:}}=N(\mathcal{W})$, for some zig-zag walk
$\mathcal{W}$ of $\mathcal{P}$. We observed that this set is unique and does
not depend on the choice of the zig-zag walk $\mathcal{W}$.

  * •

$M(\mathcal{P})\mathrel{\mathop{:}}=\\{m\in V(\mathcal{P})\ \text{as shown in
Figure~{}\ref{fig:threestepladder}, up to reflections or rotations}\\}$

  * •

$R(\mathcal{P})\mathrel{\mathop{:}}=\\{x_{a}x_{b}-x_{c}x_{d}:a,b,c,d\
\text{are as shown in Figure~{}\ref{fig:Lshape}, up to reflections or
rotations}\\}$

Define the following ideals:

${\mathfrak{p}}_{1}=I_{\mathcal{P}}+(f_{\mathcal{W}}:{\mathcal{W}}\ \text{is a
zig-zag walk of}\ {\mathcal{P}})$ ${\mathfrak{p}}_{2}=(x_{a}:a\in
N({\mathcal{P}}))+(x_{m}:m\in M({\mathcal{P}}))+(R(\mathcal{P})).$

######  Lemma 4.8.

Let $\mathcal{P}$ be a non-prime closed path polyomino. Then
$I_{\mathcal{P}}\subset{\mathfrak{p}}_{2}$.

###### Proof.

Let $[p,q]$ be an inner interval of $\mathcal{P}$, with $r,s$ as anti-diagonal
corners. We prove that $f=x_{p}x_{q}-x_{r}x_{s}$ belongs to
${\mathfrak{p}}_{2}$. Assume that, referring to Figure 8b, $q\in\
\\{a,d,\ell\\}$ or $p\in\\{e,c,b\\}$. If $f\in R(\mathcal{P})$ then
$f\in{\mathfrak{p}}_{2}$ trivially. Suppose that $f\notin R(\mathcal{P})$ and
$q=d$. Then $s=\ell$, otherwise $f\in R(\mathcal{P})$, and in particular $p$
belongs to the edge interval of $\mathcal{P}$ containing $e$ and $\ell$. It
follows that $p,s\in N(\mathcal{P})$, so
$x_{p}x_{q},x_{r}x_{s}\in{\mathfrak{p}}_{2}$, hence $f\in{\mathfrak{p}}_{2}$.
The other cases can be proved in a similar way. Now assume that, referring to
Figure 8a, $q\in\\{b,m,d_{1}\\}$ or $p\in\\{a,c,d,e,e_{1}\\}$. We may suppose
that $q=m$, because one can prove all other cases with similar arguments. In
such a case $p\in\\{c,d,e_{1}\\}$. Suppose that $p=c$. Then $r=a$ and
$s=d_{1}$. Since $q=m\in M(\mathcal{P})$ and $d_{1}\in N(\mathcal{P})$, we get
$x_{p}x_{q},x_{r}x_{s}\in{\mathfrak{p}}_{2}$, hence $f\in{\mathfrak{p}}_{2}$.
The other two cases $p=d$ and $p=e$ can be proved similarly. For the other
situations, coming from reflections or rotations of Figures 8a and 8b, the
proofs are similar to the previous ones. The last case to discuss is when no
corners of $[p,q]$ is one of the vertices in Figures 8a and 8b. In such a
case, we can have one of the following four situations: either $r,q\in
N(\mathcal{P})$, or $p,s\in N(\mathcal{P})$, or $p,r\in N(\mathcal{P})$, or
$q,s\in N(\mathcal{P})$, depending on the shape of $\mathcal{P}$. In each case
it follows that $x_{p}x_{q},x_{r}x_{s}\in{\mathfrak{p}}_{2}$, hence
$f\in{\mathfrak{p}}_{2}$. ∎

######  Proposition 4.9.

Let $\mathcal{P}$ be a non-prime closed path polyomino. Then, under the
notations of Notation 4.7, ${\mathfrak{p}}_{2}$ is a prime ideal and
$\mathrm{height}({\mathfrak{p}}_{2})=|\mathcal{P}|$.

###### Proof.

The minimal generating set of ${\mathfrak{p}}_{2}$ can be partitioned between
disjoint set of variables. So that ${\mathfrak{p}}_{2}$ admits a tensor
product decomposition, where one ideal is generated by variables and others
are principal prime ideals. Hence, ${\mathfrak{p}}_{2}$ itself is a prime
ideal.

Under the labelling of Figure 8a, the variables $x_{m}$ corresponds to the
cell $X$. Under the labelling of Figure 8b, the generator
$x_{a}x_{b}-x_{c}x_{d}$ corresponds to the cell $X$. So it suffices to show
that the number of remaining cells is equal to the cardinality of
$N({\mathcal{P}})$. Let ${\mathcal{W}}:I_{1},\ldots,I_{l}$ be a zig-zag walk
of ${\mathcal{P}}$. For $1\leq i\leq l$, the number of cells in $I_{i}$ are
equal to the number of vertices in $]v_{i},v_{i+1}]$. Therefore, the number of
remaining cells is equal to the cardinality of $N(P)$. Thus, height of
${\mathfrak{p}}_{2}$ is equal to number of cells of ${\mathcal{P}}$ ∎

In the following proposition, we show that if a polyomino ${\mathcal{P}}$
becomes simple after removing a cell, then
$\mathrm{height}(I_{\mathcal{P}})=|\mathcal{P}|$. This gives a different proof
of [7, Corollary 3.5]. As a consequence the ideal $\mathfrak{p}_{2}$ is a
minimal prime of $I_{\mathcal{P}}$.

######  Proposition 4.10.

Let $\mathcal{P}$ be a polyomino. If there exists a cell $A$ in
${\mathcal{P}}$ such that $\mathcal{P}\setminus\\{A\\}$ is simple, then
$\mathrm{height}(I_{\mathcal{P}})=|\mathcal{P}|$. In particular, it holds for
closed path polyominoes.

###### Proof.

It is known by [13, Theorem 3.1] that
$\mathrm{height}(I_{\mathcal{P}})\leq|\mathcal{P}|$ for every collection of
cells $\mathcal{P}$. Let $\mathcal{P}^{\prime}=\mathcal{P}\setminus\\{A\\}$.
Since $\mathcal{P}^{\prime}$ is simple, $I_{\mathcal{P}^{\prime}}$ is a prime
ideal (see [23]). Also, by [15, Theorem 2.1] and [14, Corollary 2.3], we have
that
$\mathrm{height}(I_{\mathcal{P}^{\prime}})=|\mathcal{P}^{\prime}|=|\mathcal{P}|-1$.
We show that
$\mathrm{height}(I_{\mathcal{P^{\prime}}})<\mathrm{height}(I_{\mathcal{P}})$.
We have that $I_{\mathcal{P^{\prime}}}\subsetneq I_{\mathcal{P}}$ and
$I_{\mathcal{P^{\prime}}}$ is prime. Let $\mathfrak{p}$ be a prime ideal such
that $I_{\mathcal{P}}\subseteq\mathfrak{p}$. In particular
$I_{\mathcal{P^{\prime}}}\subsetneq\mathfrak{p}$. Let
$\mathfrak{p}_{0}\subseteq\cdots\subseteq\mathfrak{p}_{n}=I_{\mathcal{P^{\prime}}}$
be a chain of prime ideals contained in $I_{\mathcal{P^{\prime}}}$ of length
$n=\mathrm{height}(I_{\mathcal{P^{\prime}}})$. Then
$\mathfrak{p}_{0}\subseteq\cdots\subseteq\mathfrak{p}_{n}=I_{\mathcal{P^{\prime}}}\subsetneq\mathfrak{p}$
is a chain of prime ideals contained in $\mathfrak{p}$, in particular we have
$\mathrm{height}(\mathfrak{p})\geq n+1$. Since
$\mathrm{height}(I_{\mathcal{P}}):=\min\\{\mathrm{height}(\mathfrak{q})\mid\mathfrak{q}\supseteq
I_{\mathcal{P}}\ \mbox{and}\ \mathfrak{q}\ \mbox{is prime}\\}$, we obtain
$\mathrm{height}(I_{\mathcal{P}})\geq n+1$, so
$\mathrm{height}(I_{\mathcal{P^{\prime}}})<\mathrm{height}(I_{\mathcal{P}})$
as claimed. Therefore
$|\mathcal{P}|-1=\mathrm{height}(I_{\mathcal{P}^{\prime}})<\mathrm{height}(I_{\mathcal{P}})\leq|\mathcal{P}|$.
So the only possibility is $\mathrm{height}(I_{\mathcal{P}})=|\mathcal{P}|$.

For the particular case, from [3, Lemma 3.3] we know that every closed path
polyomino $\mathcal{P}$ contains a block of rank at least three, so there
exists a cell $A$ in $\mathcal{P}$ such that $\mathcal{P}\setminus\\{A\\}$ is
simple. ∎

###  4.2. Primality of $\mathfrak{p}_{1}$ and its height

Let $\mathcal{P}$ be a polyomino. A binomial $f=f^{+}-f^{-}$ in a binomial
ideal $J\subset S_{\mathcal{P}}$ is called redundant if it can be expressed as
a linear combination of binomials in $J$ of lower degree. A binomial is called
irredundant if it is not redundant. Moreover, we denote by $V^{+}_{f}$ the set
of the vertices $v$, such that $x_{v}$ appears in $f^{+}$, and by $V^{-}_{f}$
the set of the vertices $v$, such that $x_{v}$ appears in $f^{-}$.

######  Lemma 4.11.

Let $\mathcal{P}$ be a polyomino and $\phi:S_{\mathcal{P}}\rightarrow T$ a
ring homomorphism with $T$ an integral domain. Let $J=\ker\phi$ and
$f=f^{+}-f^{-}$ be a binomial in $J$ with $\deg f\geq 3$. Suppose that
$I_{\mathcal{P}}\subseteq J$ and $\phi(x_{v})\neq 0$ for all $v\in
V(\mathcal{P})$.

  1. (1)

If there exist three vertices $p,q\in V^{+}_{f}$ and $r\in V^{-}_{f}$ such
that $p,q$ are diagonal (respectively anti-diagonal) corners of an inner
interval and $r$ is one of the anti-diagonal (respectively diagonal) corners
of the same inner interval, then $f$ is redundant in $J$. The same claim also
holds if $p,q\in V^{-}_{f}$ and $r\in V^{+}_{f}$.

  2. (2)

Suppose that $\mathcal{P}$ has a subpolyomino having a collection of cells as
in Figure 10, and that $a,e_{1}\in V_{f}^{+}$ (respectively $V_{f}^{-}$),
$d\in V_{f}^{-}$ (respectively $V_{f}^{+}$) and $e_{i},m\in V_{f}^{-}$
(respectively $V_{f}^{+}$) for some $i\in\\{2,\ldots,n\\}$. Then $f$ is
redundant.

Figure 10. Configuration examined in Lemma 4.11.

###### Proof.

For the proof of $(1)$ see [3, Lemma 2.2]. We prove $(2)$. Observe that:
$f=f^{+}-f^{-}=f^{+}-\frac{f^{-}}{x_{d}x_{m}}x_{b}x_{d_{1}}-\frac{f^{-}}{x_{d}x_{m}}(x_{d}x_{m}-x_{b}x_{d_{1}})$
and $\frac{f^{-}}{x_{d}x_{m}}(x_{d}x_{m}-x_{b}x_{d_{1}})\in
I_{\mathcal{P}}\subseteq J$. Denote
$\tilde{f}=f^{+}-\frac{f^{-}}{x_{d}x_{m}}x_{b}x_{d_{1}}$ and observe that
$\tilde{f}\in\ker\phi=J$. Moreover $e_{1}\in V_{f}^{+}$ and
$x_{d_{1}},x_{e_{i}}\in V_{\tilde{f}}^{-}$, since
$\tilde{f}^{-}=\frac{f^{-}}{x_{d}x_{m}}x_{b}x_{d_{1}}$. So $\tilde{f}$ is
redundant in $J$ by Lemma 4.11(1) and in particular also $f$ is redundant in
$J$. The same claim holds if we suppose $a,e_{1}\in V_{f}^{-}$, $d\in
V_{f}^{+}$ and $e_{i}\in V_{f}^{+}$, using the same argument. ∎

Let $\mathcal{P}$ be a closed path having a zig-zag walk and consider a sub-
polyomino of $\mathcal{P}$ having the shape as in Figure 8a, up to reflections
and rotations. Let $\\{V_{i}\\}_{i\in I}$ be the sets of the maximal vertical
edge intervals of $\mathcal{P}$ and $\\{H_{j}\\}_{j\in J}$ be the set of the
maximal horizontal edge intervals of $\mathcal{P}$. Let $\\{v_{i}\\}_{i\in I}$
and $\\{h_{j}\\}_{j\in J}$ be the set of the variables associated respectively
to $\\{V_{i}\\}_{i\in I}$ and $\\{H_{j}\\}_{j\in J}$. Let $w$ be another
variable different from $v_{i}$ and $h_{j}$, for $i\in I$ and $j\in J$. Under
the notation of Figure 8a, we define the following map:

$\displaystyle\alpha:V(\mathcal{P})$ $\displaystyle\longrightarrow
K[\\{v_{i},h_{j},w\\}:i\in I,j\in J]$ $\displaystyle r$
$\displaystyle\longmapsto v_{i}h_{j}w^{k}$

with $r\in V_{i}\cap H_{j}$, $k=0$ if $r\notin\\{a,b,c,d,e\\}$, and $k=1$, if
$r\in\\{a,b,c,d,e\\}$.
Denote $T_{\mathcal{P}}:=K[\alpha(r):r\in V(\mathcal{P})]$, that is a toric
ring. We consider the following surjective ring homomorphism

$\displaystyle\psi:S_{\mathcal{P}}$ $\displaystyle\longrightarrow
T_{\mathcal{P}}$ $\displaystyle\psi(x_{r}$ $\displaystyle)=\alpha(r)$

Let us denote the kernel of $\psi$ by $J_{\mathcal{P}}$. Observe that the
ideal $J_{\mathcal{P}}$ can be viewed as the toric ideal defined in [19,
Section 3], in particular we can deduce that
$I_{\mathcal{P}}=(J_{\mathcal{P}})_{2}$.

######  Remark 4.12.

Let $f$ be an irredundant binomial in $J_{\mathcal{P}}$ and $a\in V_{f}^{+}$
(resp. $V_{f}^{-}$). Denote by $H_{a}$ and $V_{a}$ respectively the maximal
horizontal and vertical edge intervals of $\mathcal{P}$ containing $a$. Then
there exist $a_{1}\in H_{a}$ and $a_{2}\in V_{a}$ such that $a_{1},a_{2}\in
V_{f}^{-}$ (resp. $V_{f}^{+}$). Moreover if $\mathcal{I}$ is a maximal edge
interval of $\mathcal{P}$, then
$|V_{f}^{+}\cup\mathcal{I}|=|V_{f}^{-}\cup\mathcal{I}|$ otherwise
$f\notin\ker\psi$, and $V_{f}^{+}\cap V_{f}^{-}=\emptyset$ otherwise $f$ is
redundant.

######  Lemma 4.13.

Let $\mathcal{P}$ be a closed path polyomino and $f\in J_{P}$ an irredundant
binomial with $\deg f\geq 3$. Consider a subpolyomino having a configuration
as in Figure 11.

Figure 11. Configuration examined in Lemma 11.

Suppose that $c\in V_{f}^{+},e\in V_{f}^{-}$. Then the following hold:

  1. (1)

$e_{i},d_{i}\notin V_{f}^{+}\cup V_{f}^{-}$ for all $i\in\\{1,\ldots,n-2\\}$.

  2. (2)

Exactly one of the following occurs:

     * a)

$e_{n-1}\in V_{f}^{+}$, $\ell\in V_{f}^{-}$ and $e_{n}\notin V_{f}^{+}\cup
V_{f}^{-}$.

     * b)

$e_{n}\in V_{f}^{+}$, $g\in V_{f}^{-}$ and $e_{n-1}\notin V_{f}^{+}\cup
V_{f}^{-}$.

  3. (3)

$d_{n-1},d_{n}\notin V_{f}^{-}$.

The same result holds replacing $V_{f}^{+}$ with $V_{f}^{-}$ and $V_{f}^{-}$
with $V_{f}^{+}$ in all previous occurrences.

###### Proof.

$(1).$ Firstly, for $i\in\\{2,\ldots,n-2\\}$ observe that $e_{i}\in V_{f}^{+}$
if and only if $d_{i}\in V_{f}^{-}$. In particular, if $e_{i}\in V_{f}^{+}$
and $d_{i}\in V_{f}^{-}$ we obtain a contradiction with Lemma 4.11(1)
considering $e,e_{i},d_{i}$. So, $e_{i}\notin V_{f}^{+}$ and $d_{i}\notin
V_{f}^{-}$ for all $i\in\\{2,\ldots,n-2\\}$. Moreover, if $e_{1}\in V_{f}^{+}$
then, considering its maximal vertical edge interval, we have $d_{1}\in
V_{f}^{-}$ or $m\in V_{f}^{-}$, obtaining in each case a contradiction with
Lemma 4.11(1) considering respectively $e,e_{1},d_{1}$ and $e,e_{1},m$. So
$e_{1}\notin V_{f}^{+}$. Since $e\in V_{f}^{-}$, in its maximal horizontal
edge interval there exists a vertex $X\in V_{f}^{+}$, and by the above
arguments the only possibilities is $X=e_{n-1}$ or $X=e_{n}$. In both cases
observe that, for $i\in\\{2,\ldots,n-1\\}$, $e_{i}\in V_{f}^{-}$ if and only
if $d_{i}\in V_{f}^{+}$, obtaining that $e_{i}\notin V_{f}^{-}$ and
$d_{i}\notin V_{f}^{+}$ for all $i\in\\{3,\ldots,n-1\\}$, otherwise we have a
contradiction with Lemma 4.11(1) considering $X,e_{i},d_{i}$. Suppose
$e_{1}\in V_{f}^{-}$. Then we have necessarily $e_{n-1},e_{n}\in V_{f}^{+}$
and, considering its vertical edge interval, the following two possibilities
occur: either $d_{1}\in V_{f}^{+}$ or $m\in V_{f}^{+}$. If $d_{1}\in
V_{f}^{+}$, then we have a contradiction by Lemma 4.11(1) considering
$e_{1},d_{1},e_{n-1}$. If $m\in V_{f}^{+}$, then $a\in V_{f}^{-}$ or $b\in
V_{f}^{-}$. Both cases lead to a contradiction by Lemma 4.11(1) considering
respectively $c,m,a$ and $b,m,e_{1}$. For what concern the vertex $d_{1}$, we
obtain $d_{1}\notin V_{f}^{+}\cap V_{f}^{-}$ by similar arguments.
$(2)-(3).$ We showed before that either $e_{n-1}\in V_{f}^{+}$ or $e_{n}\in
V_{f}^{+}$. Suppose that $e_{n-1}\in V_{f}^{+}$. If $d_{n-1}\in V_{f}^{-}$ we
obtain a contradiction of Lemma 4.11(1) considering $e_{n-1},d_{n-1},e$. So
$d_{n-1}\notin V_{f}^{-}$ and considering the same vertical edge interval we
obtain $\ell\in V_{f}^{-}$. If $d_{n}\in V_{f}^{-}$ then, considering its
vertical edge interval, we have $g\in V_{f}^{+}$ or $e_{n}\in V_{f}^{+}$,
obtaining in each case a contradiction with Lemma 4.11(1) considering
respectively $g,\ell,e_{n-1}$ and $d_{n},e_{n},e$. So $d_{n}\notin V_{f}^{-}$.
Moreover $e_{n}\notin V_{f}^{-}$, otherwise we have the same contradiction
considering the vertices $e_{n},e_{n-1},\ell$. Finally if $e_{n}\in V_{f}^{+}$
then, denoting with $\mathcal{I}$ the horizontal edge interval of $e$, we have
$|V_{f}^{+}\cap\mathcal{I}|=2$ and $|V_{f}^{-}\cap\mathcal{I}|=1$, that is a
contradiction. So, $e_{n}\notin V_{f}^{+}$ and we proved points (3) and the
case a) of (2). If we assume $e_{n}\in V_{f}^{+}$, with the same argument we
can prove again points (3) and the case b) of (2). By our proof, it is also
easy to argue that the same result holds exchanging $V_{f}^{+}$ with
$V_{f}^{-}$ and $V_{f}^{-}$ with $V_{f}^{+}$. ∎

######  Lemma 4.14.

Let $\mathcal{P}$ be a closed path polyomino and $f\in J_{P}$ an irredundant
binomial with $\deg f\geq 3$. Consider a subpolyomino having a configuration
as in Figure 12.

Figure 12. Configuration examined in Lemma 12.

Suppose that $b_{1},b_{2}\notin V_{f}^{-}$ and that one of the following
occurs:

  * a1)

$a_{1}\in V_{f}^{-}$, $c_{1}\in V_{f}^{+}$ and $c_{2}\notin V_{f}^{+}$.

  * a2)

$a_{2}\in V_{f}^{-}$, $c_{2}\in V_{f}^{+}$ and $c_{1}\notin V_{f}^{+}$.

Then the following hold:

  1. (1)

$a_{i},b_{i}\notin V_{f}^{+}\cup V_{f}^{-}$ for all $i\in\\{3,\ldots,n-2\\}$.

  2. (2)

Exactly one of the following occurs:

     * a)

$a_{n-1}\in V_{f}^{+}$, $d_{1}\in V_{f}^{-}$ and $a_{n}\notin V_{f}^{+}\cup
V_{f}^{-}$.

     * b)

$a_{n}\in V_{f}^{+}$, $d_{2}\in V_{f}^{-}$ and $a_{n-1}\notin V_{f}^{+}\cup
V_{f}^{-}$.

  3. (3)

$b_{1},b_{2}\notin V_{f}^{+}$ and $b_{n-1},b_{n}\notin V_{f}^{-}$.

  4. (4)

If a1) occurs then $a_{2}\notin V_{f}^{+}\cup V_{f}^{-}$, while if a2) occurs
then $a_{1}\notin V_{f}^{+}\cup V_{f}^{-}$.

The same result holds exchanging $V_{f}^{+}$ with $V_{f}^{-}$ (and vice
versa), in all occurrences above.

###### Proof.

Suppose a1) holds. Observe that, for $i\in\\{3,\ldots,n-2\\}$, $a_{i}\in
V_{f}^{+}$ if and only if $b_{i}\in V_{f}^{-}$. In particular, if $a_{i}\in
V_{f}^{+}$ and $b_{i}\in V_{f}^{-}$ we obtain a contradiction with Lemma
4.11(1) considering $a_{1},a_{i},b_{i}$. So, $a_{i}\notin V_{f}^{+}$ and
$b_{i}\notin V_{f}^{-}$ for all $i\in\\{3,\ldots,n-1\\}$. Also $a_{2}\notin
V_{f}^{+}$, otherwise we have a contradiction of Lemma 4.11(1) considering
$a_{1},a_{2},c_{1}$. Therefore, considering the horizontal edge interval of
$a_{1}$, either $a_{n-1}\in V_{f}^{+}$ or $a_{n}\in V_{f}^{+}$. We assume that
$a_{n-1}\in V_{f}^{+}$ and we prove that we obtain our result with a) in point
(2). Observe that, for $i\in\\{3,\ldots,n-2\\}$, $a_{i}\in V_{f}^{-}$ if and
only if $b_{i}\in V_{f}^{+}$. In particular, $a_{i}\notin V_{f}^{-}$ and
$b_{i}\notin V_{f}^{+}$ for all $i\in\\{3,\ldots,n-1\\}$, otherwise we obtain
a contradiction of Lemma 4.11(1) considering $a_{i},b_{i},a_{n-1}$. We have
also $a_{2}\notin V_{f}^{-}$, otherwise (since $c_{2}\notin V_{f}^{+}$) the
only possibility is $b_{2}\in V_{f}^{+}$, obtaining the same contradiction
considering $a_{2},b_{2},a_{n-1}$. As consequence, we obtain also $b_{2}\notin
V_{f}^{+}$, otherwise the only possibility is $c_{2}\in V_{f}^{-}$, that is a
contradiction of Lemma 4.11(1) considering $c_{1},a_{1},c_{2}$. Moreover
$b_{1}\notin V_{f}^{+}$, otherwise we obtain the same contradiction
considering $b_{1},a_{1},a_{n-1}$. Considering the vertical edge interval of
$a_{n-1}$, then $b_{n-1}\in V_{f}^{-}$ or $d_{1}\in V_{f}^{-}$. In the first
case we obtain the same contradiction as before, considering
$a_{n-1},b_{n-1},a_{1}$. So we obtain $d_{1}\in V_{f}^{-}$. As consequence,
$a_{n}\notin V_{f}^{-}$, otherwise we obtain the same contradiction
considering $a_{n-1},d_{1},a_{n}$. If $a_{n}\in V_{f}^{+}$, denoting with
$\mathcal{I}$ the horizontal edge interval of $a_{1}$, we have
$|V_{f}^{+}\cap\mathcal{I}|=2$ and $|V_{f}^{-}\cap\mathcal{I}|=1$, that is a
contradiction. So, $a_{n}\notin V_{f}^{+}$. Finally, $b_{n-1}\notin V_{f}^{-}$
otherwise we have the same contradiction considering $a_{n-1},b_{n-1},a_{1}$,
and $b_{n}\notin V_{f}^{+}$, otherwise the only possibility is $d_{2}\in
V_{f}^{+}$ (since $a_{n}\notin V_{f}^{+}$) and we have the same contradiction
considering $d_{2},d_{1},a_{n-1}$. So we obtained our result with a) in point
(2). With the same argument we obtain our result with b) in point (2) assuming
that $a_{n}\in V_{f}^{+}$ (instead of $a_{n-1}\in V_{f}^{+}$).
If we suppose at the beginning that condition a2) holds, we obtain our result
with the same argument. By our proof, it is also easy to argue that the same
result holds exchanging $V_{f}^{+}$ with $V_{f}^{-}$ and vice versa. ∎

######  Lemma 4.15.

Let $\mathcal{P}$ be a closed path polyomino and $f\in J_{P}$ an irredundant
binomial with $\deg f\geq 3$. Consider a subpolyomino having a configuration
as in Figure 13.

Figure 13. Configuration examined in Lemma 13.

Suppose that $b_{1},b_{2}\notin V_{f}^{-}$ and that one of the following
occurs:

  * a1)

$a_{1}\in V_{f}^{+}$, $c_{1}\in V_{f}^{-}$ and $a_{2}\notin V_{f}^{+}$.

  * a2)

$a_{2}\in V_{f}^{+}$, $c_{2}\in V_{f}^{-}$ and $a_{1}\notin V_{f}^{+}$.

Then the following hold:

  1. (1)

$b_{i},c_{i}\notin V_{f}^{+}\cup V_{f}^{-}$ for all $i\in\\{3,\ldots,n-1\\}$.

  2. (2)

If a1) occurs then $c_{2}\notin V_{f}^{+}\cup V_{f}^{-}$, while if a2) occurs
then $c_{1}\notin V_{f}^{+}\cup V_{f}^{-}$.

  3. (3)

$b_{n},h\notin V_{f}^{-}$ and $b_{1},b_{2}\notin V_{f}^{+}$.

Moreover, if $\mathcal{P}$ has a zig-zag walk and in addition $g\in
V_{f}^{-}$, then it holds that $\ell\in V_{f}^{+}$, $b_{n}\notin V_{f}^{+}$,
$c_{n},q,r,t\notin V_{f}^{+}\cup V_{f}^{-}$.
The same result holds replacing $V_{f}^{+}$ with $V_{f}^{-}$ and $V_{f}^{-}$
with $V_{f}^{+}$ in all previous occurrences.

###### Proof.

Suppose a1) holds. Observe that, for $i\in\\{3,\ldots,n-1\\}$, $c_{i}\in
V_{f}^{+}$ if and only if $b_{i}\in V_{f}^{-}$. In particular, if $c_{i}\in
V_{f}^{+}$ and $b_{i}\in V_{f}^{-}$ we obtain a contradiction with Lemma
4.11(1) considering $c_{1},c_{i},b_{i}$. So, $c_{i}\notin V_{f}^{+}$ and
$b_{i}\notin V_{f}^{-}$ for all $i\in\\{3,\ldots,n-1\\}$. Moreover
$c_{2}\notin V_{f}^{+}$, otherwise we obtain the same contradiction
considering $c_{1},a_{1},c_{2}$. It follows that, since $c_{1}\in V_{f}^{-}$,
considering the horizontal edge interval of $c_{1}$ there exists a vertex
$X\in V_{f}^{+}$ such that $X\in\\{c_{n},\ell\\}$. Therefore, for
$i\in\\{3,\ldots,n-1\\}$, observing that $c_{i}\in V_{f}^{-}$ if and only if
$b_{i}\in V_{f}^{+}$, with the same argument above we obtain $c_{i}\notin
V_{f}^{-}$ and $b_{i}\notin V_{f}^{+}$ for all $i\in\\{3,\ldots,n-1\\}$,
otherwise we contradict Lemma 4.11(1) considering $c_{i},b_{i},X$. Moreover,
considering the vertical edge interval of $c_{2}$, if $c_{2}\in V_{f}^{-}$
then $b_{2}\in V_{f}^{+}$ (since $a_{2}\notin V_{f}^{+}$), that is a
contradiction with Lemma 4.11(1) considering $c_{2},b_{2},X$. Therefore, at
this point, we proved points (1) and (2).

Suppose $b_{2}\in V_{f}^{+}$, then $a_{2}\in V_{f}^{-}$ (since $c_{2}\notin
V_{f}^{-}$), but this is a contradiction with Lemma 4.11(1) considering
$a_{1},a_{2},b_{2}$. Moreover also $b_{1}\notin V_{f}^{+}$, otherwise we
obtain the same contradiction considering $b_{1},c_{1},X$. Now suppose
$b_{n}\in V_{f}^{-}$. If $c_{n}\in V_{f}^{+}$ we have the same contradiction
considering $b_{n},c_{n},c_{1}$. Hence, the only possibility is $q\in
V_{f}^{+}$ but in such case we have also $r\in V_{f}^{-}$ or $t\in V_{f}^{-}$,
obtaining in each case a contradiction with Lemma 4.11(1). So $b_{n}\notin
V_{f}^{-}$. Suppose $h\in V_{f}^{-}$. If $\ell\in V_{f}^{+}$ we have the same
contradiction with Lemma 4.11(1) considering $h,\ell,c_{1}$. It follows that
the only possibility, for $V_{f}^{+}$ in the horizontal edge interval of
$c_{1}$, is $c_{n}\in V_{f}^{+}$. To avoid the same contradiction, in the
vertical edge interval of $c_{n}$ we have to consider $q\in V_{f}^{-}$, and as
consequence in the horizontal edge interval of $q$ we have to consider $t\in
V_{f}^{+}$. But in this case we contradict Lemma 4.11(2), considering the
vertices $c_{n},t\in V_{f}^{+}$ and $h,c_{1}\in V_{f}^{-}$. So, we obtain
$h\notin V_{f}^{-}$. At this point, we proved also (3).

Suppose now $g\in V_{f}^{-}$ and $\mathcal{P}$ has a zig-zag walk. By the
above arguments, we showed that either $c_{n}\in V_{f}^{+}$ or $\ell\in
V_{f}^{+}$. If $c_{n}\in V_{f}^{+}$, then either $b_{n}\in V_{f}^{-}$ or $q\in
V_{f}^{-}$. In the first case we obtain a contradiction with Lemma 4.11(1)
considering $b_{n},c_{n},c_{1}$, in the second case we have either $r\in
V_{f}^{+}$ or $t\in V_{f}^{+}$, obtaining again the same contradiction
considering respectively $c_{n},q,r$ and $q,t,g$. So $c_{n}\notin V_{f}^{+}$
and $\ell\in V_{f}^{+}$. Moreover also $c_{n}\notin V_{f}^{-}$, otherwise
either $b_{n}\in V_{f}^{+}$ or $q\in V_{f}^{+}$, obtaining in each case the
same contradiction together with $\ell\in V_{f}^{+}$. As consequence also
$b_{n}\notin V_{f}^{+}$, otherwise the only possibility is $q\in V_{f}^{-}$,
obtaining the same contradiction considering $b_{n},q,g$. At this point we
have $c_{n},b_{n}\notin V_{f}^{+}\cup V_{f}^{-}$, and considering their
vertical edge interval we obtain also $q\notin V_{f}^{+}\cup V_{f}^{-}$.
Finally if we suppose the assumption $r,t\notin V_{f}^{+}\cup V_{f}^{-}$ is
not true, at this point the only possibility is either $r\in V_{f}^{-}$ and
$t\in V_{f}^{+}$, or $r\in V_{f}^{+}$ and $t\in V_{f}^{-}$. In the first case
we obtain a contradiction with Lemma 4.11(1) considering $r,t,g$. In the
second case, consider that there exists a vertex $X$ belonging to the same
vertical edge interval of $t$ such that $X\in V_{f}^{+}$, and since
$\mathcal{P}$ has a zig-zag walk then $X,t,r$ are vertices of an inner 2-minor
of $\mathcal{P}$, obtaining again a contradiction with Lemma 4.11(1).
We have just proved our results in the case condition a1) holds. It is easy to
understand that the same arguments hold also in case condition a2) is
satisfied. By our proof, it is also easy to argue that the same result holds
exchanging $V_{f}^{+}$ with $V_{f}^{-}$ and vice versa. ∎

By the following result we can state that the ideal ${\mathfrak{p}}_{1}$,
defined in the previous section, is a prime ideal.

######  Theorem 4.16.

Let $\mathcal{P}$ be a closed path polyomino having zig-zag walks and let
$Z_{\mathcal{P}}$ be the ideal generated by the binomials $f_{\mathcal{W}}$,
for every zig-zag walk $\mathcal{W}$ in $\mathcal{P}$. Then
$I_{\mathcal{P}}+Z_{\mathcal{P}}=J_{\mathcal{P}}$.

###### Proof.

$\supseteq)$ Let $f=f^{+}-f^{-}\in J_{\mathcal{P}}$ be an irredundant
binomial. Since $(J_{\mathcal{P}})_{2}=I_{\mathcal{P}}$, if $\deg f=2$ then
$f\in I_{\mathcal{P}}$. So, suppose $\deg f\geq 3$. We prove that $f\in
Z_{\mathcal{P}}$ and in particular $f=\pm f_{\mathcal{W}}$ for some zig-zag
walk $\mathcal{W}$ in $\mathcal{P}$.
Consider the unique sub-polyomino of $\mathcal{P}$ having configuration as in
Figure 8a and such that the points $a,b,c,d,e$ involve the variable $w$ in the
toric ring that define $J_{\mathcal{P}}$. We first show that $a,b,d,m\notin
V_{f}^{+}\cup V_{f}^{-}$.
If $b\in V_{f}^{+}$ then $a\in V_{f}^{-}$ or $m\in V_{f}^{-}$. In the first
case, there exists a vertex $F\in V_{f}^{+}$ in the same vertical edge
interval of $a$ and since $\mathcal{P}$ has zig-zag walks then $a,F,b$ are the
corners of an inner 2-minor, that is a contradiction of Lemma 4.11(1). In the
second case, there exists a vertex $F\in V_{f}^{+}$ in the same vertical edge
interval of $m$ such that $m,F,b$ are the corners of an inner 2-minor,
obtaining the same contradiction. So $b\notin V_{f}^{+}$ and with the same
argument we can prove $b\notin V_{f}^{-}$. Suppose that $a\in V_{f}^{+}$,
since $b\notin V_{f}^{+}\cup V_{f}^{-}$ then $m\in V_{f}^{-}$ and in order to
avoid contradictions the only possibility is $e_{1}\in V_{f}^{+}$ and
$c,e\notin V_{f}^{-}$. It follows that there exists a vertex $F\in V_{f}^{-}$
in the same horizontal edge interval of $e$ (with $F\neq e,e_{1}$), and since
$\mathcal{P}$ has zig-zag walks then $d,e,F$ are the corners of an inner
2-minor. Moreover, considering $a\in V_{f}^{+}$, then
$w\in\mathrm{supp}(\psi(f^{+}))=\mathrm{supp}(\psi(f^{-}))$, and since
$b,c,e\notin V_{f}^{-}$ the only possibility is $d\in V_{f}^{-}$. So, we
obtain a contradiction of Lemma 4.11(2), considering the vertices
$a,d,e_{1},m,F$. Therefore $a\notin V_{f}^{+}$ and with the same argument we
can prove $a\notin V_{f}^{-}$. Suppose $d\in V_{f}^{+}$, then
$w\in\mathrm{supp}(\psi(f^{+}))=\mathrm{supp}(\psi(f^{-}))$ and since
$a,b\notin V_{f}^{+}\cup V_{f}^{-}$ then $c\in V_{f}^{-}$ or $e\in V_{f}^{-}$.
Since $\mathcal{P}$ has zig-zag walks, in the first case there exist e vertex
$F\in V_{f}^{+}$ such that $F$ belong in the same vertical edge interval of
$c$ and $c,F,d$ are corners of the same inner 2-minor, while in the second
case $F$ belong in the same horizontal edge interval of $e$ and $F,e,d$ are
corners of the same inner 2-minor. In both cases we obtain a contradiction of
Lemma 4.11(1). Therefore $d\notin V_{f}^{+}$ and with the same argument we can
prove $d\notin V_{f}^{-}$. Finally, since $a,b\notin V_{f}^{+}\cup V_{f}^{-}$,
considering their horizontal edge interval we obtain $m\notin V_{f}^{+}\cup
V_{f}^{-}$.So, $(V_{f}^{+}\cup
V_{f}^{-})\cap\\{a,b,c,d,e,m\\}\subseteq\\{c,e\\}$.
Suppose that $(V_{f}^{+}\cup V_{f}^{-})\cap\\{a,b,c,d,e,m\\}=\emptyset$. Then
all variables of $f$ are contained in $S^{\prime}=K[x_{v}\mid v\in
V(\mathcal{P})\setminus\\{a,b,c,d,e,m\\}]$. Consider $\psi^{\prime}$ the
restriction of $\psi$ on $S^{\prime}$ and $\mathcal{P}^{\prime}$ the polyomino
obtained by $\mathcal{P}$ removing the cells in $[c,m]$ and $[e,m]$. It is
verified that $f\in\ker\psi^{\prime}$ and,by the results contained in [23], we
have $\ker\psi^{\prime}=I_{\mathcal{P}^{\prime}}$. So $f$ is an irredundant
binomial in $I_{\mathcal{P}^{\prime}}$, but this means that $\deg f=2$, that
contradicts our assumption. So, $(V_{f}^{+}\cup
V_{f}^{-})\cap\\{c,e\\}\neq\emptyset$ and it is not possible $|(V_{f}^{+}\cup
V_{f}^{-})\cap\\{c,e\\}|=1$, since
$w\in\mathrm{supp}(\psi(f^{+}))=\mathrm{supp}(\psi(f^{-}))$. Then, the only
possibility is either $c\in V_{f}^{+}$ and $e\in V_{f}^{-}$ or $c\in
V_{f}^{-}$ and $e\in V_{f}^{+}$. Without loss of generality we can suppose
that the first possibility occurs. Now we continue our argument following the
structure of $\mathcal{P}$.

Since $\mathcal{P}$ has zig-zag walks then (continuing on the “east” part) we
can continue considering a subpolyomino having the same shape as one in Figure
14. Let $\mathcal{V}_{1}$ be the set of vertices of such a subpolyomino. In
particular, considering Figure 14a and Figure 14b, we set $i$ varying on
$\\{1,\ldots,n\\}$, where $b_{1}^{(1)}$ belong to the same horizontal edge
interval of $e$ and $n$ is the index such that $a_{n}^{(1)}$ belongs to the
same horizontal edge interval of $c_{1}$.

(a)

(b)

(c)

Figure 14. Possible configurations of $\mathcal{C}_{k}$, up to reflections or
rotations.

By Lemma 11, Lemma 12 and Lemma 13 observe that, in the horizontal edge
interval of $c$, all vertices different from $c$ and $e_{1}$ does not belong
to $V^{+}\cup V^{-}$. So, since $c\in V_{f}^{+}$, it is verified that
$e_{1}\in V_{f}^{-}$. Therefore, by the same lemmas we can argue that

$V^{+}\cap\mathcal{V}_{1}=\\{c,c_{1},b_{i}^{(j_{i})}\mid
i\in\\{1,\ldots,n\\},\mbox{and either $j_{i}=1$ or $j_{i}=2$}\\},$
$V^{-}\cap\mathcal{V}_{1}=\\{e,e_{1},a_{i}^{(j_{i})}\mid
i\in\\{1,\ldots,n\\},\mbox{and either $j_{i}=1$ or $j_{i}=2$}\\}.$

Observe that $\mathcal{P}$ can be built as union of the configurations in
Figure 14. In particular we set
$\mathcal{P}=\bigcup_{k=1}^{t}\mathcal{C}_{k}$, where $\mathcal{C}_{k}$, for
all $k\in\\{1,\ldots,t\\}$, is a configuration as in Figure 14a or Figure 14b
or Figure 14c. Let $\mathcal{V}_{k}$ the set of the vertices of
$\mathcal{C}_{k}$ that are highlighted with a black point in the picture.
Denote with $c_{k},e_{k},a_{i,k}^{(j_{i})},b_{i,k}^{(j_{i})}$, for
$j_{i}\in\\{1,2\\}$, the vertices in $\mathcal{C}_{k}$ corresponding to the
vertices $c_{1},e_{1},a_{i}^{(j_{i})},b_{i}^{(j_{i})}$ in the picture, and let
$n_{k}$ be the index such that $a_{n_{k},k}^{(1)}$ belong to the same
horizontal edge interval of $c_{k}$. Observe that, since $\mathcal{P}$ is a
closed path, $c_{t}=c$ and $e_{t}=e$. Moreover, starting from $c_{1}\in
V_{f}^{+}$ and $e_{1}\in V_{f}^{-}$, that we proved before, and considering
the vertices in $\mathcal{C}_{1}$ not belonging to $V_{f}^{+}\cup V_{f}^{-}$
(as also we have shown before), by Lemma 11, Lemma 12, Lemma 13 and using the
same arguments we can obtain that, for all $k\in\\{2,\ldots,t\\}$:

$V^{+}\cap\mathcal{V}_{k}=\\{c_{k-1},c_{k},b_{i,k}^{(j_{i})}\mid
i\in\\{1,\ldots,n_{k}\\},\mbox{and either $j_{i}=1$ or $j_{i}=2$}\\},$
$V^{-}\cap\mathcal{V}_{k}=\\{e_{k-1},e_{k},a_{i,k}^{(j_{i})}\mid
i\in\\{1,\ldots,n_{k}\\},\mbox{and either $j_{i}=1$ or $j_{i}=2$}\\}.$

This means that:

$f=\prod_{k=1}^{t}\left(x_{c_{k}}\prod_{i=1}^{n_{k}}x_{b_{i,k}^{(j_{i})}}\right)-\prod_{k=1}^{t}\left(x_{e_{k}}\prod_{i=1}^{n_{k}}x_{a_{i,k}^{(j_{i})}}\right)$

One can see that (following also the mentioned figures) that the variable
involved in $f$ allow to obtain a zig-zag walk $\mathcal{W}$ of $\mathcal{P}$
and that the structure of $f$ corresponds exactly to $f=f_{\mathcal{W}}$ or
$f=-f_{\mathcal{W}}$ (the symbol + or - depends on the convention on $z_{i}$
and $u_{i}$, with reference to the definition of $f_{\mathcal{W}}$ mentioned
in the preliminaries of this paper). So, we can conclude $f\in
Z_{\mathcal{P}}$.
$\subseteq)$ We know that $I_{\mathcal{P}}\subseteq J_{\mathcal{P}}$. Let
$\mathcal{W}$ be a zig-zag walk and
$f_{\mathcal{W}}=\prod_{i=1}^{\ell}x_{z_{i}}-\prod_{i=1}^{\ell}x_{u_{i}}$ the
related binomial. Set $f^{+}=\prod_{i=1}^{\ell}x_{z_{i}}$ and
$f^{-}=\prod_{i=1}^{\ell}x_{u_{i}}$, we prove that $\psi(f^{+})=\psi(f^{-})$.
Let $j\in\\{1,\ldots,\ell\\}$, $x_{z_{j}}\in\mathrm{supp}(f^{+})$ and $V_{j}$
and $H_{j}$ be respectively the vertical and horizontal edge interval of
$z_{j}$, so $v_{j},h_{j}\in\mathrm{supp}(\psi(f^{+}))$. We prove that
$v_{j},h_{j}\in\mathrm{supp}(\psi(f^{-}))$. Consider that $\mathcal{P}$ can be
built as union of the configurations in Figure 14. Using the same construction
of the previous part of the proof, we consider
$\mathcal{P}=\bigcup_{k=1}^{t}\mathcal{C}_{k}$ where $\mathcal{C}_{k}$, for
all $k\in\\{1,\ldots,t\\}$, is a configuration as in Figure 14a, Figure 14b or
Figure 14c, and denote with $c_{k},e_{k},a_{i,k}^{(j_{i})},b_{i,k}^{(j_{i})}$,
for $j_{i}\in\\{1,2\\}$, the vertices in $\mathcal{C}_{k}$ corresponding to
the vertices $c_{1},e_{1},a_{i}^{(j_{i})},b_{i}^{(j_{i})}$. Let $n_{k}$ be the
index such that $a_{n_{k},k}^{(1)}$ belong to the same horizontal edge
interval of $c_{k}$. It is easy to see that
$z_{j}\in\\{c_{k},e_{k},a_{i,k}^{(j_{i})},b_{i,k}^{(j_{i})}\\}$ for some
$k\in\\{1,\ldots,t\\}$. In each case we obtain that
$v_{j},h_{j}\in\mathrm{supp}(\psi(f^{-}))$, as we explain in the following:

  * •

if $z_{j}=c_{k}$, we can consider $a_{n_{k},k}^{(j_{i})},e_{k+1}\in f^{-}$,
for $j_{i}=1$ or $j_{i}=2$, with either $a_{n_{k},k}^{(j_{i})}\in V_{i}$ and
$e_{k+1}\in H_{j}$ or $a_{n_{k},k}^{(j_{i})}\in H_{j}$ and $e_{k+1}\in V_{j}$;

  * •

if $z_{j}=e_{k}$, we can consider $b_{1,k}^{(j_{i})},c_{k-1}\in f^{-}$, for
$j_{i}=1$ or $j_{i}=2$, with either $b_{1,k}^{(j_{i})}\in V_{j}$ and
$c_{k-1}\in H_{j}$ or $b_{1,k}^{(j_{i})}\in H_{j}$ and $c_{k-1}\in V_{j}$;

  * •

if $z_{j}=a_{i,k}^{(j_{i})}$ with $i\neq n_{k}$, we can consider
$b_{i,k}^{(j_{i})},b_{i+1,k}^{(j_{i+1})}\in f^{-}$ with either
$b_{i,k}^{(j_{i})}\in V_{j}$ and $b_{i+1,k}^{(j_{i+1})}\in H_{j}$ or
$b_{i,k}^{(j_{i})}\in H_{j}$ and $b_{i+1,k}^{(j_{i+1})}\in V_{j}$;

  * •

if $z_{j}=a_{n_{k},k}^{(j_{i})}$ we can consider
$b_{n_{k},k}^{(j_{i})},c_{k}\in f^{-}$ with either $b_{n_{k},k}^{(j_{i})}\in
V_{j}$ and $c_{k}\in H_{j}$ or $b_{n_{k},k}^{(j_{i})}\in H_{j}$ and $c_{k}\in
V_{j}$;

  * •

if $z_{j}=b_{i,k}^{(j_{i})}$ with $i\neq 1$, we can consider
$a_{i,k}^{(j_{i})},a_{i-1,k}^{(j_{i-1})}\in f^{-}$ with either
$a_{i,k}^{(j_{i})}\in V_{j}$ and $a_{i-1,k}^{(j_{i-1})}\in H_{j}$ or
$a_{i,k}^{(j_{i})}\in H_{j}$ and $a_{i-1,k}^{(j_{i-1})}\in V_{j}$;

  * •

if $z_{j}=b_{1,k}^{(j_{i})}$ we can consider $a_{1,k}^{(j_{i})},e_{k-1}\in
f^{-}$ with either $a_{1,k}^{(j_{i})}\in V_{j}$ and $e_{k-1}\in H_{j}$ or
$a_{1,k}^{(j_{i})}\in H_{j}$ and $e_{k_{1}}\in V_{j}$.

Finally , with reference to Figure 8a we easily obtain that either
$x_{c}\in\mathrm{supp}(f^{+})$ and $x_{e}\in\mathrm{supp}(f^{-})$ or
$x_{c}\in\mathrm{supp}(f^{-})$ and $x_{e}\in\mathrm{supp}(f^{+})$. So we can
conclude that $\psi(f^{+})=w\prod_{i=1}^{\ell}v_{i}h_{i}=\psi(f^{-})$, that is
$f_{\mathcal{W}}\in J_{\mathcal{P}}$. ∎

So we have proved that the ideal $\mathfrak{p}_{1}$, defined in the previous
section, is prime. By the following general result we can also argue that
$\mathrm{height}(\mathfrak{p}_{1})=|\mathcal{P}|$

######  Proposition 4.17.

Let $\mathcal{P}$ be a collection of cells having zig-zag walks and let
$Z_{\mathcal{P}}$ be the ideal generated by the binomials $f_{\mathcal{W}}$,
for every zig-zag walk $\mathcal{W}$ in $\mathcal{P}$. Denoted
$J=I_{\mathcal{P}}+Z_{\mathcal{P}}$, then
$\mathrm{height}(J)\leq|\mathcal{P}|$. Moreover if $J$ is unmixed or
$\mathrm{height}(I_{\mathcal{P}})=|\mathcal{P}|$ then equality holds.

###### Proof.

Let $V_{J}$ be the $\mathbb{Q}$-vector space generated by the set
$\\{\mathbf{v}-\mathbf{w}\in\mathbb{Q}^{n}\mid
x^{\mathbf{v}}-x^{\mathbf{w}}\in J\\}$. By [13, Theorem 1.1] then
$\mathrm{height}(J)\leq\dim_{\mathbb{Q}}V_{J}$, and if $J$ is unmixed equality
holds. We prove that $\dim_{\mathbb{Q}}V_{J}=|\mathcal{P}|$. Observe that
$V_{J}$ is a subspace of $\mathbb{Q}^{|V(\mathcal{P})|}$. For $a\in
V(\mathcal{P})$, we denote by $\mathbf{v}_{a}$ the vector in
$\mathbb{Q}^{|V(\mathcal{P})|}$ whose $a$-th component is 1, while its other
components are 0 (in particular, the set $\\{\mathbf{v}_{a}\mid a\in
V(\mathcal{P})\\}$ is the canonical basis of $\mathbb{Q}^{|V(\mathcal{P})|}$).
Moreover if $I=[a,b]$ is an inner interval of $\mathcal{P}$, with diagonal
corners $a,b$ and anti-diagonal corners $c,d$, we denote
$\mathbf{v}_{I}=\mathbf{v}_{a}+\mathbf{v}_{b}-\mathbf{v}_{c}-\mathbf{v}_{d}$.
Let $\mathcal{B}=\\{\mathbf{v}_{C}\mid C\ \mbox{is a cell of}\
\mathcal{P}\\}$, by the proof of [13, Theorem 3.1] we know that the vectors in
$\mathcal{B}$ are linearly independent and if $I$ is an inner interval of
$\mathcal{P}$ then $\mathbf{v}_{I}\in\langle\mathcal{B}\rangle$. Moreover, if
$f_{\mathcal{W}}=\prod_{i=1}^{l}x_{z_{i}}-\prod_{i=1}^{l}x_{u_{i}}$ is the
binomial associated to a zig-zag walk $\mathcal{W}:I_{1},\ldots,I_{l}$, it is
not difficult to check that
$\sum_{i=1}^{r}\mathbf{v}_{z_{i}}-\sum_{i=1}^{r}\mathbf{v}_{u_{i}}=\sum_{i=1}^{r}(-1)^{i+1}\mathbf{v}_{I_{i}}\in\langle\mathcal{B}\rangle$.
So $V_{J}=\langle\mathcal{B}\rangle$ and in particular
$\dim_{\mathbb{Q}}V_{J}=|\mathcal{P}|$. Finally, if
$|\mathcal{P}|=\mathrm{height}(I_{\mathcal{P}})$, then
$|\mathcal{P}|=\mathrm{height}(I_{\mathcal{P}})\leq\mathrm{height}(J)\leq|\mathcal{P}|$.
Therefore $\mathrm{height}(J)=|\mathcal{P}|$. ∎

###  4.3. Final result

######  Lemma 4.18.

Let ${\mathcal{P}}$ be a non-prime closed path polyomino and let
${\mathfrak{p}}$ be a minimal prime of $I_{{\mathcal{P}}}$. If
$f_{\mathcal{W}}\notin{\mathfrak{p}}$ for some zig-zag walk ${\mathcal{W}}$ of
${\mathcal{P}}$, then ${\mathfrak{p}}={\mathfrak{p}}_{2}$, where
${\mathfrak{p}}_{2}$ is as defined in Notation 4.7.

###### Proof.

Suppose
$f_{\mathcal{W}}=\prod_{i=1}^{l}x_{z_{i}}-\prod_{i=1}^{l}x_{u_{i}}\notin{\mathfrak{p}}$
for some $\mathcal{W}$. First we show that under the labelling of Figure 8a,
$x_{m}\in{\mathfrak{p}}$. Note that
$x_{m}x_{e}-x_{b}x_{e_{1}},x_{m}x_{c}-x_{a}x_{d_{1}}\in{\mathfrak{p}}$. Since,
by Lemma 4.3 and Remark 4.6, $x_{b},x_{d_{1}}\in{\mathfrak{p}}$, we get
$x_{m}x_{e},x_{m}x_{c}\in{\mathfrak{p}}$. If $x_{m}\notin{\mathfrak{p}}$ we
obtain that $x_{e},x_{c}\in{\mathfrak{p}}$, and consequently
$f_{\mathcal{W}}\in{\mathfrak{p}}$ by (3) of Remark 4.6, which is a
contradiction. Therefore $x_{m}\in{\mathfrak{p}}$. Under the assumption, by
Lemma 4.3, $x_{v}\in{\mathfrak{p}}$ for all $v\in N({\mathcal{P}})$.
Therefore, we have ${\mathfrak{p}}_{2}\subseteq{\mathfrak{p}}$. Since
${\mathfrak{p}}$ is a minimal prime of $I_{\mathcal{P}}$, ${\mathfrak{p}}_{2}$
is a prime ideal (see Lemma 4.9) and $I_{\mathcal{P}}\subseteq p_{2}$ by Lemma
4.8, we get ${\mathfrak{p}}={\mathfrak{p}}_{2}$. Hence the proof. ∎

We are now ready to prove our main theorem. We recall that an ideal $I$ of
$K[x_{1},\dots,x_{n}]$ is called unmixed if all associated prime ideals of $I$
have the same height.

######  Theorem 4.19.

Let ${\mathcal{P}}$ be a non-prime closed path polyomino. Then
$I_{\mathcal{P}}={\mathfrak{p}}_{1}\cap{\mathfrak{p}}_{2}$, where
${\mathfrak{p}}_{1}$ and ${\mathfrak{p}}_{2}$ are as defined in Notation 4.7.
In particular, $I_{\mathcal{P}}$ is unmixed.

###### Proof.

First, note that ${\mathfrak{p}}_{1}$ and ${\mathfrak{p}}_{2}$ are prime
ideals by Proposition 4.9 and Theorem 4.16. Clearly,
$I_{\mathcal{P}}\subset{\mathfrak{p}}_{1}$ and by Lemma 4.8,
$I_{\mathcal{P}}\subset{\mathfrak{p}}_{2}$. Since
$\mathrm{height}({\mathfrak{p}}_{1})=\mathrm{height}({\mathfrak{p}}_{2})=\mathrm{height}(I_{\mathcal{P}})$
by Propositions 4.10, 4.9 and 4.17, ${\mathfrak{p}}_{1}$ and
${\mathfrak{p}}_{2}$ are minimal prime ideals containing $I_{\mathcal{P}}$.
Moreover, by [5] there exists a Gröbner basis of $I_{{\mathcal{P}}}$ with
square-free initial terms, so $I_{\mathcal{P}}$ is radical (see the proof of
[12, Corollary 2.2]). Therefore, every associated prime of $I_{{\mathcal{P}}}$
is minimal. So it suffices to show that ${\mathfrak{p}}_{1}$ and
${\mathfrak{p}}_{2}$ are the only minimal prime ideals containing
$I_{\mathcal{P}}$. Let ${\mathfrak{p}}$ be a minimal prime of
$I_{\mathcal{P}}$. By [3, Theorem 6.2], ${\mathcal{P}}$ contain zig-zag walks.
If there exists a zig-zag walk ${\mathcal{W}}$ of ${\mathcal{P}}$ such that
$f_{\mathcal{W}}\notin{\mathfrak{p}}$, then
${\mathfrak{p}}={\mathfrak{p}}_{2}$ by Lemma 4.18. If for all zig-zag walk
${\mathcal{W}}$ of ${\mathcal{P}}$, $f_{\mathcal{W}}\in{\mathfrak{p}}$, then
${\mathfrak{p}}_{1}\subseteq{\mathfrak{p}}$. Since ${\mathfrak{p}}_{1}$ is a
prime ideal, we get that ${\mathfrak{p}}={\mathfrak{p}}_{1}$. Hence the proof.
∎

##  5\. Insights and open questions

Let $\mathcal{P}$ be a closed path polyomino. Comparing Theorem 4.19 with
Theorem 3.13 we can observe that
$\mathfrak{p}_{1}=J_{\emptyset}=L_{\mathcal{P}}$. Moreover
$Y=N({\mathcal{P}})\cup M({\mathcal{P}})$ is an admissible set of
$\mathcal{P}$ and ${\mathfrak{p}}_{2}=J_{Y}$. In such a particular case the
polyocollection $\mathcal{P}^{(Y)}$ consists of the disjoint intervals related
to the binomials in $R(\mathcal{P})$. See for instance the closed path in
Figure 15, where the vertices in the set $Y$ are highlighted in green and
$\mathcal{P}^{(Y)}$ consists of the red intervals.

Figure 15. A polyomino $\mathcal{P}$, with the admissible set $Y$ in green and
in red the polyocollection $\mathcal{P}^{(Y)}$

Observe that not all admissible sets $X$ of $\mathcal{P}$ are related to a
minimal prime of $I_{\mathcal{P}}$. In fact, consider in Figure 16a the same
polyomino with the highlighted vertices $a$ and $b$. Then the set $\\{a,b\\}$
is an admissible set of $\mathcal{P}$, but $J_{\\{a,b\\}}$ is not a minimal
prime of $I_{\mathcal{P}}$, since $J_{\emptyset}\subsetneq J_{\\{a,b\\}}$. The
related polyocollection $\mathcal{P}^{(\\{a,b\\})}$ is pictured in Figure 16b.

(a)

(b)

Figure 16. A polyomino $\mathcal{P}$ and the polyocollection
$\mathcal{P}^{(\\{a,b\\})}$

Some questions arise from the results obtained in this work:

  * (a)

If $\mathcal{P}$ is a collection of cells it is well know that some properties
of the ideal $I_{\mathcal{P}}$ are related to the combinatorics of
$\mathcal{P}$. For instance it is well known that if $\mathcal{P}$ is simple
then $I_{\mathcal{P}}$ is a normal Cohen-Macaulay domain of dimension
$V(\mathcal{P})-|\mathcal{P}|$ (see [15, 23]). Also the Gorenstein property is
known for some classes of polyominoes (see [2, 6, 21, 22, 24]). Moreover,
referring to Figure 16, note that $I_{\mathcal{P}^{(\\{a,b\\})}}$ is not
prime. In fact, if $\mathcal{W}$ is a zig-zag walk of the polyomino in Figure
16a, it is not difficult to see that the binomial $f_{\mathcal{W}}$ is a zero
divisor of $I_{\mathcal{P}^{(\\{a,b\\})}}$ (by the same arguments explained at
the beginning of Section 4). Furthermore, $I_{\mathcal{P}^{(\\{a,b\\})}}$ is
radical, since the set of generators forms the reduced Gröbner basis with
respect to a suitable order obtained as done in [5]. In light of the previous
considerations, we wonder if some results holding for polyominoes can be
extended also for polyocollections.

  * (b)

For a collection of cells $\mathcal{P}$, it is an open question whether the
ideal $I_{\mathcal{P}}$ is always radical. If this is true, then Theorem 3.13
always provides a primary decomposition of it. Such a question can be extended
naturally for the ideal $I_{\mathcal{C}}$ related to a polyocollection.
Another property that could be investigated is the unmixedness for the ideal
related to a polyomino (and in general for a polyocollection), that we find
for closed paths.

  * (c)

In order to have a better description of the primary decomposition provided in
Theorem 3.13, we should have a better understanding of the ideal
$L_{\mathcal{C}}$ for a polyocollection $\mathcal{C}$. For a closed path
$\mathcal{P}$ we have proved that
$L_{\mathcal{P}}=I_{\mathcal{P}}+Z_{\mathcal{P}}$, where $Z_{\mathcal{P}}$ is
the ideal generated by the binomials $f_{\mathcal{W}}$ related to zig-zag
walks. Anyway we know that this is not true for all polyominoes, as shown in
[19, Example 3.8]. In general, for a polyomino $\mathcal{P}$,
$L_{\mathcal{P}}=I_{\mathcal{P}}+T_{\mathcal{P}}$ where $T_{\mathcal{P}}$ is a
binomial ideal and if $\mathcal{W}$ is a zig-zag walk then $f_{\mathcal{W}}\in
T_{\mathcal{P}}$. It would be interesting to know the (combinatorial)
structure of the generators of $T_{\mathcal{P}}$ for other classes of
polyominoes (or polyocollections) $\mathcal{P}$.

  * (d)

For a closed path $\mathcal{P}$, we find that the minimal prime ideal of
$I_{\mathcal{P}}$, different to $L_{\mathcal{P}}$, is related to an admissible
set containing the (unique) necklace of all zig-zag walks. Anyway, in general
a polyomino with zig-zag walks has different necklaces related to different
zig-zag walks. The concept of zig-zag walk can be formulated also for
polyocollections. In general, if $\mathcal{C}$ is a polyocollection, we ask if
in the primary decomposition provided in Theorem 3.13, the minimal primes of
$I_{\mathcal{C}}$ different to $L_{\mathcal{C}}$ are ideals of kind $J_{X}$
where $X$ is related in some way to a necklace of a zig-zag walk. Examples
with Macaulay2 suggest such a behavior for some polyomino ideals.

### Acknowledgments

The first and the second author wish to thank Giancarlo Rinaldo for inspiring
the studying of the primary decomposition of closed path polyominoes. All the
authors wish to thank Jürgen Herzog, who encouraged them to study the primary
decomposition for polyominoes in the “EMS Summer School of Combinatorial
Commutative Algebra” held in Gebze (Turkey) during the SCALE conference, where
the authors met. The first author want to thank also the group GNSAGA of
INDAM, that partially supported by a grant his participation to the
conference. The last author was supported by a grant of IIT Gandhinagar and
was partly supported by an Infosys Foundation fellowship.

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