# Polarization of neural codes

Katie Christensen  Department of Mathematics
University of Louisville
Louisville, KY 40292, USA<EMAIL_ADDRESS>and  Hamid
Kulosman† Department of Mathematics
University of Louisville
Louisville, KY 40292, USA<EMAIL_ADDRESS>
###### Abstract.

The neural rings and ideals as an algebraic tool for analyzing the intrinsic
structure of neural codes were introduced by C. Curto et al. in 2013. Since
then they were investigated in several papers, including the 2017 paper by
Güntürkün et al., in which the notion of polarization of neural ideals was
introduced. In this paper we extend their ideas by introducing the notions of
polarization of motifs and neural codes. We show that the notions that we
introduced have very nice properties which could allow the studying of the
intrinsic structure of neural codes of length $n$ via the square free monomial
ideals in $2n$ variables and interpreting the results back in the original
neural code ambient space.

In the last section of the paper we introduce the notions of inactive neurons,
partial neural codes, and partial motifs, as well as the notions of
polarization of these codes and motifs. We use these notions to give a new
proof of a theorem from the paper by Güntürkün et al. that we mentioned above.

###### Key words and phrases:

Neural code; Neural ideal; Canonical form; Minimal prime ideals; Motifs;
Polarization; Pseudomonomial ideals; Square free monomial ideals

###### 2010 Mathematics Subject Classification:

Primary 13B25, 13F20, 13P25; Secondary 92B20, 94B60

${\dagger}$ the corresponding author

##  1\. Introduction

The neural rings and ideals as an algebraic tool for analyzing the intrinsic
structure of neural codes were introduced by C. Curto et al. in 2013 in the
pioneering paper [3]. In order to make our paper self-contained, we will give
in this section all the definitions and facts from [3] that we are going to
use, which are related to neural codes. All other notions and facts (that we
assume are well-known) can be found either in [3], or in the standard
references [1] and [2].

######  Definition and basic facts 1.1 ([3]).

An element ${\mathbf{w}}=w_{1}\dots w_{n}$ of ${\mathbb{F}}_{2}^{n}$ is called
a word (of length $n$). A set $\mathcal{C}\subseteq{\mathbb{F}}_{2}^{n}$ is
called a neural code, shortly code (of length $n$). We also call the subsets
of ${\mathbb{F}}_{2}^{n}$ varietes in ${\mathbb{F}}_{2}^{n}$. The code
${\mathcal{D}}={\mathbb{F}}_{2}^{n}\setminus{\mathcal{C}}$ is called the
complement of the code ${\mathcal{C}}$ and is denoted by
${}^{c}{\mathcal{C}}$. We denote ${\mathbb{M}}=\\{0,1,*\\}$. We say that this
set is the set of motifs of length $1$. We define a partial order on
${\mathbb{M}}$ by declaring that $0<*$ and $1<*$. A sequence
${\mathbf{a}}=a_{1}\dots a_{n}\in{\mathbb{M}}^{n}$ is called a motif (of
length $n$).We define a partial order on the set ${\mathbb{M}}^{n}$ by
declaring that ${\mathbf{a}}\leq{\mathbf{b}}$ if $a_{i}\leq b_{i}$ for every
$i\in[n]$. In other words, ${\mathbf{a}}\leq{\mathbf{b}}$ if for each
$i\in[n]$, $b_{i}=0$ (resp. $1$) implies $a_{i}=0$ (resp. $1$). We have

${\mathbf{a}}\leq{\mathbf{b}}\;\Leftrightarrow\;V_{\mathbf{a}}\subseteq
V_{\mathbf{b}}.$

For ${\mathbf{a}}\in{\mathbb{M}}^{n}$, the subset $V_{\mathbf{a}}$ of
${\mathbb{F}}_{2}^{n}$ consisiting of all the words ${\mathbf{w}}$ obtained by
replacing the stars of ${\mathbf{a}}$ by elements of ${\mathbb{F}}_{2}$ is
called the variety of ${\mathbf{a}}$.

For a code $\mathcal{C}\subseteq{\mathbb{F}}_{2}^{n}$, a motif ${\mathbf{a}}$
of length $n$ is called a motif of $\mathcal{C}$ if
$V_{\mathbf{a}}\subseteq\mathcal{C}$. The set of all motifs of $\mathcal{C}$
is denoted by $\mathrm{Mot}(\mathcal{C})$. A motif
${\mathbf{a}}\in\mathrm{Mot}(\mathcal{C})$ is called a maximal motif of
$\mathcal{C}$ if for any motif ${\mathbf{b}}\in\mathrm{Mot}(\mathcal{C})$,
${\mathbf{a}}\leq{\mathbf{b}}$ implies ${\mathbf{a}}={\mathbf{b}}$. The set of
all maximal motifs of $\mathcal{C}$ is denoted by
$\mathrm{MaxMot}(\mathcal{C})$. For any
${\mathbf{a}}\in\mathrm{Mot}(\mathcal{C})$ there is a
${\mathbf{b}}\in\mathrm{MaxMot}(\mathcal{C})$ such that
${\mathbf{a}}\leq{\mathbf{b}}$. We have ${\mathcal{C}}=\emptyset$ if and only
if $\mathrm{MaxMot}(\mathcal{C})=\emptyset$. Moreover, for any two codes
${\mathcal{C}}^{1}$ and ${\mathcal{C}}^{2}$,

${\mathcal{C}}^{1}={\mathcal{C}}^{2}\Leftrightarrow\mathrm{MaxMot}({\mathcal{C}}^{1})=\mathrm{MaxMot}({\mathcal{C}}^{2}).$

######  Remark 1.2 ([3, pages 1593 and 1594]).

We have

$\mathcal{C}=\cup\,\\{V_{\mathbf{a}}\;:\;{\mathbf{a}}\in\mathrm{MaxMot}(\mathcal{C})\\},$

however it can happen that for a proper subset $M$ of
$\mathrm{MaxMot}(\mathcal{C})$ we still have

$\mathcal{C}=\cup\,\\{V_{\mathbf{a}}\;:\;{\mathbf{a}}\in M\\}.$

For example, consider the neural code
$\mathcal{C}=\\{000,001,011,111\\}\subseteq{\mathbb{F}}_{2}^{3}$. Then
$\mathrm{MaxMot}(\mathcal{C})=\\{00*,0\\!*\\!1,*11\\}$, however
$\mathcal{C}=V_{00*}\cup V_{*11}$.

######  Definition 1.3 ([2], [3]).

For a variety $V\subseteq{\mathbb{F}}_{2}^{n}$ we define the ideal of $V$,
$\mathcal{I}(V)\subseteq{\mathbb{F}}_{2}[X_{1},\dots,X_{n}]$, in the following
way:

$\mathcal{I}(V)=\\{f\in{\mathbb{F}}_{2}[X_{1},\dots,X_{n}]\;:\>f({\mathbf{w}})=0\text{
for all }{\mathbf{w}}\in V\\}.$

(Note that for any variety $V$ in ${\mathbb{F}}_{2}^{n}$ we have
$\mathcal{I}(V)\supseteq\mathcal{B}$, where
$\mathcal{B}=(X_{1}^{2}-X_{1},\dots,X_{n}^{2}-X_{n})$ is the Boolean ideal of
${\mathbb{F}}_{2}[X_{1},\dots,X_{n}]$. Moreover, for
$V\subseteq{\mathbb{F}}_{2}^{n}$ we have $\mathcal{I}(V)=\mathcal{B}$ if and
only if $V={\mathbb{F}}_{2}^{n}$.)

For an ideal $I\subseteq{\mathbb{F}}_{2}[X_{1},\dots,X_{n}]$ we define the
variety of $I$, $\mathcal{V}(I)\subseteq{\mathbb{F}}_{2}^{n}$, in the
following way:

$\mathcal{V}(I)=\\{{\mathbf{w}}\in{\mathbb{F}}_{2}^{n}\;:\>f({\mathbf{w}})=0\text{
for all }f\in I\\}.$

######  Theorem 1.4 ([3], [4]).

For every variety $V\subseteq{\mathbb{F}}_{2}^{n}$ we have

$\mathcal{V}(\mathcal{I}(V))=V.$

For every ideal $I\subseteq{\mathbb{F}}_{2}[X_{1},\dots,X_{n}]$ we have

$\mathcal{I}(\mathcal{V}(I))=\sqrt{I}=I+\mathcal{B}.$

The second formula in the previous theorem is called the Hilbert’s
Nullstellensatz for ${\mathbb{F}}_{2}$.

######  Definition 1.5 ([3], [4]).

For a motif ${\mathbf{a}}\in{\mathbb{M}}^{n}$ we define the Lagrange
polynomial of ${\mathbf{a}}$,
$L_{\mathbf{a}}\in{\mathbb{F}}_{2}[X_{1},\dots,X_{n}]$, in the following way:

$L_{\mathbf{a}}=\prod_{a_{i}=1}X_{i}\prod_{a_{j}=0}(1-X_{j}).$

Note that for any word ${\mathbf{w}}\in{\mathbb{F}}_{2}^{n}$,
$L_{\mathbf{a}}({\mathbf{w}})=1$ if and only if ${\mathbf{w}}\in
V_{\mathbf{a}}$ (i.e., $L_{\mathbf{a}}({\mathbf{w}})=0$ if and only if
${\mathbf{w}}\notin V_{\mathbf{a}}$).

######  Definition 1.6 ([3, page 1582]).

For a neural code ${\mathcal{C}}\subseteq{\mathbb{F}}_{2}^{n}$ we define the
neural ideal of ${\mathcal{C}}$,
$J_{\mathcal{C}}\subseteq{\mathbb{F}}_{2}[X_{1},\dots,X_{n}]$, in the
following way:

$J_{\mathcal{C}}=(\\{L_{\mathbf{w}}\;:\;{\mathbf{w}}\in\,^{c}{\mathcal{C}}\\}).$

######  Proposition 1.7 ([3, Lemma 3.2]).

For a neural code ${\mathcal{C}}\subseteq{\mathbb{F}}_{2}^{n}$ we have:

$\displaystyle\mathcal{V}(J_{\mathcal{C}})$ $\displaystyle={\mathcal{C}},$
$\displaystyle\mathcal{I}({\mathcal{C}})$
$\displaystyle=J_{\mathcal{C}}+\mathcal{B}.$

######  Definition 1.8 ([3, page 1585]).

A polynomial $f\in{\mathbb{F}}_{2}[X_{1},\dots,X_{n}]$ is called a pseudo-
monomial if it has the form

$f=\prod_{i\in\sigma}X_{i}\,\prod_{j\in\tau}(1-X_{j})$

for some $\sigma,\tau\subseteq[n]=\\{1,\dots,n\\}$ with
$\sigma\cap\tau=\emptyset$.

An ideal $I\subseteq{\mathbb{F}}_{2}[X_{1},\dots,X_{n}]$ is called a pseudo-
monomial ideal if $I$ can be generated by a finite set of pseudo-monomials.

######  Definition 1.9 ([3, page 1585]).

Let $I$ be an ideal in ${\mathbb{F}}_{2}[X_{1},\dots,X_{n}]$ and $f\in I$ a
pseudo-monomial. We say that $f$ is a minimal pseudo-monomial of $I$ if there
does not exist another pseudo-monomial $g\in I$ such that $\deg(g)<\deg(f)$
and $g\mid f$ in $F_{2}[X_{1},\dots,X_{n}]$.

######  Definition 1.10 ([3, page 1585]).

Let $I$ be a pseudo-monomial ideal in $F_{2}[X_{1},\dots,X_{n}]$. We call the
(finite) set $CF(I)$, consisting of all minimal pseudo-monomials of $I$, the
canonical form of $I$.

######  Remark 1.11 ([3, page 1585]).

Clearly, for any pseudo-monomial ideal $I$ of $F_{2}[X_{1},\dots,X_{n}]$ ,
$CF(I)$ is unique and $I=(CF(I))$. On the other hand, $CF(I)$ is not
necessarily a minimal generating set of $I$. For example, consider the ideal
$I=(X_{1}(1-X_{2}),X_{2}(1-X_{3}))$. This ideal contains a third minimal
pseudo-monomial:
$X_{1}(1-X_{3})=(1-X_{3})\cdot[X_{1}(1-X_{2})]+X_{1}\cdot[X_{2}(1-X_{3})]$, so
that $CF(I)=\\{(X_{1}(1-X_{2}),X_{2}(1-X_{3}),X_{1}(1-X_{3})\\}$, which is not
a minimal generating set of $I$.

######  Proposition 1.12 ([3, Proposition 4.5]).

Let ${\mathcal{C}}\subseteq{\mathbb{F}}_{2}^{n}$ be a code in
${\mathbb{F}}_{2}^{n}$ and ${\mathcal{D}}$ its complement. Let

$\mathrm{MaxMot}({\mathcal{C}})=\\{{\mathbf{a}}^{1},\dots,{\mathbf{a}}^{l}\\}.$

Then

$\displaystyle\mathrm{MaxMot}({\mathcal{D}})=\,$
$\displaystyle\\{{\mathbf{b}}=b_{1}\dots b_{n}\;:\;$ $\displaystyle[\,(\forall
b_{i}\neq\ast)(\exists{\mathbf{a}}^{j})\;b_{i}=\overline{{\mathbf{a}}^{j}_{i}}\,]\;\;\mathrm{and}$
$\displaystyle[\,(\forall{\mathbf{a}}^{j}\neq\ast\dots\ast)(\exists
b_{i}\neq\ast)\;b_{i}=\overline{{\mathbf{a}}^{j}_{i}}\,]\;\;\mathrm{and}$
$\displaystyle[\,\text{${\mathbf{b}}$ is maximal with respect to these two
properties}\,]\\}.$

In particular, if $\mathrm{MaxMot}({\mathcal{C}})=\\{\ast\dots\ast\\}$, then
$\mathrm{MaxMot}({\mathcal{D}})=\emptyset$.

###### Proof.

The proposition follows from Proposition 4.5 and Corollary 5.5 from [3]. ∎

######  Proposition 1.13 ([3, Lemma 5.7]).

Let ${\mathcal{C}}\subseteq{\mathbb{F}}_{2}^{n}$ be a neural code and
$J_{\mathcal{C}}$ the neural ideal of ${\mathcal{C}}$. Then

$CF(J_{\mathcal{C}})=\\{L_{\mathbf{a}}\;:\;{\mathbf{a}}\in\mathrm{MaxMot}(^{c}{\mathcal{C}})\\}.$

######  Remark 1.14 ([3, page 1594]).

Note that it can happen that
$J_{\mathcal{C}}=(\\{L_{\mathbf{a}}\;:\;{\mathbf{a}}\in M\\})$, where $M$ is a
proper subset of $\mathrm{MaxMot}(^{c}{\mathcal{C}})$. For example, for the
neural code $\mathcal{C}=\\{000,001,011,111\\}\subseteq{\mathbb{F}}_{2}^{3}$,
we have ${}^{c}{\mathcal{C}}=\\{100,010,110,101\\}$, so that
$\mathrm{MaxMot}(^{c}{\mathcal{C}})=\\{10*,*10,1\negthinspace*0\\}$. Hence
$CF(J_{\mathcal{C}})=\\{L_{10*},L_{*10},L_{1*0}\\}$. However,
$J_{\mathcal{C}}=(L_{10*},L_{*10})$.

######  Definition 1.15 ([3, page 1594]).

For a motif ${\mathbf{a}}\in\\{0,1,*\\}^{n}$ we define a prime ideal of
${\mathbf{a}}$,
${\mathfrak{p}}_{\mathbf{a}}\subseteq{\mathbb{F}}_{2}[X_{1},\dots,X_{n}]$, in
the following way:

${\mathfrak{p}}_{\mathbf{a}}=(\\{X_{i}\;:\;a_{i}=0\\}\cup\\{1-X_{j}\;:\;a_{j}=1\\}).$

If a prime ideal ${\mathfrak{p}}$ in ${\mathbb{F}}_{2}[X_{1},\dots,X_{n}]$ is
equal to ${\mathfrak{p}}_{\mathbf{a}}$ for some motif ${\mathbf{a}}$, we say
that ${\mathfrak{p}}$ is a motivic prime ideal.

######  Proposition 1.16 ([3, page 1594]).

Let ${\mathbf{a}},{\mathbf{b}}\in\\{0,1,*\\}^{n}$ be two motifs of length $n$.
We have:

$\displaystyle V_{\mathbf{a}}\subseteq V_{\mathbf{b}}$
$\displaystyle\Leftrightarrow{\mathfrak{p}}_{\mathbf{b}}\subseteq{\mathfrak{p}}_{\mathbf{a}},$
$\displaystyle\mathcal{I}(V_{\mathbf{a}})$
$\displaystyle\,={\mathfrak{p}}_{\mathbf{a}}+\mathcal{B},$
$\displaystyle\mathcal{V}({\mathfrak{p}}_{\mathbf{a}})$
$\displaystyle\,=V_{\mathbf{a}}.$

For an ideal $I$ in ${\mathbb{F}}_{2}[X_{1},\dots,X_{n}]$ we denote by
$\mathrm{Min}(I)$ the set of all minimal prime ideals of $I$.

######  Proposition 1.17 ([3, Lemma 5.1, Lemma 5.3 and Corollary 5.5]).

Let ${\mathcal{C}}\subseteq{\mathbb{F}}_{2}^{n}$ be a neural code and
${\mathbf{a}}\in{\mathbb{M}}^{n}$ a motif. We have:

$\displaystyle{\mathbf{a}}\in\mathrm{Mot}({\mathcal{C}})$
$\displaystyle\Leftrightarrow{\mathfrak{p}}_{\mathbf{a}}\supseteq
J_{\mathcal{C}},$ $\displaystyle{\mathbf{a}}\in\mathrm{MaxMot}({\mathcal{C}})$
$\displaystyle\Leftrightarrow{\mathfrak{p}}_{\mathbf{a}}\in\mathrm{Min}(J_{\mathcal{C}}).$

Moreover,

(1)
$\mathrm{Min}(J_{\mathcal{C}})=\\{{\mathfrak{p}}_{\mathbf{a}}\;:\;{\mathbf{a}}\in\mathrm{MaxMot}({\mathcal{C}})\\}.$

######  Proposition 1.18 ([3, Corollary 5.5]).

Let ${\mathcal{C}}\subseteq{\mathbb{F}}_{2}^{n}$ be a non-empty neural code.
Then

$J_{\mathcal{C}}=\cap\,\\{{\mathfrak{p}}_{\mathbf{a}}\;:\;{\mathbf{a}}\in\mathrm{MaxMot}({\mathcal{C}})\\}$

is the unique irredundant primary decomposition of $J_{\mathcal{C}}$.

The notions of polarization of pseudo-monomials and pseudo-monomial ideals
were introduced in 2017 in the paper [5] by Güntürkün et al.

######  Definition 1.19 ([5, page 6]).

For a pseudo-monomial

$f=\prod_{i\in\sigma}X_{i}\,\prod_{j\in\tau}(1-X_{j})\in{\mathbb{F}}_{2}[X_{1},\dots,X_{n}],$

where $\sigma,\tau$ are two disjoint subsets of $[n]$, we define its
polarization $f^{p}$ to be the square-free monomial

$f^{p}=\prod_{i\in\sigma}X_{i}\,\prod_{j\in\tau}Y_{j}\in{\mathbb{F}}_{2}[X_{1},\dots,X_{n},Y_{1},\dots,Y_{n}].$

######  Proposition 1.20 ([5, Lemma 3.1]).

Let $f,g\in{\mathbb{F}}_{2}[X_{1},\dots,X_{n}]$ be two pseudomonomials. Then

$f\mid g\;\Leftrightarrow\;f^{p}\mid g^{p}.$

######  Definition 1.21 ([5, Definition 3.3]).

Let $J$ be a pseudo-monomial ideal in ${\mathbb{F}}_{2}[X_{1},\dots,X_{n}]$
and let $CF(J)=\\{f_{1},\dots,f_{l}\\}$ be its canonical form. We define the
polarization of $J$ to be the ideal

$J^{p}=(f_{1}^{p},\dots,f_{l}^{p})\subseteq{\mathbb{F}}_{2}[X_{1},\dots,X_{n},Y_{1},\dots,Y_{n}].$

Note that then

$CF(J^{p})=\\{f_{1}^{p},\dots,f_{l}^{p}\\}.$

######  Remark 1.22 (about definitions and notation related to
${\mathbb{F}}_{2}^{2n}$ vs those related to ${\mathbb{F}}_{2}^{n}$).

Square-free monomial ideals are the ideals generated by square-free monomials
and they are easier to deal with than the pseudo-monomial ideals. The previous
two definitions show that, in order to get some conclusions about the pseudo-
monomial ideals in $n$ variables $X_{1},\dots,X_{n}$, we can consider some
related square-free monomial ideals in $2n$ variables, which, however, are not
denoted by $X_{1},\dots,X_{2n}$, but by $X_{1},\dots,X_{n},Y_{1},\dots,Y_{n}$.
Because of this difference in the notation for variables, we should be aware
that, for example, a pseudomonomial in
${\mathbb{F}}_{2}[X_{1},\dots,X_{n},Y_{1},\dots,Y_{n}]$ is a polynomial of the
form

$f=\prod_{i\in\sigma}X_{i}\,\prod_{j\in\tau}(1-X_{j})\prod_{k\in\mu}Y_{k}\,\prod_{l\in\nu}(1-Y_{l}),$

where $\sigma,\tau,\mu,\nu\subseteq[n]$, $\sigma\cap\tau=\emptyset$,
$\mu\cap\nu=\emptyset$. Similarly, we have, for example, that for a motif
${\mathbf{a}}=b_{1}\dots b_{n}\,c_{1}\dots c_{n}\in\mathrm{Mot}(2n)$, the
Lagrange polynomial $L_{\mathbf{a}}$ of ${\mathbf{a}}$ and the prime ideal
${\mathfrak{p}}_{\mathbf{a}}$ of ${\mathbf{a}}$ are respectively given in the
following way:

(2)
$L_{\mathbf{a}}=\prod_{a_{i}=1}X_{i}\,\prod_{a_{j}=0}(1-X_{j})\prod_{b_{i}=1}Y_{i}\,\prod_{b_{j}=0}(1-Y_{j}),$
(3)
${\mathfrak{p}}_{\mathbf{a}}=(\\{X_{i}\,:\,b_{i}=0\\}\cup\\{1-X_{j}\,:\,b_{j}=1\\}\cup\\{Y_{i}\,:\,c_{i}=0\\}\cup\\{1-Y_{j}\,:\,c_{j}=1\\}).$

So the definitions of these notions with respect to ${\mathbb{F}}_{2}^{2n}$
are the same as the ones with respect to ${\mathbb{F}}_{2}^{n}$, we just need
to take into account the notation for the variables. This works for other
notions as well (like, for example, minimal pseudo-monomials in an ideal, the
neural ideal of a code, the canonical form of a pseudo-monomial ideal, etc.),
while some notions (like, for example, minimal primes of an ideal, etc.) can
be given in the form that does not depend on the notation for the variables.

From now on we have the following convention: if the length of motifs and
codes is denoted by $n$, then the associated rings and ideals will always be
in $n$ variables $X_{1},\dots,X_{n}$, while for the length denoted by $2n$ the
associated rings and ideals will always be in $2n$ variables
$X_{1},\dots,X_{n},Y_{1},\dots,Y_{n}$. For lengths given by concrete numbers
it will always be clear from the context if the number is $n$ or $2n$.

##  2\. Definitions of the polarizations of motifs and codes

We would like to define the motif ${\mathbf{a}}^{p}$ which is the polarization
of a motif ${\mathbf{a}}=a_{1}\dots a_{n}\in{\mathbb{M}}^{n}$. Since for the
motifs from ${\mathbb{M}}^{n}$ we have that the Lagrange polynomials and the
prime ideals of motifs are in $n$ variables, and the polarizations of those
Lagrange polynomials and prime ideals of motifs are in $2n$ variables, it is
natural to try to define ${\mathbf{a}}^{p}$ to be an element of
${\mathbb{M}}^{2n}$. After defining the polarization of the motifs, we would
define the polarization of a neural code
${\mathcal{C}}\subseteq{\mathbb{F}}_{2}^{n}$ in the following way:

${\mathcal{C}}^{p}=\cup\,\\{V_{{\mathbf{a}}^{p}}\;:\;{\mathbf{a}}\in\mathrm{MaxMot}({\mathcal{C}})\\}\subseteq{\mathbb{F}}_{2}^{2n}.$

This would imply that

(4) $\mathrm{MaxMot}^{p}({\mathcal{C}})=\mathrm{MaxMot}({\mathcal{C}}^{p}).$

Here, and in the rest of the paper, we use the notation

$M^{p}=\\{{\mathbf{a}}^{p}\;:\;{\mathbf{a}}\in M\\}$

for any $M\subseteq{\mathbb{M}}^{n}$. We would also like that the formula

(5) $\mathrm{Min}(J_{{\mathcal{C}}^{p}})=\mathrm{Min}^{p}(J_{\mathcal{C}})$

holds. We will first try to find the formula for ${\mathbf{a}}^{p}$ in an
example. So we will assume that the relations (4), and (5) hold. In addition
to that, we will be using the relation (1):

$\mathrm{Min}(J_{\mathcal{C}})=\\{{\mathfrak{p}}_{\mathbf{a}}\;:\;{\mathbf{a}}\in\mathrm{MaxMot}({\mathcal{C}})\\}.$

######  Example 2.1.

Suppose that the relations (1), (4) and (5) hold. Let
${\mathcal{C}}=\\{10\\}\subseteq{\mathbb{F}}_{2}^{2}$. Then

$\text{MaxMot}({\mathcal{C}})=\\{10\\}.$

By (5) we have

$\mathrm{MaxMot}({\mathcal{C}}^{p})=\\{10^{p}\\}.$

By (1),

$\mathrm{Min}(J_{\mathcal{C}})=\\{(1-X_{1},X_{2})\\}={\mathfrak{p}}_{10},$

hence by (5),

$\mathrm{Min}(J_{{\mathcal{C}}^{p}})=\\{(1-X_{1},X_{2})^{p}\\}=\\{X_{2},Y_{1}\\}={\mathfrak{p}}_{*00*}.$

Hence by (1),

$\mathrm{MaxMot}({\mathcal{C}}^{p})=\\{*00*\\}=\mathrm{MaxMot}^{p}({\mathcal{C}})=\\{10^{p}\\}.$

Thus

(6) $10^{p}=*00\\!*.$

Now ${\mathcal{D}}=\,^{c}{\mathcal{C}}=\\{00,01,11\\}$, so that

$\mathrm{MaxMot}({\mathcal{D}})=\\{0*,*1\\}.$

Hence by (4),

$\mathrm{MaxMot}({\mathcal{D}}^{p})=\\{0*^{p},*1^{p}\\}.$

By (1),

$\mathrm{Min}(J_{\mathcal{D}})=\\{(X_{1}),(1-X_{2})\\},$

hence by (5),

$\mathrm{Min}(J_{{\mathcal{D}}^{p}})=\\{(X_{1}),(1-X_{2})\\}^{p}=\\{(X_{1}),(Y_{2})\\}=\\{{\mathfrak{p}}_{0***},{\mathfrak{p}}_{***0}\\}.$

Hence by (1),

$\mathrm{MaxMot}({\mathcal{D}}^{p})=\\{0\\!*\\!**,\,*\\!*\\!*0\\}=\mathrm{MaxMot}^{p}({\mathcal{D}})=\\{0*^{p},*1^{p}\\}.$

Hence we have either

(7) $0*^{p}=0\\!*\\!**,\quad*1^{p}=*\\!*\\!*0,$

or

(8) $*1^{p}=*\\!*\\!*0,\quad 0*^{p}=0\\!*\\!**.$

Now if we have

$(a_{1}a_{2})^{p}=b_{1}b_{2}|c_{1}c_{2},$

then the formula (6) would suggest that

$\displaystyle a_{1}=1$ $\displaystyle\Rightarrow b_{1}=*,\,c_{1}=0,$
$\displaystyle a_{2}=0$ $\displaystyle\Rightarrow b_{2}=0,\,c_{2}=*,$

and the formula (7) would suggest that

$\displaystyle a_{1}=0$ $\displaystyle\Rightarrow b_{1}=0,\,\,c_{1}=*,$
$\displaystyle a_{2}=*$ $\displaystyle\Rightarrow b_{2}=*,\,\,c_{2}=*.$

This would suggest that if

$(a_{1}\dots a_{n})^{p}=b_{1}\dots b_{n}|c_{1}\dots c_{n},$

then

$\displaystyle a_{i}=0$ $\displaystyle\Rightarrow b_{i}=0,\,\,c_{i}=*,$
$\displaystyle a_{i}=1$ $\displaystyle\Rightarrow b_{i}=*,\,\,c_{i}=0,$
$\displaystyle a_{i}=*$ $\displaystyle\Rightarrow b_{i}=*,\,\,c_{i}=*.$

(This, in particular, suggests that the formulas (7) are the correct way to
polarize and not the formulas (8).)

Having completed this example, based on the reasoning in it, we introduce the
following definition.

######  Definition 2.2.

Let ${\mathbf{a}}=a_{1}\dots a_{n}\in{\mathbb{M}}^{n}$. We define its
polarization

${\mathbf{a}}^{p}=b_{1}\dots b_{n}|c_{1}\dots c_{n}$

in the following way:

$\displaystyle\text{if }a_{i}=0\text{ in }{\mathbf{a}},$ $\displaystyle\text{
then }b_{i}=0,\;c_{i}=*\text{ in }{\mathbf{a}}^{p};$ $\displaystyle\text{if
}a_{i}=1\text{ in }{\mathbf{a}},$ $\displaystyle\text{ then
}b_{i}=*,\;c_{i}=0\text{ in }{\mathbf{a}}^{p};$ $\displaystyle\text{if
}a_{i}=*\text{ in }{\mathbf{a}},$ $\displaystyle\text{ then
}b_{i}=*,\;c_{i}=*\text{ in }{\mathbf{a}}^{p}.$

Schematically:

$\displaystyle\dots 0\dots\quad$ $\displaystyle\mapsto\quad\dots
0\dots|\dots*\dots$ (9) $\displaystyle\dots 1\dots\quad$
$\displaystyle\mapsto\quad\dots*\dots|\dots 0\dots$
$\displaystyle\dots*\dots\quad$
$\displaystyle\mapsto\quad\dots*\dots|\dots*\dots$

We now define the polarization of a code.

######  Definition 2.3.

For any code ${\mathcal{C}}\subseteq{\mathbb{F}}_{2}^{n}$ we define its
polarization ${\mathcal{C}}^{p}\subseteq{\mathbb{F}}_{2}^{2n}$ in the
following way:

${\mathcal{C}}^{p}=\cup\\{V_{{\mathbf{a}}^{p}}\;|\;{\mathbf{a}}\in\text{MaxMot}({\mathcal{C}})\\}.$

######  Example 2.4.

Let’s now determine ${\mathcal{C}}^{p}$ and ${\mathcal{D}}^{p}$ for
${\mathcal{C}}$ and ${\mathcal{D}}=^{c}\\!{\mathcal{C}}$ from Example 2.1:

$\displaystyle{\mathcal{C}}^{p}$ $\displaystyle=\\{10\\}^{p}$
$\displaystyle=V_{(10)^{p}}$ $\displaystyle=V_{*00*}$
$\displaystyle=\\{0000,1000,0001,1001\\},$ $\displaystyle{\mathcal{D}}^{p}=$
$\displaystyle\\{00,01,11\\}^{p}$ $\displaystyle=$ $\displaystyle
V_{(0*)^{p}}\cup V_{(*1)^{p}}$ $\displaystyle=$ $\displaystyle V_{0***}\cup
V_{***0}$ $\displaystyle=$ $\displaystyle\\{0000,0100,0010,0110,0001,0101,$
$\displaystyle\;\;0011,0111,1000,1100,1010,1110\\}.$

Note that here ${\mathcal{C}}^{p}\cap{\mathcal{D}}^{p}=\\{0000\\}$ and
${\mathcal{C}}^{p}\cup{\mathcal{D}}^{p}={\mathbb{F}}_{2}^{4}\setminus\\{1111\\}$.
In general, ${\mathcal{C}}^{p}\cap{\mathcal{D}}^{p}$ can contain several words
and the complement of ${\mathcal{C}}^{p}\cup{\mathcal{D}}^{p}$ in
${\mathbb{F}}_{2}^{2n}$ can, as well, contain several words.

######  Definition 2.5.

We say that a motif ${\mathbf{b}}\in{\mathbb{M}}^{2n}$ is a polar motif if
there is a motif ${\mathbf{a}}\in{\mathbb{M}}^{n}$ such that
${\mathbf{b}}={\mathbf{a}}^{p}$. The motif ${\mathbf{a}}$ is unique and we
then denote ${\mathbf{a}}={\mathbf{b}}^{d}$.

Note that we have

$\displaystyle{\mathbf{a}}^{pd}={\mathbf{a}}$ $\displaystyle\text{\quad for
every ${\mathbf{a}}\in{\mathbb{M}}^{n}$},$
$\displaystyle{\mathbf{b}}^{dp}={\mathbf{b}}$ $\displaystyle\text{\quad for
every polar motif ${\mathbf{b}}\in{\mathbb{M}}^{2n}$}.$

##  3\. Properties of the polarization of motifs and codes

######  Proposition 3.1.

Let ${\mathbf{a}},{\mathbf{b}}\in{\mathbb{M}}^{n}$. Then

${\mathbf{a}}\leq{\mathbf{b}}\;\Leftrightarrow\;{\mathbf{a}}^{p}\leq{\mathbf{b}}^{p}.$

###### Proof.

Suppose ${\mathbf{a}}\leq{\mathbf{b}}$. Let $i\in[n]$. If
$({\mathbf{b}}^{p})_{i}=0$, then $({\mathbf{b}}^{p})_{n+i}=*$ and $b_{i}=0$,
hence $a_{i}=0$, hence $({\mathbf{a}}^{p})_{i}=0$. If
$({\mathbf{b}}^{p})_{n+i}=0$, then $({\mathbf{b}}^{p})_{i}=*$ and $b_{i}=1$,
hence $a_{i}=1$, hence $({\mathbf{a}}^{p})_{n+i}=0$. Thus
${\mathbf{a}}^{p}\leq{\mathbf{b}}^{p}$.

Suppose ${\mathbf{a}}^{p}\leq{\mathbf{b}}^{p}$. Let $i\in[n]$. If $b_{i}=0$,
then $({\mathbf{b}}^{p})_{i}=0$, hence $({\mathbf{a}}^{p})_{i}=0$, hence
$a_{i}=0$. If $b_{i}=1$, then $({\mathbf{b}}^{p})_{n+i}=0$, hence
$({\mathbf{a}}^{p})_{n+i}=0$, hence $a_{i}=1$. Thus
${\mathbf{a}}\leq{\mathbf{b}}$. ∎

######  Corollary 3.2.

For any code ${\mathcal{C}}\subseteq{\mathbb{F}}_{2}^{n}$ we have

$\mathrm{Mot}^{p}({\mathcal{C}})\;\subseteq\;\mathrm{Mot}({\mathcal{C}}^{p}).$

###### Proof.

Let ${\mathbf{a}}\in\mathrm{Mot}({\mathcal{C}})$ and let
${\mathbf{b}}\in\mathrm{MaxMot}({\mathcal{C}})$ such that
${\mathbf{a}}\leq{\mathbf{b}}$. By Proposition 3.1,
${\mathbf{a}}^{p}\leq{\mathbf{b}}^{p}$. Since (by the definition of
${\mathcal{C}}^{p}$) ${\mathbf{b}}^{p}\in\mathrm{Mot}({\mathcal{C}}^{p})$, we
have ${\mathbf{a}}^{p}\in\mathrm{Mot}({\mathcal{C}}^{p})$. ∎

######  Theorem 3.3.

For any code ${\mathcal{C}}\subseteq{\mathbb{F}}_{2}^{n}$ we have

$\mathrm{MaxMot}({\mathcal{C}}^{p})=\mathrm{MaxMot}^{p}({\mathcal{C}}).$

###### Proof.

Claim 1. $\mathrm{MaxMot}({\mathcal{C}}^{p})\subseteq\\{0,*\\}^{2n}$.

Proof of Claim 1. Suppose to the contrary, i.e., that
${\mathbf{b}}\in\mathrm{MaxMot}({\mathcal{C}}^{p})$ has a component
$b_{\alpha}=1$ for some $\alpha\in[2n]$. Let ${\mathbf{w}}\in V_{\mathbf{b}}$.
Then $w_{\alpha}=1$. We have ${\mathbf{w}}\in V_{{\mathbf{a}}^{p}}$ for some
${\mathbf{a}}\in\mathrm{MaxMot}({\mathcal{C}})$. Then
$({\mathbf{a}}^{p})_{\alpha}=*$, hence the word ${\mathbf{w}}^{\prime}$
obtained by replacing $w_{\alpha}$ in ${\mathbf{w}}$ by $0$ is also in
$V_{{\mathbf{a}}^{p}}$, hence in ${\mathcal{C}}$. Hence the motif
${\mathbf{b}}^{\prime}$ obtained by replacing $b_{\alpha}$ by $*$ is also a
motif of ${\mathcal{C}}^{p}$, contradicting to the maximality of
${\mathbf{b}}$. Claim 1 is proved.

Claim 2. Let ${\mathbf{b}}\in\mathrm{MaxMot}({\mathcal{C}}^{p})$. Then there
is no $i\in[n]$ such that $b_{i}=b_{n+i}=0$.

Proof of Claim 2. Suppose to the contrary. Let

$\displaystyle A$ $\displaystyle=\\{j\in[n]\;:\;b_{j}=b_{n+j}=*\\},$
$\displaystyle B$ $\displaystyle=\\{j\in[n]\;:\;b_{j}=0,\,b_{n+j}=*\\},$
$\displaystyle C$ $\displaystyle=\\{j\in[n]\;:\;b_{j}=*,\,b_{n+j}=0\\},$
$\displaystyle D$ $\displaystyle=\\{j\in[n]\;:\;b_{j}=b_{n+j}=0\\}.$

Then the sets $A,B,C,D$ form a partition of $[n]$ and $i\in D$. Let
${\mathbf{w}}\in V_{\mathbf{b}}$ be defined in the following way:

$\displaystyle(\forall\,j\in A)\;$ $\displaystyle w_{j}=w_{n+j}=1;$
$\displaystyle(\forall\,j\in B)\;$ $\displaystyle w_{j}=0,\,w_{n+j}=1;$
$\displaystyle(\forall\,j\in C)\;$ $\displaystyle w_{j}=1,\,w_{n+j}=0;$
$\displaystyle(\forall\,j\in D)\;$ $\displaystyle w_{j}=w_{n+j}=0.$

Since ${\mathbf{w}}\in{\mathcal{C}}^{p}$, there is an
${\mathbf{a}}\in\mathrm{MaxMot}({\mathcal{C}})$ such that ${\mathbf{w}}\in
V_{{\mathbf{a}}^{p}}$. Since ${\mathbf{a}}^{p}$ is a polar motif, we have:

$\displaystyle(\forall\,j\in A)\;$
$\displaystyle({\mathbf{a}}^{p})_{j}=({\mathbf{a}}^{p})_{n+j}=*;$
$\displaystyle(\forall\,j\in B)\;$
$\displaystyle({\mathbf{a}}^{p})_{j}=0\text{ or
}*,\,({\mathbf{a}}^{p})_{n+j}=*;$ $\displaystyle(\forall\,j\in C)\;$
$\displaystyle({\mathbf{a}}^{p})_{j}=*,\,({\mathbf{a}}^{p})_{n+j}=0\text{ or
}*;$ $\displaystyle(\forall\,j\in D)\;$ $\displaystyle\text{at least one of
$({\mathbf{a}}^{p})_{j},\,({\mathbf{a}}^{p})_{n+j}$ is $*$, the other one is
$0$ or $*$}.$

Since $D$ contains at least one element, these relations imply
${\mathbf{a}}^{p}>{\mathbf{b}}$, contradicting to the maximality of
${\mathbf{b}}$. Claim 2 is proved.

Claim 3.
$\mathrm{MaxMot}({\mathcal{C}}^{p})\subseteq\mathrm{MaxMot}^{p}({\mathcal{C}})$.

Proof of Claim 3. Let ${\mathbf{b}}\in\mathrm{MaxMot}({\mathcal{C}}^{p})$. By
the claims 1 and 2, for each $i\in[n]$ we have one the following three cases:
$b_{i}=b_{n+i}=*$, or, $b_{i}=0$, $b_{n+i}=*$, or, $b_{i}=*$, $b_{n+i}=0$. Let
${\mathbf{w}}\in V_{\mathbf{b}}$ be a word defined in the following way: if
$b_{i}=b_{n+i}=*$, then $w_{i}=w_{n+i}=1$; if $b_{i}=0$ and $b_{n+i}=*$, then
$w_{i}=0$, $w_{n+i}=1$; if $b_{i}=*$ and $b_{n+i}=0$, then $w_{i}=1$,
$w_{n+i}=0$. This word belongs to some $V_{{\mathbf{a}}^{p}}$, where
${\mathbf{a}}\in\mathrm{MaxMot}({\mathcal{C}})$. Since ${\mathbf{a}}^{p}$ is a
polar motif, we have the following cases: when $w_{i}=w_{n+i}=1$, then
$({\mathbf{a}}^{p})_{i}=({\mathbf{a}}^{p})_{n+i}=*$; when $w_{i}=0$ and
$w_{n+i}=1$, then $({\mathbf{a}}^{p})_{i}=0$, $({\mathbf{a}}^{p})_{n+i}=*$;
when $w_{i}=1$ and $w_{n+i}=0$, then $({\mathbf{a}}^{p})_{i}=*$,
$({\mathbf{a}}^{p})_{n+i}=0$. Hence ${\mathbf{a}}^{p}\geq{\mathbf{b}}$. Since
${\mathbf{a}}^{p}\in\mathrm{Mot}({\mathcal{C}})$ and
${\mathbf{b}}\in\mathrm{MaxMot}({\mathcal{C}})$, we have
${\mathbf{b}}={\mathbf{a}}^{p}$. Claim 3 is proved.

Claim 4.
$\mathrm{MaxMot}^{p}({\mathcal{C}})\subseteq\mathrm{MaxMot}({\mathcal{C}}^{p})$.

Proof of Claim 4. Suppose to the contrary. Let
${\mathbf{a}}\in\mathrm{MaxMot}({\mathcal{C}})$ such that
${\mathbf{a}}^{p}\notin\mathrm{MaxMot}({\mathcal{C}}^{p})$. By the definition
of ${\mathcal{C}}^{p}$, ${\mathbf{a}}^{p}\in\mathrm{Mot}({\mathcal{C}}^{p})$,
hence there is a ${\mathbf{b}}\in\mathrm{MaxMot}({\mathcal{C}}^{p})$ such that
${\mathbf{a}}^{p}<{\mathbf{b}}$. By Claim 3, ${\mathbf{b}}={\mathbf{c}}^{p}$
for some ${\mathbf{c}}\in\mathrm{MaxMot}({\mathcal{C}})$. Now by Proposition
3.1, from ${\mathbf{a}}^{p}<{\mathbf{c}}^{p}$ we get
${\mathbf{a}}<{\mathbf{c}}$, which is a contradiction since both
${\mathbf{a}}$ and ${\mathbf{c}}$ are maximal motifs of ${\mathcal{C}}$. Claim
4 is proved.

Now the statement of the theorem follows from Claim 3 and Claim 4. ∎

######  Definition 3.4.

For a motif ${\mathbf{a}}=a_{1}\dots a_{n}\in{\mathbb{M}}^{n}$ we define
$\overline{{\mathbf{a}}}$ to be the motif ${\mathbf{b}}=b_{1}\dots
b_{n}\in{\mathbb{M}}^{n}$ which satisfies the following condition

$\text{for }i=1,2\dots,n,\text{ if }a_{i}\neq*,\text{ then
}b_{i}=\overline{a_{i}}=1-a_{i}.$

######  Example 3.5.

$\overline{1*01}\,=\,0*10$.

Note that for two motifs ${\mathbf{a}},{\mathbf{b}}\in{\mathbb{M}}^{n}$ we
have

(10)
${\mathbf{b}}=\overline{\overline{{\mathbf{a}}}^{p}}\;\Leftrightarrow\;{\mathbf{a}}=\overline{\overline{{\mathbf{b}}}^{d}}.$

Also, if for any code ${\mathcal{C}}\subseteq{\mathbb{F}}_{2}^{n}$ and any
$M\subseteq\mathrm{Mot}({\mathcal{C}})$ we denote
$\overline{M}=\\{\overline{{\mathbf{a}}}\;:\;{\mathbf{a}}\in M\\}$, then

(11) $\displaystyle\mathrm{Mot}(\overline{{\mathcal{C}}})$
$\displaystyle=\overline{\mathrm{Mot}({\mathcal{C}})},$ (12)
$\displaystyle\mathrm{MaxMot}(\overline{{\mathcal{C}}})$
$\displaystyle=\overline{\mathrm{MaxMot}({\mathcal{C}})}.$

######  Proposition 3.6.

For any motif ${\mathbf{a}}\in{\mathbb{M}}^{n}$ we have

$L_{\mathbf{a}}^{p}=L_{\overline{\overline{{\mathbf{a}}}^{\,p}}}.$

###### Proof.

This follows from the definitions (1.19), (2), (2.2), and (3.4). ∎

######  Example 3.7.

Let $n=4$ and let ${\mathbf{a}}=11*0\in\text{Mot}(4)$. Then

$L_{{\mathbf{a}}}=X_{1}X_{2}(1-X_{4}).$

Hence

$L_{{\mathbf{a}}}^{p}=X_{1}X_{2}Y_{4}.$

On the other side we have:

$\displaystyle\overline{\overline{{\mathbf{a}}}^{p}}$
$\displaystyle=\overline{\overline{11*0}^{p}}$
$\displaystyle=\overline{00*11^{p}}$ $\displaystyle=\overline{00*0|***0}$
$\displaystyle=11**|***1.$

Hence

$L_{\overline{\overline{{\mathbf{a}}}^{p}}}=X_{1}X_{2}Y_{4}=L_{{\mathbf{a}}}^{p}.$

######  Definition 3.8.

We say that two motifs ${\mathbf{a}},{\mathbf{b}}\in{\mathbb{M}}^{n}$ are
disjoint if there is an $i\in[n]$ such that $a_{i}=\overline{b_{i}}$.

######  Definition 3.9.

On the set ${\mathbb{M}}$ of motifs of length $1$ we introduce a commutative
operation of addition in the following way:

$\displaystyle 0+0$ $\displaystyle=0,$ $\displaystyle 0+1$ $\displaystyle=1,$
$\displaystyle 1+1$ $\displaystyle=0,$ $\displaystyle 0+*$ $\displaystyle=*,$
$\displaystyle 1+*$ $\displaystyle=*,$ $\displaystyle*+*$ $\displaystyle=*.$

The first three lines represent the arithmetic in the field
${\mathbb{F}}_{2}$, while the last three lines represent the max-arithmetic.
We then the addition in ${\mathbb{M}}^{n}$ by adding two motifs componentwise.

It is easy to verify that with this operation and the partial order that we
introduced before, ${\mathbb{M}}^{n}$ is a partially ordered monoid.

The importance of above definition lies in the fact that the sum
${\mathbf{a}}+{\mathbf{b}}$ of two motifs
${\mathbf{a}},{\mathbf{b}}\in{\mathbb{M}}^{n}$ has an $1$-component if and
only if the motifs ${\mathbf{a}}$ and ${\mathbf{b}}$ are disjoint. Thus we can
recognize the disjointness of two motifs algebraically by considering their
sum.

######  Proposition 3.10.

Let ${\mathbf{a}}\in\mathrm{Mot}({\mathcal{C}})$ for some code
${\mathcal{C}}\subseteq{\mathbb{F}}_{2}^{n}$ and
${\mathbf{b}}\in{\mathbb{M}}^{n}$. Then
${\mathbf{b}}\in\mathrm{Mot}(^{c}{\mathcal{C}})$ if and only if ${\mathbf{b}}$
is disjoint with ${\mathbf{a}}$. Moreover, the maximal motifs of
${}^{c}{\mathcal{C}}$ are the motifs ${\mathbf{b}}$ that are maximal among the
motifs from ${\mathbb{M}}^{n}$ that are disjoint with all the maximal motifs
of ${\mathcal{C}}$.

###### Proof.

Easy to see. ∎

######  Proposition 3.11.

Let ${\mathbf{a}},{\mathbf{b}},{\mathbf{b}}^{\prime}\in{\mathbb{M}}^{n}$. If
${\mathbf{a}}+{\mathbf{b}}$ has an $1$-component and
${\mathbf{b}}^{\prime}\leq{\mathbf{b}}$, then
${\mathbf{a}}+{\mathbf{b}}^{\prime}$ has an $1$-component.

In particular, if ${\mathcal{C}}$ is a code in ${\mathbb{M}}^{n}$, the maximal
motifs of ${}^{c}{\mathcal{C}}$ are the maximal elements
${\mathbf{b}}\in{\mathbb{M}}^{n}$ such that each ${\mathbf{a}}+{\mathbf{b}}$
(${\mathbf{a}}\in\mathrm{MaxMot}({\mathcal{C}})$) has an $1$-component.

###### Proof.

Easy to see. ∎

######  Corollary 3.12.

Let ${\mathcal{C}}\subseteq{\mathbb{F}}_{2}^{n}$ be a code in
${\mathbb{F}}_{2}^{n}$. If
$\textbf{b}\in\mathrm{MaxMot}(^{c}({\mathcal{C}}^{p}))$, then every $b_{i}$
which is different than $\ast$ is equal to $1$.

###### Proof.

The statement follows from the previous proposition as each $0$ could be
replaced by a $*$, which would result in a strictly bigger motif disjoint with
all maximal motifs of ${\mathcal{C}}^{p}$. ∎

######  Proposition 3.13.

The motifs ${\mathbf{a}}$ and ${\mathbf{b}}$ from ${\mathbb{M}}^{n}$ are
disjoint if and only if the motifs ${\mathbf{a}}^{p}$ and
$\overline{\overline{{\mathbf{b}}}^{p}}$ from ${\mathbb{M}}^{2n}$ are
disjoint.

###### Proof.

$\Rightarrow)$ Suppose that ${\mathbf{a}}$ and ${\mathbf{b}}$ are disjoint. We
first consider the case $a_{i}=1$, $b_{i}=0$ for some $i\in[n]$. Then
$({\mathbf{a}}^{p})_{i}=*$ and $({\mathbf{a}}^{p})_{n+i}=0$, while
$(\overline{\overline{{\mathbf{b}}}^{p}})_{i}=*$ and
$(\overline{\overline{{\mathbf{b}}}^{p}})_{n+i}=1$. Hence ${\mathbf{a}}^{p}$
and $\overline{\overline{{\mathbf{b}}}^{p}}$ are disjoint. The case $a_{i}=0$,
$b_{i}=1$ for some $i\in[n]$ is similar.

$\Leftarrow)$ Suppose that ${\mathbf{a}}^{p}$ and
$\overline{\overline{{\mathbf{b}}}^{p}}$ are disjoint. We first consider the
case $({\mathbf{a}}^{p})_{i}=0$,
$(\overline{\overline{{\mathbf{b}}}^{p}})_{i}=1$ for some $i\in[n]$. Then
$a_{i}=0$ and $(\overline{{\mathbf{b}}}^{p})_{i}=0$, hence
$(\overline{{\mathbf{b}}})_{i}=0$. Hence $b_{i}=1$, so that ${\mathbf{a}}$ and
${\mathbf{b}}$ are disjoint. The case $({\mathbf{a}}^{p})_{i}=1$,
$(\overline{\overline{{\mathbf{b}}}^{p}})_{i}=0$ for some $i\in[n]$ is
similar. ∎

######  Proposition 3.14.

Let ${\mathcal{C}},{\mathcal{D}}$ be two codes in ${\mathbb{F}}_{2}^{n}$.
Then:

${\mathcal{D}}\,\subseteq\,^{c}{\mathcal{C}}\;\Leftrightarrow\;\overline{\overline{{\mathcal{D}}}^{p}}\,\subseteq\,^{c}({\mathcal{C}}^{p}).$

###### Proof.

The next equivalences follow from Proposition 3.10, Proposition 3.13, Theorem
3.3, and Proposition 3.10, respectively.

$\displaystyle{\mathcal{D}}\subseteq\,^{c}{\mathcal{C}}$
$\displaystyle\Leftrightarrow(\forall\,{\mathbf{a}}\in\mathrm{MaxMot}({\mathcal{C}}))(\forall\,{\mathbf{b}}\in\mathrm{MaxMot}({\mathcal{D}}))\text{
${\mathbf{a}}$ and ${\mathbf{b}}$ are disjoint}$
$\displaystyle\Leftrightarrow(\forall\,{\mathbf{a}}\in\mathrm{MaxMot}({\mathcal{C}}))(\forall\,{\mathbf{b}}\in\mathrm{MaxMot}({\mathcal{D}}))\text{
${\mathbf{a}}^{p}$ and $\overline{\overline{{\mathbf{b}}}^{p}}$ are disjoint}$
$\displaystyle\Leftrightarrow(\forall\,{\mathbf{c}}\in\mathrm{MaxMot}({\mathcal{C}}^{p}))(\forall\,{\mathbf{d}}\in\mathrm{MaxMot}(\overline{\overline{{\mathcal{D}}}^{p}}))\text{
${\mathbf{c}}$ and ${\mathbf{d}}$ are disjoint}$
$\displaystyle\Leftrightarrow\overline{\overline{{\mathcal{D}}}^{p}}\,\subseteq\,^{c}({\mathcal{C}}^{p}).$

∎

######  Remark 3.15.

Note that in the previous proposition the equality on the left hand side is
not equivalent with the equality on the right hand side, as we are going to
see in Example 3.20.

######  Corollary 3.16.

Let ${\mathcal{D}}=\,^{c}{\mathcal{C}}$. Then

${\mathcal{C}}^{p}\,\subseteq\,^{{}^{{}^{c}}}\overline{\overline{{\mathcal{D}}}^{p}}.$

###### Proof.

Follows immediately from the previous proposition. ∎

The reason for giving the next definition and using the terminology introduced
in it will become clear later, after Theorem 3.19 and Example 3.20.

######  Definition 3.17.

Let ${\mathcal{C}}$ be a code in ${\mathbb{F}}_{2}^{n}$ and let
${\mathcal{D}}$ be its complement. We call the code ${\mathcal{C}}^{[p]}$,
defined by

(13)
${\mathcal{C}}^{[p]}=\,^{{}^{{}^{c}}}\overline{\overline{{\mathcal{D}}}^{p}},$

the formal polarization of the code ${\mathcal{C}}$.

######  Proposition 3.18.

Let ${\mathcal{C}}$ be a code in ${\mathbb{F}}_{2}^{n}$ and ${\mathcal{D}}$
its complement. We have:

(14) $\displaystyle\mathrm{MaxMot}({\mathcal{C}}^{p})$
$\displaystyle\subseteq\mathrm{MaxMot}({\mathcal{C}}^{[p]}),$ (15)
$\displaystyle\mathrm{MaxMot}(^{c}({\mathcal{C}}^{[p]}))$
$\displaystyle=\overline{\overline{\mathrm{MaxMot}({\mathcal{D}})}^{\,p}}\subseteq\mathrm{MaxMot}(^{c}({\mathcal{C}}^{p})).$

###### Proof.

By Theorem 3.3 and the formula (13), this is equivalent with showing that

$(\forall\,{\mathbf{a}}\in\mathrm{MaxMot}({\mathcal{C}}))(\forall\,{\mathbf{b}}\in\mathrm{MaxMot}({\mathcal{D}}))\text{
${\mathbf{a}}^{p}$ and $\overline{\overline{{\mathbf{b}}}^{p}}$ are
disjoint},$

which is true by Proposition 3.13.

We now show (15). Let
${\mathbf{d}}\in\mathrm{MaxMot}(^{c}({\mathcal{C}}^{[p]}))=\mathrm{MaxMot}(\overline{\overline{{\mathcal{D}}}^{p}})$.
Then ${\mathbf{d}}=\overline{\overline{{\mathbf{b}}}^{p}}$ for some
${\mathbf{b}}\in\mathrm{MaxMot}({\mathcal{D}})$. Let
${\mathbf{e}}\in\mathrm{MaxMot}(^{c}({\mathcal{C}}^{p}))$ such that
${\mathbf{d}}\leq{\mathbf{e}}$, i.e.
$\overline{\overline{{\mathbf{b}}}^{p}}\leq{\mathbf{e}}$. Hence
$\overline{{\mathbf{b}}}^{p}\leq\overline{{\mathbf{e}}}$. Since ${\mathbf{e}}$
is bigger than or equal to a polar motif, ${\mathbf{e}}$ is a polar motif too,
so ${\mathbf{e}}=\overline{{\mathbf{f}}}^{p}$ for some motif ${\mathbf{f}}$.
Hence $\overline{{\mathbf{b}}}^{p}\leq\overline{{\mathbf{f}}}^{p}$, so that
$\overline{\overline{{\mathbf{b}}}^{p}}\leq\overline{\overline{{\mathbf{f}}}^{p}}={\mathbf{e}}$.
Hence, since ${\mathbf{e}}$ is disjoint with all the maximal motifs of
${\mathcal{C}}^{p}$, then by Proposition 3.13, ${\mathbf{f}}$ is disjoint with
all the maximal motifs of ${\mathcal{C}}$ and ${\mathbf{f}}\geq{\mathbf{b}}$,
where ${\mathbf{b}}$ is one of the maximal motifs among the motifs that are
disjoint with all maximal motifs of ${\mathcal{C}}$. Hence
${\mathbf{f}}={\mathbf{b}}$, so that ${\mathbf{d}}={\mathbf{e}}$. ∎

######  Theorem 3.19.

Let ${\mathcal{C}}$ be a code in ${\mathbb{F}}_{2}^{n}$. We have:

$CF(J_{\mathcal{C}}^{p})=CF^{p}(J_{\mathcal{C}})=CF(J_{{\mathcal{C}}^{[p]}})\subseteq
CF(J_{{\mathcal{C}}^{p}}).$

###### Proof.

Let $CF(J_{\mathcal{C}})=\\{f_{1},\dots,f_{k}\\}$. By definition,
$J_{\mathcal{C}}^{p}=(f_{1}^{p},\dots,f_{k}^{p})$. Here $f_{1}^{p}$, …,
$f_{k}^{p}$ are square-free monomials. By [6, Corrolary 1.10], the set
$\\{f_{1}^{p},\dots,f_{k}^{p}\\}$ contains a minimal susbest $S$ (with respect
to inclusion) which generates $J_{\mathcal{C}}^{p}$. By [6, Corrolary 1.8], if
$f_{i}^{p}\notin S$, then $f_{i}^{p}\mid f_{j}^{p}$ for some $f_{j}^{p}\in S$.
Then by Proposition 1.20, $f_{i}\mid f_{j}$, a contradiction. Thus
$S=\\{f_{1}^{p},\dots,f_{k}^{p}\\}$. Hence by [6, Proposition 1.11],
$CF(J_{\mathcal{C}}^{p})=\\{f_{1}^{p},\dots,f_{k}^{p}\\}=CF^{p}(J_{\mathcal{C}})$.

Let ${\mathcal{D}}$ be the complement of ${\mathcal{C}}$. We have:

$\displaystyle CF^{p}(J_{\mathcal{C}})$
$\displaystyle=\\{L_{\mathbf{a}}^{p}\;:\;{\mathbf{a}}\in\text{MaxMot}({\mathcal{D}})\\}\hskip
59.75095pt\text{(by Proposition \ref{CF_max_mot_compl})}$
$\displaystyle=\\{L_{\overline{\overline{{\mathbf{a}}}^{p}}}\;:\;{\mathbf{a}}\in\text{MaxMot}({\mathcal{D}})\\}\hskip
56.9055pt\text{(by Proposition \ref{pol_of_LP})}$
$\displaystyle=\\{L_{\mathbf{b}}\;:\;\overline{\overline{{\mathbf{b}}}^{d}}\in\text{MaxMot}({\mathcal{D}})\\}\hskip
54.06006pt\text{(by (\ref{p_d}))}$
$\displaystyle=\\{L_{\mathbf{b}}\;:\;\overline{{\mathbf{b}}}^{d}\in\text{MaxMot}(\overline{{\mathcal{D}}})\\}$
$\displaystyle=\\{L_{\mathbf{b}}\;:\;\overline{{\mathbf{b}}}\in\text{MaxMot}(\overline{{\mathcal{D}}}^{p})\\}$
$\displaystyle=\\{L_{\mathbf{b}}\;:\;{\mathbf{b}}\in\text{MaxMot}(\overline{\overline{{\mathcal{D}}}^{p}})\\}$
$\displaystyle=\,CF(J_{{\mathcal{C}}^{[p]}}).\hskip 132.30513pt\text{(by
Proposition \ref{CF_max_mot_compl})}$

Finally, the inclusion in the statement of the theorem follows from
Proposition 3.18 and Proposition 1.13. ∎

######  Example 3.20.

Consider the neural codes

${\mathcal{C}}=\\{000,100,110,011\\}\text{ and
}{\mathcal{D}}=^{c}{\mathcal{C}}=\\{001,010,101,111\\}$

in ${\mathbb{F}}_{2}^{3}$. We have

$\mathrm{MaxMot}({\mathcal{C}})=\\{*00,1\\!*\\!0,011\\}\text{ and
}\mathrm{MaxMot}(DC)=\\{*01,1\\!*\\!1,111\\}.$

Then by Theorem 3.3,

(16) $\displaystyle\mathrm{MaxMot}({\mathcal{C}}^{p})$
$\displaystyle=\\{*00\\!*\\!**,\,*\\!*\\!00\\!*\\!*,\,0\\!*\\!*\\!*\\!00\\},$
(17) $\displaystyle\mathrm{MaxMot}(\overline{\overline{{\mathcal{D}}}^{p}})$
$\displaystyle=\\{*\\!*\\!1\\!*\\!1*,\,1\\!*\\!1\\!*\\!**,\,*1\\!*\\!1\\!*\\!1\\}.$

By Proposition 3.14 we have
$\overline{\overline{{\mathcal{D}}}^{p}}\,\subseteq\,^{c}({\mathcal{C}}^{p})$.
From (17) we have by Proposition 1.13,

$CF(J_{{\mathcal{C}}^{[p]}})=\\{X_{3}Y_{2},\,X_{1}X_{3},\,X_{2}Y_{1}Y_{3}\\}=CF^{p}(J_{\mathcal{C}}).$

By Proposition 3.10, a motif ${\mathbf{b}}\in{\mathbb{M}}^{6}$ is a maximal
motif of ${}^{c}({\mathcal{C}}^{p})$ if and only if it is a maximal motif from
${\mathbb{M}}^{6}$ disjoint with all the maximal motifs of
${\mathcal{C}}^{p}$. The sets

$A_{1}=\\{2,3\\},\;\;A_{2}=\\{3,4\\},\;\;A_{3}=\\{1,5,6\\},$

are the sets of coordinates which the maximal motifs
${\mathbf{a}}^{1}=*00\\!*\\!**,\,{\mathbf{a}}^{2}=*\\!*\\!00\\!*\\!*,\,\text{
and }{\mathbf{a}}^{3}=0\\!*\\!*\\!*\\!00$ of ${\mathcal{C}}^{p}$ have zeros
at, respectively. To get a set $B$ of coordinates where a motif
${\mathbf{b}}\in\mathrm{MaxMot}(^{c}({\mathcal{C}}^{p}))$ has ones, we need to
take one element from each of the sets $A_{1},A_{2},A_{3}$, and then, out of
all sets $B$ obtained in that way ($2\times 2\times 3=12$ of them) select the
minimal ones with respect to inclusion. In that way we get the sets
$B_{1}=\\{2,4,1\\},\;B_{2}=\\{2,4,5\\},\;B_{3}=\\{2,4,6\\},\;B_{4}=\\{3,1\\},\;B_{5}=\\{3,5\\},$
and $B_{6}=\\{3,6\\}$. For each of these sets $B_{i}$ we get an element
${\mathbf{b}}^{i}\in\mathrm{MaxMot}(^{c}({\mathcal{C}}^{p}))$ by putting ones
at all the coordinates of $B_{i}$ and stars at all other coordinates. Thus

$\displaystyle\mathrm{MaxMot}(^{c}({\mathcal{C}}^{p}))$
$\displaystyle=\\{*\\!*\\!1\\!*\\!1*,\;\;1\\!*\\!1\\!*\\!**,\;\;*1\\!*\\!1\\!*\\!1,\;\;*\\!*\\!1\\!*\\!*1,$
$\displaystyle\quad\;\;\,*\\!1\\!*\\!11*,\;\;11\\!*\\!1\\!*\\!*\\}.$

Hence by Proposition 1.13,

$CF(J_{{\mathcal{C}}^{p}})=\\{X_{3}Y_{2},\;X_{1}X_{3},\;X_{2}Y_{1}Y_{3},\;X_{3}Y_{3},\;X_{2}Y_{1}Y_{2},\;X_{1}X_{2}Y_{1}\\}.$

In particular,

${\mathcal{C}}^{p}\;\subset\;{\mathcal{C}}^{[p]}.$

(One can check that ${\mathcal{C}}^{p}$ has $29$ words, while
${\mathcal{C}}^{[p]}$ has $35$ words.)

######  Example 3.21.

Consider again the code ${\mathcal{C}}=\\{10\\}\subseteq{\mathbb{F}}_{2}^{n}$
and its complement ${\mathcal{D}}=\\{00,01,11\\}$ from Examples 2.1 and 2.4.
We have:

$\displaystyle\mathrm{MaxMot}({\mathcal{C}})$ $\displaystyle=\\{10\\},$
$\displaystyle\mathrm{MaxMot}({\mathcal{D}})$ $\displaystyle=\\{0*,\;*1\\},$
$\displaystyle CF(J_{\mathcal{C}})$ $\displaystyle=\\{1-X_{1},\;X_{2}\\},$
$\displaystyle CF(J_{{\mathcal{C}}^{[p]}})$
$\displaystyle=CF^{p}(J_{\mathcal{C}})=\\{X_{2},\;Y_{1}\\},$
$\displaystyle{\mathcal{C}}^{p}$ $\displaystyle=\\{0000,1000,0001,1001\\},$
$\displaystyle\mathrm{MaxMot}(^{c}({\mathcal{C}}^{p}))$
$\displaystyle=\\{*1\\!*\\!*,\;*\\!*\\!1*\\},$ $\displaystyle
CF(J_{{\mathcal{C}}^{p}})$ $\displaystyle=\\{X_{2},\;Y_{1}\\}.$

Thus in this example ${\mathcal{C}}^{p}={\mathcal{C}}^{[p]}$.

######  Definition 3.22.

The prime ideals
${\mathfrak{p}}\subseteq{\mathbb{F}}_{2}[X_{1},\dots,X_{n},Y_{1},\dots,Y_{n}]$
such that ${\mathfrak{p}}={\mathfrak{p}}_{{\mathbf{a}}^{p}}$ for some
${\mathbf{a}}\in{\mathbb{M}}^{n}$ are called polar motivic primes.

For polar motivic primes we have the following formula:

(18) ${\mathfrak{p}}_{{\mathbf{a}}^{p}}={\mathfrak{p}}_{\mathbf{a}}^{p}.$

Indeed, if ${\mathbf{a}}=a_{1}\dots a_{n}\in\mathrm{MaxMot}({\mathcal{C}})$
and ${\mathbf{a}}^{p}=b_{1}\dots b_{n}c_{1}\dots c_{n}$, then

$\displaystyle{\mathfrak{p}}_{{\mathbf{a}}^{p}}$
$\displaystyle=(\\{X_{i}\;:\;b_{i}=0\\}\cup\\{Y_{j}\;:\;c_{j}=0\\})$
$\displaystyle=(\\{X_{i}\;|\;a_{i}=0\\}\cup\\{Y_{j}\;:\;a_{j}=1\\})$
$\displaystyle={\mathfrak{p}}_{\mathbf{a}}^{p}.$

######  Theorem 3.23.

For any code ${\mathcal{C}}\subseteq{\mathbb{F}}_{2}^{n}$ we have:

$\mathrm{Min}(J_{{\mathcal{C}}^{p}})=\mathrm{Min}^{p}(J_{\mathcal{C}})\subseteq\mathrm{Min}(J_{{\mathcal{C}}^{[p]}}).$

###### Proof.

We have

$\displaystyle\mathrm{Min}(J_{{\mathcal{C}}^{p}})$
$\displaystyle=\\{{\mathfrak{p}}_{\mathbf{d}}\;:\;\mathbf{d}\in\mathrm{MaxMot}({\mathcal{C}}^{p})\\}\hskip
38.69576pt\text{(by Proposition \ref{Min_max_motifs})}$
$\displaystyle=\\{{\mathfrak{p}}_{{\mathbf{a}}^{p}}\;:\;{\mathbf{a}}\in\textrm{MaxMot}({\mathcal{C}})\\}\hskip
40.68723pt\text{(by Theorem \ref{maximal_motifs})}$
$\displaystyle=\\{{\mathfrak{p}}_{{\mathbf{a}}^{p}}\;:\;{\mathfrak{p}}_{\mathbf{a}}\in\mathrm{Min}(J_{\mathcal{C}})\\}\hskip
54.9136pt\text{(by Proposition \ref{Min_max_motifs})}$
$\displaystyle=\\{{\mathfrak{p}}_{\mathbf{a}}^{p}\;:\;{\mathfrak{p}}_{\mathbf{a}}\in\mathrm{Min}(J_{\mathcal{C}})\\}.\hskip
56.9055pt\text{(by the formula (\ref{formula20}))}$

Hence

$\mathrm{Min}(J_{{\mathcal{C}}^{p}})=\mathrm{Min}^{p}(J_{\mathcal{C}}).$

The inclusion part of the statement follows from (14) and the relation (1)
from Proposition 1.17. ∎

######  Example 3.24.

We continue Example 3.20. By (17) we have

$\mathrm{MaxMot}(^{c}({\mathcal{C}}^{[p]}))=\\{*\\!*\\!1\\!*\\!1*,\,1\\!*\\!1\\!*\\!**,\,*1\\!*\\!1\\!*\\!1\\}.$

Now using the same technique as in Example 3.20 (for finding
$\mathrm{MaxMot}(^{c}({\mathcal{C}}^{p}))$ given
$\mathrm{MaxMot}({\mathcal{C}}^{p})$) we find here that

$\displaystyle\mathrm{MaxMot}({\mathcal{C}}^{[p]})$
$\displaystyle=\\{*00\\!*\\!**,\;\;*\\!*\\!00\\!*\\!*,\;\;*\\!*\\!0\\!*\\!*0,\;\;00\\!*\\!*0*,$
$\displaystyle\quad\;\;\;\;0\\!*\\!*00*,\;\;0\\!*\\!*\\!*\\!00\\}.$

Hence by Proposition 1.17 we have:

$\displaystyle\mathrm{Min}(J_{{\mathcal{C}}^{p}})$
$\displaystyle=\\{(X_{2},X_{3}),\;(X_{3},Y_{1}),\;(X_{1},Y_{2},Y_{3})\\}=\mathrm{Min}^{p}(J_{\mathcal{C}}),$
$\displaystyle\mathrm{Min}(J_{{\mathcal{C}}^{[p]}})$
$\displaystyle=\\{(X_{2},X_{3}),\;(X_{3},Y_{1}),\;(X_{1},Y_{2},Y_{3}),$
$\displaystyle\hskip
73.97733pt(X_{3},Y_{3}),\;(X_{1},X_{2},Y_{2}),\;(X_{1},Y_{1},Y_{2})\\}.$

The minimal prime ideals of $J_{{\mathcal{C}}}^{p}$ (i.e.,
$J_{{\mathcal{C}}^{[p]}}$) were also calculated in [5, Example 5.4] in a
different way.

Note that among the minimal primes of $J_{{\mathcal{C}}^{[p]}}$ we have, in
addition to all the minimal primes of $J_{{\mathcal{C}}^{p}}$, three non-polar
minimal primes, namely ${\mathfrak{p}}_{**0**0}=(X_{3},Y_{3})$,
${\mathfrak{p}}_{00**0*}=(X_{1},X_{2},Y_{2})$, and
$p_{0**00*}=(X_{1},Y_{1},Y_{2})$. A natural question to ask is the following
one: if for an ${\mathbf{a}}\in{\mathbb{M}}^{2n}$ we have
${\mathfrak{p}}_{\mathbf{a}}\in\mathrm{Min}(J_{{\mathcal{C}}^{[p]}})$, how is
then the motif ${\mathbf{a}}$ related to ${\mathcal{C}}$? A statement related
to this question is given in the next section in Theorem 4.5, which is [5,
Theorem 5.1]. We will give a different proof of this theorem.

######  Theorem 3.25.

For any code ${\mathcal{C}}\subseteq{\mathbb{F}}_{2}^{n}$, the ideal
$J_{{\mathcal{C}}^{p}}$ has the unique irredundant primary decomposition and
it is obtained by polarizing the prime ideals from the unique irredundant
primary decomposition of $J_{\mathcal{C}}$.

###### Proof.

By [3, Corollary 5.5], the ideals $J_{\mathcal{C}}$ and
$J_{{\mathcal{C}}^{p}}$ have the unique irredundant primary decompositions

$\displaystyle J_{\mathcal{C}}$
$\displaystyle=\cap\\{{\mathfrak{p}}_{\mathbf{a}}\;:\;{\mathbf{a}}\in\mathrm{MaxMot}({\mathcal{C}})\\},$
$\displaystyle J_{\mathcal{C}}$
$\displaystyle=\cap\\{{\mathfrak{p}}_{\mathbf{b}}\;:\;{\mathbf{b}}\in\mathrm{MaxMot}({\mathcal{C}}^{p})\\}.$

Hence the statement follows from Theorem 3.3 and the formula (18). ∎

##  4\. Partial motifs

######  Definition 4.1.

We denote ${\mathbb{PM}}=\\{0,1,*,{\textunderscore}\\}$. We say that this set
is the set of partial motifs of length $1$. We define a partial order on
${\mathbb{PM}}$ by declaring that $0<*$ and $1<*$. Note that
${\textunderscore}$ is comparable only with itself (the same holds for $0$ and
$1$). We define a partial order on the set ${\mathbb{PM}}^{n}$ by declaring
that ${\mathbf{a}}\leq{\mathbf{b}}$ if $a_{i}\leq b_{i}$ for every $i\in[n]$.
A partial motif (of length $n$) is an element of ${\mathbb{PM}}^{n}$. A
partial word (of length $n$) is an element of
${\mathbb{PW}}^{n}=\\{0,1,{\textunderscore}\\}^{n}$. The neurons $i\in[n]$ for
which $w_{i}={\textunderscore}$ are said to be inactive. A partial code (of
length $n$) is a subset of ${\mathbb{PW}}^{n}$. The variety of a partial motif
${\mathbf{a}}$ is the set of all partial words obtained by replacing all the
stars in ${\mathbf{a}}$ by zeros and ones. It is denoted by $V_{\mathbf{a}}$.
If ${\mathcal{C}}\subseteq{\mathbb{PW}}^{n}$ is a a partial code, then
${\mathbf{a}}\in{\mathbb{PM}}^{n}$ is a partial motif of ${\mathcal{C}}$ if
$V_{\mathbf{a}}\subseteq{\mathcal{C}}$. The set of all partial motifs of a
partial code ${\mathcal{C}}$ is denoted by $\mathrm{ParMot}({\mathcal{C}})$.
The set of all maximal partial motifs of a partial code ${\mathcal{C}}$ is
denoted by $\mathrm{MaxParMot}({\mathcal{C}})$.

######  Example 4.2.

We can think of the partial word
${\mathbf{w}}={\textunderscore}01{\textunderscore}00{\textunderscore}1$ as of
a statement that the neurons $3$ and $8$ are firing, the neurons $2,5$ and $6$
are not firing, and the neurons $1,4,$ and $7$ are inactive.

The set of all partial motifs ${\mathbf{a}}\in{\mathbb{PM}}^{n}$ (resp.
partial words ${\mathbf{w}}\in{\mathbb{PW}}^{n}$) such that
$a_{i_{1}}=\dots=a_{i_{k}}={\textunderscore}$ (resp.
$w_{i_{1}}=\dots=w_{i_{k}}={\textunderscore}$) and all the remaining neurons
are active, is denoted by ${\mathbb{PM}}^{n}_{i_{1},\dots,i_{k}}$ (resp.
${\mathbb{PW}}^{n}_{i_{1},\dots,i_{k}})$. It is naturally in a bijective
correspondence with the set ${\mathbb{M}}^{n-k}$ (resp.
${\mathbb{F}}_{2}^{n-k}$). If ${\mathbf{a}}$ (resp. ${\mathbf{w}}$) is a motif
(resp. word), then the partial motif (resp. partial word) obtained by
replacing each $a_{i}$ (resp. $w_{i}$), $i=i_{1},\dots,i_{k}$, by
${\textunderscore}$ is called the partial motif (resp. partial word) obtained
by deactivating the neurons $i_{1},\dots,i_{k}$ and is denoted by
$a^{\textunderscore}_{i_{1},\dots,i_{k}}$ (resp.
$w^{\textunderscore}_{i_{1},\dots,i_{k}}$).

If ${\mathcal{C}}\subseteq{\mathbb{F}}_{2}^{n}$ is a code and
$\\{i_{1},\dots,i_{k}\\}\subseteq[n]$, then the code obtained by replacing
each $w_{i_{r}}$ $(r=1,\dots,k)$ by ${\textunderscore}$ in each word
${\mathbf{w}}\in{\mathcal{C}}$ is called the partial code obtained by
deactivating the neurons $i_{1},\dots,i_{k}$ and is denoted by
${\mathcal{C}}^{{}^{\textunderscore}}_{i_{1},\dots,i_{k}}$. The partial code
${\mathcal{C}}^{{}^{\textunderscore}}_{i_{1},\dots,i_{k}}$ is naturally in a
bijective correspondence with the code ${\mathcal{C}}_{i_{1},\dots,i_{k}}$
obtained by deleting the neurons $i_{1},\dots,i_{k}$.

######  Proposition 4.3.

Let ${\mathcal{C}}$ be a code in ${\mathbb{F}}_{2}^{n}$ and let
${\mathbf{w}}\in{\mathbb{PW}}^{n}_{i_{1},\dots,i_{k}}$. If ${\mathbf{w}}$ is
not an element of ${\mathcal{C}}^{{}^{\textunderscore}}_{i_{1},\dots,i_{k}}$,
then the motif ${\mathbf{a}}$ obtained from ${\mathbf{w}}$ by replacing each
${\textunderscore}$ by $*$ belongs to $\mathrm{Mot}(^{c}{\mathcal{C}})$.

###### Proof.

Easy to see. ∎

######  Definition 4.4.

If ${\mathbf{a}}\in{\mathbb{PM}}^{n}$, then we define its polarization
${\mathbf{a}}^{p}=b_{1}\dots b_{n}b_{n+1}\dots b_{2n}\in{\mathbb{PM}}^{2n}$
defined by:

$\displaystyle b_{i}=0,\;b_{n+1}=*,\quad\text{when }a_{i}=0;$ $\displaystyle
b_{i}=*,\;b_{n+1}=0,\quad\text{when }a_{i}=1;$ $\displaystyle
b_{i}=*,\;b_{n+1}=*,\quad\text{when }a_{i}=*;$ $\displaystyle
b_{i}={\textunderscore},\;\;b_{n+1}={\textunderscore},\quad\,\text{when
}a_{i}={\textunderscore}.$

A partial motif ${\mathbf{b}}\in{\mathbb{PM}}^{2n}$ is called a polar partial
motif if ${\mathbf{b}}={\mathbf{a}}^{p}$ for some partial motif
${\mathbf{a}}\in{\mathbb{PM}}^{n}$. Then ${\mathbf{a}}={\mathbf{b}}^{d}$ is
called the depolarization of the polar partial motif ${\mathbf{b}}$.

The next theorem is a slight reformulation of Theorem 5.1 from [5]. We give a
different proof.

######  Theorem 4.5 ([5, Theorem 5.1]).

Let ${\mathbf{c}}\in{\mathbb{M}}^{2n}$ and let
${\mathbf{a}}\in{\mathbb{M}}^{2n}$ be the motif obtained by replacing all ones
in ${\mathbf{c}}$ by stars. Let $\\{i_{1},\dots,i_{k}\\}$ be the set of all
elements $i$ of $[n]$ such that $a_{i}=a_{i+n}$. Then
${\mathfrak{p}}_{\mathbf{c}}\supseteq J_{{\mathcal{C}}^{[p]}}$ if and only if
${\mathbf{a}}^{{\textunderscore}\,d}_{i_{1},\dots,i_{k},i_{1}+n,\dots,i_{k}+n}\in\mathrm{ParMot}({\mathcal{C}}^{{}^{{\textunderscore}}}_{i_{1},\dots,i_{k}})$.

###### Proof.

Let ${\mathcal{D}}=\,^{c}{\mathcal{C}}$. Since

$\mathrm{MaxMot}(^{c}({\mathcal{C}}^{[p]}))=\\{\,\overline{\overline{{\mathbf{b}}}^{p}}\;:\;{\mathbf{b}}\in\mathrm{MaxMot}({\mathcal{D}})\\},$

the motifs ${\mathbf{c}}$ of ${\mathcal{C}}^{[p]}$ are the motifs from
${\mathbb{M}}^{2n}$ that are disjoint with all
$\overline{\overline{{\mathbf{b}}}^{p}}$,
${\mathbf{b}}\in\mathrm{MaxMot}({\mathcal{D}})$. They give the primes
${\mathfrak{p}}_{\mathbf{c}}$, and these are all the motivic primes that
contain $J_{{\mathcal{C}}^{[p]}}$. Note, however, that
${\mathfrak{p}}_{\mathbf{c}}\supset J_{{\mathcal{C}}^{[p]}}$ if and only if
${\mathfrak{p}}_{\mathbf{a}}\supset J_{{\mathcal{C}}^{[p]}}$. It follows that
a motivic prime ${\mathfrak{p}}_{\mathbf{c}}$ contains
$J_{{\mathcal{C}}^{[p]}}$ if and only if
${\mathbf{a}}+\overline{\overline{{\mathbf{b}}}^{p}}$ has at least one
component equal to $1$, or, equivalently, such that

(19) ${\mathbf{a}}+\overline{\overline{{\mathbf{b}}}^{p}}<*\dots*$

(as ${\mathbf{a}}\in\\{0,*\\}^{2n}$ and
$\overline{\overline{{\mathbf{b}}}^{p}}\in\\{1,*\\}^{2n}$) for every
${\mathbf{b}}\in\mathrm{MaxMot}({\mathcal{D}})$. Let $\\{i_{1},\dots,i_{k}\\}$
be the set of elements $i$ of $[n]$ such that $a_{i}=a_{i+n}=0$. (This
includes the possibility $k=0$.) Let ${\mathbf{a}}^{{\textunderscore}\,d}$
denote
${\mathbf{a}}^{{\textunderscore}\,d}_{i_{1},\dots,i_{k},i_{1}+n,\dots,i_{k}+n}$.
The statement of the theorem follows if we now justify the claim that
${\mathbf{a}}$ satisfies (19) if and only if each partial word
${\mathbf{w}}\leq{\mathbf{a}}^{{\textunderscore}\,d}$ belongs to
${\mathcal{C}}^{{}^{{\textunderscore}}}_{i_{1},\dots,i_{k}}$, i.e., if and
only if
${\mathbf{a}}^{{\textunderscore}\,d}\in\mathrm{ParMot}({\mathcal{C}}^{{}^{{\textunderscore}}}_{i_{1},\dots,i_{k}})$.
The necessity is clear. For the sufficiency, there would otherwise be a
partial word ${\mathbf{w}}\leq{\mathbf{a}}^{{\textunderscore}\,d}$ which is
not coming from any partial word in
${\mathcal{C}}^{{}^{{\textunderscore}}}_{i_{1},\dots,i_{k}}$. Let
${\mathbf{w}}^{\ast}$ be the motif obtained by replacing each
${\textunderscore}$ in ${\mathbf{w}}$ by $*$. Then
${\mathbf{w}}^{\ast}\in\mathrm{Mot}({\mathcal{D}})$, hence
$\overline{\overline{{\mathbf{w}}^{\ast}}^{p}}\in\mathrm{Mot}(^{c}({\mathcal{C}}^{[p]}))$.
This motif would have stars at all the components at which ${\mathbf{a}}$ has
zeros and ones or stars at all other components. Hence
${\mathbf{a}}+\overline{\overline{{\mathbf{w}}^{*}}^{p}}=*\dots*$,
contradicting to the asumption that ${\mathbf{a}}$ satisfies (19). ∎

######  Example 4.6.

In the context of the examples (3.20) and (3.24), let
${\mathbf{c}}=00\\!*\\!*0*\in{\mathbb{M}}^{6}$. Then
${\mathbf{a}}={\mathbf{c}}$ and

${\mathbf{a}}^{{\textunderscore}\,d}_{2,2}\;=\;0{\textunderscore}\\!*\\!*{\textunderscore}*^{d}\;=\;0{\textunderscore}\\!*.$

Also

${\mathcal{C}}^{{}^{\textunderscore}}_{2}=\\{000,\,100,\,110,\,011\\}^{{}^{\textunderscore}}_{2}=\\{0{\textunderscore}0,\,1{\textunderscore}0,\,0{\textunderscore}1\\}.$

Hence

$\mathrm{MaxParMot}({\mathcal{C}}^{{}^{\textunderscore}}_{2})\;=\;\\{*{\textunderscore}0,\,0{\textunderscore}*\\}.$

Thus

${\mathbf{a}}^{{\textunderscore}\,d}_{2,2}\in\mathrm{MaxParMot}({\mathcal{C}}^{{}^{\textunderscore}}_{2}),$

so that

${\mathfrak{p}}_{\mathbf{c}}=(X_{1},X_{2},Y_{2})\supseteq
J_{{\mathcal{C}}^{[p]}}.$

In fact we have

${\mathfrak{p}}_{\mathbf{c}}=(X_{1},X_{2},Y_{2})\in\mathrm{Min}(J_{{\mathcal{C}}^{[p]}}).$

Let now ${\mathbf{c}}=0\\!*\\!0\\!*\\!*0\in{\mathbb{M}}^{6}$. Then
${\mathbf{a}}={\mathbf{c}}$ and

${\mathbf{a}}^{{\textunderscore}\,d}_{3,3}\;=\;0\\!*\\!{\textunderscore}\\!*\\!*{\textunderscore}^{d}\;=\;0\\!*\\!{\textunderscore}.$

Also

${\mathcal{C}}^{{}^{\textunderscore}}_{3}=\\{000,\,100,\,110,\,011\\}^{{}^{\textunderscore}}_{3}=\\{00{\textunderscore},\,10{\textunderscore},\,11{\textunderscore},\,01{\textunderscore}\\}.$

Hence

$\mathrm{MaxParMot}({\mathcal{C}}^{{}^{\textunderscore}}_{3})\;=\;\\{*\\!*\\!{\textunderscore}\\}.$

Thus

${\mathbf{a}}^{{\textunderscore}\,d}_{3,3}\in\mathrm{ParMot}({\mathcal{C}}^{{}^{\textunderscore}}_{3}),$

so that

${\mathfrak{p}}_{\mathbf{c}}=(X_{1},X_{3},Y_{3})\supseteq
J_{{\mathcal{C}}^{[p]}}.$

Note that

${\mathfrak{p}}_{\mathbf{c}}=(X_{1},X_{3},Y_{3})\notin\mathrm{Min}(J_{{\mathcal{C}}^{[p]}})$

since it was shown in Example 3.24 that the prime $(X_{3},Y_{3})$ is a minimal
prime of $J_{{\mathcal{C}}^{[p]}}$.

Finally, let ${\mathbf{c}}=100\\!*\\!0*\in{\mathbb{M}}^{6}$. Then
${\mathbf{a}}=*00\\!*\\!0*$ and

${\mathbf{a}}^{{\textunderscore}\,d}_{2,2}\;=\;*{\textunderscore}0\\!*\\!{\textunderscore}*^{d}\;=\;*{\textunderscore}0.$

Also

${\mathcal{C}}^{{}^{\textunderscore}}_{2}=\\{000,\,100,\,110,\,011\\}^{{}^{\textunderscore}}_{2}=\\{0{\textunderscore}0,\,1{\textunderscore}0,\,0{\textunderscore}1\\}.$

Hence

$\mathrm{MaxParMot}({\mathcal{C}}^{{}^{\textunderscore}}_{2})\;=\;\\{*{\textunderscore}0,\;0{\textunderscore}1\\}.$

Thus

${\mathbf{a}}^{{\textunderscore}\,d}_{2,2}\in\mathrm{MaxParMot}({\mathcal{C}}^{{}^{\textunderscore}}_{2}),$

so that

${\mathfrak{p}}_{\mathbf{a}}=(X_{2},X_{3},Y_{2})\supseteq
J_{{\mathcal{C}}^{[p]}}$

and

${\mathfrak{p}}_{\mathbf{c}}=(1-X_{1},X_{2},X_{3},Y_{2})\supseteq
J_{{\mathcal{C}}^{[p]}}.$

Note that

${\mathfrak{p}}_{\mathbf{a}}\notin\mathrm{Min}(J_{{\mathcal{C}}^{[p]}})$

even though
${\mathbf{a}}^{{\textunderscore}\,d}_{2,2}\in\mathrm{MaxParMot}({\mathcal{C}}^{{}^{\textunderscore}}_{2})$
since it was shown in Example 3.24 that the prime $(X_{2},X_{3})$ is a minimal
prime of $J_{{\mathcal{C}}^{[p]}}$.

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