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\title{Uniqueness of the decompositon of the rank function of a 2-polimatroid\footnote{Research partially supported by Conselho Nacional de Desenvolvimento Cient\'\i fico e Tecnol\'ogico (CNPq, Brazil).}}

\author{Manoel Lemos\\
        Departamento de Matem\'atica\\
        Universidade Federal de Pernambuco\\
        Cidade Universit\'aria\\
        Recife, PE, 50740-540, Brazil}

\date{e-mail: manoel@dmat.ufpe.br}

\begin{document}

\maketitle

\begin{abstract}
In [5], Murty and Simon asked the following question: when the rank function of a polimatroid can be decomposed as the sum of rank funcitons of matroids? Another natural question to ask is the following: when this decomposition is unique? In this paper, we shall adress the second question. 
\end{abstract}

\section{Introduction}

We shall denote the multiset $\{X_1,X_2,\dots,X_m\}$ of matroids $X_1,X_2,\dots,X_m$ by $\{X\}_m$. We say that a multiset of matroids $\{M\}_m$ is {\it admissible}, when $M_1,M_2,\dots,$ $M_m$ are matroids defined over the same ground set, which is denoted by $E(\{M\}_m)$, and
$$r_{M_1}(\{e\})+r_{M_2}(\{e\})+\cdots+r_{M_m}(\{e\})\le 2,$$
for every $e\in E(\{M\}_m)$. That is, the sum of the rank function of all the matroids belonging to $\{M\}_m$ is equal to the rank function of a 2-polimatroid $P$. We say that $\{M\}_m$ {\it defines} $P$ or that $\{M\}_m$ is a {\it decomposition} of $P$. When this decompositon exists, $P$ is said to be {\it matroidal}. We say that $(\{M\}_m,\{N\}_n)$ is a {\it pair}, when $\{M\}_m$ and $\{N\}_n$ are admissible multisets of matroids, $E(\{M\}_m)=E(\{N\}_n)$ and
$$r_{M_1}(X)+r_{M_2}(X)+\cdots+r_{M_m}(X)=r_{N_1}(X)+r_{N_2}(X)+\cdots+r_{N_n}(X),$$
for every $X\subseteq E(\{M\}_m)$. That is, $\{M\}_m$ and $\{N\}_n$ defines the same 2-polimatroid. 

A pair $(\{M\}_m,\{N\}_n)$ is said to be {\it connected}, when, for every partition $\{X,Y\}$ of $E(\{M\}_m)$ such that $\min\{|X|,|Y|\}\ge 1$,
$$r_{M_i}(X)+r_{M_i}(Y)-r(M_i)\ge 1,$$
for some $i\in\{1,2,\dots,k\}$. That is, the matroids belonging to $\{M\}_m$ do not have a common 1-separation. Thus, $\{M\}_m$ defines a connected 2-polimatroid. 

When $M$ is a matroid and $N$ a connected component of $M$, we denote by $\overline{N}$ the matroid such that $E(\overline{N})=E(M)$, $\overline{N}|E(N)=N$ and $r(\overline{N}|E(M)\setminus E(N))=0$. We define 
$${\cal A}(M)=\{\overline{N}: N\hbox{ is a connected component of $M$ and $r(N)\not=0$}\}.$$
Observe that
$$r_{M}(X)=\sum_{H\in{\cal A}(M)}r_H(X),$$
for every $X\subseteq E$. For a multiset $\{M\}_m$, consider the following multiset:
$${\cal A}(\{M\}_m)=\bigcup_{i=1}^m{\cal A}(M_i).$$
Observe that $(\{M\}_m,{\cal A}(\{M\}_m))$ is a pair. Two admissible multisets of matroids $\{M\}_m$ and $\{N\}_n$ are said to be {\it equivalent}, when 
$${\cal A}(\{M\}_m)={\cal A}(\{N\}_n).$$
We say that a pair $(\{M\}_m,\{N\}_n)$ is {\it trivial}, when $\{M\}_m$ is equivalent to $\{N\}_n$. When $(\{M\}_m,\{N\}_n)$ is a trivial pair, we can get $\{N\}_n$ from $\{M\}_m$ by moving around connected components and ignoring some loops. Thus, we shall not consider $\{M\}_m$ and $\{N\}_n$ different decompositons for the 2-polimatroid defined by $\{M\}_m$.

The {\it size} of an admissible multiset of matroids $\{M\}_m$ is defined as
$$size(\{M\}_m)=|{\cal A}(\{M\}_m)|.$$
We define the {\it size} of a 2-polimatroid $P$ as 
$$size(P)=\max\{size(\{M\}_m):\{M\}_m\hbox{ decomposes $P$}\}.$$

In this paragraph, we shall discuss an ideia originated on Seymour's paper [8]. For a matroid $M$, we define
$${\cal C}_{\ge 2}(M)=\{C\in{\cal C}(M):|C|\ge 2\},$$
that is, ${\cal C}_{\ge 2}(M)$ is the set of circuits of $M$ which are not loops. For an admissible multiset of matroids $\{M\}_m$, we define the following multiset:
$${\cal C}_{\ge 2}(\{M\}_m)=\bigcup_{i=1}^m{\cal C}_{\ge 2}(M_i).$$
We say that a pair $(\{M\}_m,\{N\}_n)$ is {\it preserving} when
$${\cal C}_{\ge 2}(\{M\}_m)={\cal C}_{\ge 2}(\{N\}_n).$$ 

In section three, we shall study the non-preserving connected pairs and conclude that they are very few. Essentially, there is just one of them, since all the others can be obtained from it making 2-sums of appropriated matroids, as described in the next theorem. To avoid a more cumbersome statement for it, we need to extend the 2-sum operation for matroids having more than one element. When $N$ and $M$ are matroids such that $E(N)\cap E(M)=\{e\}$ and $E(N)\setminus E(M)=\{f\}$, the 2-sum of $N$ with $M$ is the matroid obtained from $M$ by renaming $e$ by $f$. 
\vspace{0.3cm}

\noindent {\bf Theorem 1.} Let $H_1,H_2,K_1,K_2,K_3$ be matroids defined over the same ground set $\{a,b,c\}$ such that 
$$H_1\simeq U_{2,3},H_2\simeq U_{1,3},K_i\simeq U_{1,2}\oplus U_{0,1},\hbox{ for every }i\in\{1,2,3\},$$
and $a,b,c$ are loops of $K_1,K_2,K_3$, respectively. When $\{a,b,c\}\cap E(\{M\}_m)=\emptyset$, $(\{M\}_m,\{N\}_n)$ is a non-pre\-ser\-ving connected pair if and only if there are connected matroids $P_a,P_b,P_c$ such that $x\in P_x,$ $|E(P_x)|\ge 2$, for every $x\in\{a,b,c\}$, $\{P_a\setminus\{a\},P_b\setminus\{b\},P_c\setminus\{c\}\}$ is a partition of $E(\{M\}_m)$, $\{M\}_m$ or $\{N\}_n$ is equivalent to 
$$\{H_1\oplus_2 N_a\oplus_2 N_b\oplus_2 N_c,H_2\oplus_2 N_a\oplus_2 N_b\oplus_2 N_c\}$$ 
and the other admissible multiset of matroids in this pair is equivalent to 
$$\{(L_a\oplus K_1\backslash a)\oplus_2 N_b\oplus_2 N_c,(L_b\oplus K_2\backslash b)\oplus_2 N_a\oplus_2 N_c,(L_c\oplus K_3\backslash c)\oplus_2 N_a\oplus_2 N_b\},$$
where $L_x$ is a matroid over $P_x\setminus\{x\}$ such that $r(L_x)=0$, for every $x\in\{a,b,c\}$.
\vspace{0.05cm}

As the non-preserving pairs are rare and well described, the rest of the paper shall be dedicated to prove results about preserving pairs. In section four, we shall prove the following theorem:
\vspace{0.3cm}

\noindent {\bf Theorem 2.} If $(\{M\}_m,\{N\}_n)$ is a non-trivial preserving connected pair, then there is a pair $(\{M'\}_2,\{N'\}_2)$ such that $\{M\}_m$ is equivalent to $\{M'\}_2$ and $\{N\}_n$ is equivalent to $\{N'\}_2$.
\vspace{0.3cm}

Thus, when a connected 2-polimatroid has non-equivalent decompositons, we can reorganize the matroids on these decompositons moving around connected components and ignoring loops, in such way that at the end we get only two matroids in each decompositon, unless these decompositions forms a non-preserving pair. 

Now one example to show that theorem 2 is quite surprising: for each vertex $v$ of the complete graph $K_n$, let $M_v$ be a matroid such that $E(M_v)=E(K_n),M_v|st(v)$ is a connected component of $M$ and $r_{M_v}(E(K_n)\setminus st(v))=0$. When $n\ge 2$, then $\{M_v:v\in V(K_n)\}$ is an admissible multiset of matroids which is not equivalent to one having less that $n$ matroids.

%${\cal H}$ a family of subsets of $E$ such that each element of $E$ belongs to at most two elements of ${\cal H}$ and $H\cap H'\not=\emptyset$, for every $H,H'\in{\cal H}$. For each $H\in{\cal H}$, let $M_H$ be a  matroid having $E$ as its ground set, $H$ as a minimal separating set and $r_{M_H}(E\setminus H)=0$. Thus, $\{M_H:H\in{\cal H}\}$ is an admissible multiset of matroids which is not equivalent to one having less than $|{\cal H}|$ matroids.

A vector ${\bf M}=(M_1,M_2,M_3,M_4)$ with four matroids defined on the same ground set $E$ is called a {\it quad} when these matroids satisfies:
$$r_{M_1}(X)+r_{M_2}(X)=r_{M_3}(X)+r_{M_4}(X),$$
for every $X\subseteq E$. ${\bf M}$ is said to be {\it non-trivial}, when $\{M_1,M_2\}\not=\{M_3,M_4\}$. So, theorem 1 and theorem 2 reducess the problem of uniqueness of decomposition of a connected 2-polimatroid to the study of non-trivial quads. 

In [3], Lemos and Mota shall present an algorithm to decide when a vector with two matroids $(M_1,M_2)$ can beginning a non-trivial quad. Moreover, this algorithm gives all the possibilities for the others two matroids that completes that quad. Thus, the algorithm, theorem 1 and theorem 2 solves the uniqueness problem of decomposition of the rank function of a 2-polimatroid as the sum of rank function of matroids.

We shall have the following corollaries from these theorems, which guaranties uniqueness in many cases:
\vspace{0.3cm}

\noindent {\bf Corollary 1.} If $P$ is a connected matroidal 2-polimatroid such that $size(P)\ge 4$, then the decompositon of $P$ is unique.
\vspace{0.3cm}

\noindent {\bf Corollary 2.} Suppose that $P=(E,\rho)$ is a connected matroidal 2-polimatroid such that $\rho(E)$ is odd. If the decomposition of $P$ is not unique, then $P$ has only two non-equivalent decompositions which forms a non-preserving pair.
\vspace{0.3cm}

\noindent {\bf Corollary 3.} Suppose that $P=(E,\rho)$ is a connected matroidal 2-polimatroid. If $\rho(e)=1$, for some $e\in E$, then the decomposition of $P$ is unique.
\vspace{0.3cm}

We say that a 2-polimatroid is {\it binary} when it decomposes as an admissible multiset of binary matroids. 
\vspace{0.3cm}

\noindent {\bf Corollary 4.} If $P$ is a connected binary 2-polimatroid, then it has at most two non-equivalent decompositions.
\vspace{0.3cm}

Now, we give an example to show that a connected 2-polimatroid can have arbitrary large number of non-equivalent decompositions: let ${\cal H}$ be a family of subsets of a set $E$ such that $|H|=r$ and $|H\cup H'|\ge r+2$, for every $H,H'\in{\cal H}$ such that $H\not=H'$. For ${\cal Z}\subseteq {\cal H}$, let $M({\cal Z})$ be the matroid having $E$ as ground set and every subset of $E$ with $r$ elements not belonging to ${\cal Z}$ as a basis. When $\{{\cal X},{\cal Y}\}\not=\{{\cal H},\emptyset\}$ is a partition of ${\cal H}$, then $(\{M({\cal H}),M({\emptyset})\},\{M({\cal X}),M({\cal Y})$\}) is a non-trivial pair.

\section{Tutte's geometry}

In this section, we shall present some results from Tutte's geometry. We shall use Tutte's geometry, since it is natural to work with it when one is dealing with circuits. 

We say that $L$ is a {\it Tutte-line} of a matroid $M$, when $L$ is the uniun of circuits of $M$ and $r^*(M|L)=2$. $P$ is said to be a {\it Tutte-plane} of $M$, when $P$ is the uniun of circuits of $M$ and $r^*(M|P)=3$. Every Tutte-line has a partition $\{L_1,L_2,\dots,L_k\}$, which is called {\it canonical}, such that $C$ is a circuit of $M$ contained in $L$ if and only if $C=L\setminus L_i$, for some $i\in\{1,2,\dots,k\}$. We say that a Tutte-line $L$ or a Tutte-plane $P$ is {\it connected}, when $M|L$ or $M|P$ is connected, respectively. When a line $L$ is connected, its canonical partition has at least three sets. Now, we shall present some results of Tutte's geometry.

\bteo\rotulo{ta}
(4.26 of [9]) Suppose that $P$ is a connected Tutte-plane of a matroid $M$. If $C$ is a circuit of $M$ contained in $P$, then there are at least two connected Tutte-lines of $M$ contained in $P$ which contains $C$.\qeds

\bteo\rotulo{tb}
(4.171 of [9]) Suppose that $P$ is a Tutte-plane of a matroid $M$. If $L$ and $L'$ are different Tutte-lines of $M$ contained in $P$, then $L\cap L'$ contains exactly one circuit of $M$.\qeds

\bteo\rotulo{tc}
(4.35 of [9]) Suppose that $M$ is a connected matroid. If $C$ and $D$ are different circuits of $M$, then there is a sequence 
$$C=C_0,C_1,C_2,\dots,C_n=D$$
of circuits of $M$ such that $C_{i-1}\cup C_i$ is a connected Tutte-line of $M$, for every $i\in\{1,2,\dots,n\}$.\qeds
\eteo

Now, we shall present a result of Lehman. Before the introduction of its statement, we remember that the set of all circuits of a matroid $M$ which contains an element $e$ is denoted by ${\cal C}_e(M)$.

\bteo\rotulo{lem}
(4.3.2 of [6]) Suppose that $M$ and $N$ are connected matroids defined on the same ground set. If ${\cal C}_e(M)={\cal C}_e(N)$, then $N=M$.\qeds
\eteo

\section{The non-preserving case}

Throughout this section, we shall suppose that $(\{M\}_m,\{N\}_n)$ is a pair. When $X$ and $Y$ are disjoint subsets of $E(\{M\}_m)$, then
$$\hbox{$\{M\backslash X/Y\}_m$ denotes the multiset }\{M_1\backslash X/Y,M_2\backslash X/Y,\dots,M_m\backslash X/Y\}.$$
We shall define a {\it minor} of $(\{M\}_m,\{N\}_n)$ as
$$(\{M\}_m,\{N\}_n)\backslash X/Y=(\{M\backslash X/Y\}_m,\{N\backslash X/Y\}_n).$$
We denote $(\{M\}_m,\{N\}_n)\backslash(E(\{M\}_m)\setminus X)$ and $\{M\backslash(E(\{M\}_m)\setminus X)\}_m$  by $(\{M\}_m,\{N\}_n)|X$ and $\{M|X\}_m$, respectively. We shall use the next result along this paper without refering to it.

\bteo\rotulo{aa}
$(\{M\}_m,\{N\}_n)\backslash X/Y$ is a pair.
\eteo

\noindent {\bf Proof:} When $Z\subseteq E(\{M\}_m)\setminus (X\cup Y)$, we have that
\begin{center}
\begin{eqnarray*}
\sum_{i=1}^mr_{M_i\backslash X/Y}(Z)&=&\sum_{i=1}^m[r_{M_i}(Z\cup Y)-r_{M_i}(Y)]\\
&=&\sum_{i=1}^mr_{M_i}(Z\cup Y)-\sum_{i=1}^mr_{M_i}(Y)\\
&=&\sum_{i=1}^nr_{N_i}(Z\cup Y)-\sum_{i=1}^nr_{N_i}(Y)\\
&=&\sum_{i=1}^n[r_{N_i}(Z\cup Y)-r_{N_i}(Y)]=\sum_{i=1}^nr_{N_i\backslash X/Y}(Z)\hbox{\qeds}
\end{eqnarray*}
\end{center}

\bteo\rotulo{ab}
If $M_i|X$ is loopless, for some $i\in\{1,2,\dots,m\}$, then $\{M|X\}_k$ is equivalent to $\{M_i|X,K\}$, for some matroid $K$.
\eteo

\noindent {\bf Proof:} As $r_{M_i}(\{e\})=1$, for every $e\in X$, it follows that 
$$r_{M_1}(\{e\})+\cdots+r_{M_{i-1}}(\{e\})+r_{M_{i+1}}(\{e\})+\cdots+r_{M_m}(\{e\})\le 1.$$
Thus, there is a matroid $K$ such that every connected component $N$ of $M_j$ satisfying $r(N)\not=0$, for $j\not=i$, is also a connected component of $K$. \qed

Observe that every chain of elements of ${\cal C}_{\ge2}(\{M\}_m)$ has size one or two, otherwise there are three elements $C_1,C_2$ and $C_3$ of  ${\cal C}_{\ge 2}(\{M\}_m)$ such that $C_1\subset C_2\subset C_3$. Thus, 
$$r_{M_1}(\{e\})+r_{M_2}(\{e\})+\cdots+r_{M_m}(\{e\})\ge 3,$$
for every element $e\in C_1$, since $e$ cannot be a loop in $M_i$, when $C_j\in{\cal C}(M_i)$. Moreover, an element of 
${\cal C}_{\ge 2}(\{M\}_m)$ can have multiplicity at most two. When this happens, it does not contains or it is not contained in another element of ${\cal C}_{\ge 2}(\{M\}_m)$. We denote by ${\cal C}_{min}(\{M\}_m)$ the multiset of minimal elements of ${\cal C}_{\ge2}(\{M\}_m)$. Note that an element has multiplicity two in ${\cal C}_{min}(\{M\}_m)$ if and only if is has multiplicity two in ${\cal C}_{\ge 2}(\{M\}_m)$. 

For a matroid $K$, we shall define
$$\delta(K)=r(K)-\sum_{e\in E(K)}r_K(\{e\}).$$
The following observation is fundamental in the proof of the next lemma:

\bteo\rotulo{obs}
For a matroid $K$, $\delta(K)\le 0$. Moreover, $\delta(K)=0$ if and only if ${\cal C}_{\ge 2}(K)=\emptyset$ and $\delta(K)=-1$ if and only if $|{\cal C}_{\ge 2}(K)|=1$.\qeds
\eteo

\bteo\rotulo{ac}
${\cal C}_{min}(\{M\}_m)={\cal C}_{min}(\{N\}_n).$
\eteo

\noindent {\bf Proof:} By symmetry, we need only to prove that ${\cal C}_{min}(\{M\}_m)\subseteq {\cal C}_{min}(\{N\}_n)$. Let $C$ be an element of ${\cal C}_{min}(\{M\}_m)$. For $X\subseteq C$, observe that
\begin{center}
\begin{eqnarray*}
\sum_{i=1}^m\delta(M_i|X)&=&\sum_{i=1}^m\left(r_{M_i}(X)-\sum_{x\in X}r_{M_i}(\{x\})\right)\\
&=&\sum_{i=1}^mr_{M_i}(X)-\sum_{i=1}^m\sum_{x\in X}r_{M_i}(\{x\})\\
&=&\sum_{i=1}^mr_{M_i}(X)-\sum_{x\in X}\sum_{i=1}^mr_{M_i}(\{x\})\\
&=&\sum_{i=1}^nr_{N_i}(X)-\sum_{x\in X}\sum_{i=1}^nr_{N_i}(\{x\})\\
&=&\sum_{i=1}^n\left(r_{N_i}(X)-\sum_{x\in X}r_{N_i}(\{x\})\right)=\sum_{i=1}^n\delta(N_i|X)
\end{eqnarray*}
\end{center}
Now, we shall prove that:
\vspace{0.3cm}
 
\noindent {\bf Step 1:} ${\cal C}_{\ge 2}(\{N|X\}_n)=\emptyset$, for every $X\subset C$.
\vspace{0.3cm}

When $X\subset C$, ${\cal C}_{\ge 2}(M_i|X)=\emptyset$, for every $i\in\{1,2,\dots,m\}$, since $C\in{\cal C}_{min}(\{M\}_m)$. By \recorde{obs}, we have that
$$0=\sum_{i=1}^m\delta(M_i|X)=\sum_{i=1}^n\delta(N_i|X).$$
Thus, by \recorde{obs}, $\delta_{N_i}(X)=0$ and ${\cal C}_{\ge 2}(N_i|X)=\emptyset$, for every $i\in\{1,2,\dots,n\}$.
\vspace{0.3cm}

So, for every $i\in\{1,2,\dots,m\}$ and $j\in\{1,2,\dots,n\}$, ${\cal C}_{\ge 2}(M_i|C)\in\{\emptyset,\{C\}\}$ and ${\cal C}_{\ge 2}(N_j|C)\in\{\emptyset,\{C\}\}$. Thus, the result follows when we stabilish:
\vspace{0.3cm}

\noindent {\bf Step 2:} $|{\cal C}_{\ge 2}(\{M|C\}_m)|=|{\cal C}_{\ge 2}(\{N|C\}_n)|.$
\vspace{0.3cm}

From \recorde{obs} and step 1, we have that 
$$-|{\cal C}_{\ge 2}(\{M|C\}_m)|=\sum_{i=1}^m\delta(M_i|X)=\sum_{i=1}^n\delta(N_i|X)=-|{\cal C}_{\ge 2}(\{N|C\}_n)|$$
and the result follows.\qed

We define the following multiset 
$${\cal C}_{uni}(\{M\}_m)={\cal C}_{\ge 2}(\{M\}_m)\setminus {\cal C}_{\ge 2}(\{N\}_n).$$
We define ${\cal C}_{uni}(\{N\}_n)$ similarly. In {\bf (3.7)}, we shall describe the nice behaviour of the elements of ${\cal C}_{uni}(\{M\}_m)$. To do this, we need to prove some preliminary lemmas. 

\bteo\rotulo{ad}
If $C\in {\cal C}_{uni}(\{M\}_m)$ and $C'\in{\cal C}_{uni}(\{N\}_n)$, then $C\not\subseteq C'$ and $C'\not\subseteq C$.
\eteo

\noindent {\bf Proof:} We shall argue by contradiction and we assume that the result fails. Without loss of generality, we may suppose that $C\subseteq C'$. By definition of these multisets, we have that $C\subset C'$. By \recorde{ab}, there are matroids $K$ and $K'$ such that $\{M|C\}_m$ and $\{N|C\}_n$ are equivalent to $\{M_i|C,K\}$ and $\{N_j|C,K'\}$, respectively, where $C$ is a circuit of $M_i$ and $C'$ is a circuit of $N_j$. If $C''\in {\cal C}_{\ge 2}(\{M\}_m)$ is contained in $C$ and $C'''\in {\cal C}_{\ge 2}(\{N\}_n)$ is contained in $C'$, then $C''\in{\cal C}_{min}(\{M\}_m)$ and $C'''\in{\cal C}_{min}(\{N\}_n)$. So, by \recorde{ac}, $C''$ and $C'''$ belongs to ${\cal C}_{\ge 2}(\{M\}_m)\cap{\cal C}_{\ge 2}(\{N\}_n)$. Thus, $K$ and $K'$ have the same circuits and $K=K'$. We arrive at a contradiction, since $M_i|C\not=M_j|C$, and the result follows.\qeds

\bteo\rotulo{af}
For $i\in\{1,2,\dots,m\}$, if $L$ is a Tutte-line of $M_i$ having canonical partition $\{L_1,L_2,L_3\}$ such that $C_j=L\setminus L_j$ is a circuit of $N_j$, for every $j\in\{1,2,3\}$, then $L\in{\cal C}_{uni}(\{M\}_m)$.
\eteo

\noindent {\bf Proof:} By \recorde{ab}, there is a matroid $K$ such that $E(K)=L$ and $\{M|L\}_m$ is equivalent to $\{M_i|L,K\}$. For $j\in\{1,2,3\}$, observe that $r(N_j|L_j)=0$ because $r_{N_k}(\{e\})=1$, for every $e\in L_j$ and $k\in\{1,2,3\}\setminus\{j\}$. Thus, ${\cal C}_{\ge 2}(N_i|L)=\{C_i\}$. Hence ${\cal C}_{\ge 2}(\{N|L\}_n)=\{C_1,C_2,C_3\}$. Observe that
$$r_{K}(L)=\left(\sum_{j=1}^3r_{N_j}(L)\right)-r_{M_i}(L)=\left(\sum_{j=1}^3(|L\setminus L_j|-1)\right)-(|L|-2)=|L|-1.$$
Hence ${\cal C}_{\ge 2}(\{N|L\}_n)\bigtriangleup{\cal C}_{\ge 2}(\{M_i|L,K\})=\{C\}$, for some circuit $C$ of $K$. So, $C\in{\cal C}_{uni}(\{M\}_m)$. When $L=C$, then the result follows. We may suppose that there is $e\in L\setminus C$. In this case, we arrive at a contradiction, since $r_H(L\setminus\{e\})=r_H(L)$, for every $H\in\{K,N_1,N_2,N_3\}$ and $r_{M_i}(L)=r_{M_i}(L\setminus\{e\})+1$.\qeds

\bteo\rotulo{ae}
If $C\in {\cal C}_{uni}(\{M\}_m)$, then $C$ is a Tutte-line of $M_i$ containing exactly three circuits of $M_i$, for some $i\in\{1,2,\dots,m\}$, and these circuits are circuits of different $N_j$.
\eteo

\noindent {\bf Proof:} We do not loose generality when we suppose that $C$ is a circuit of $M_1$. By \recorde{ab}, $\{M|C\}_m$ is equivalent to $\{M_1|C,K\}$. By \recorde{ac} and \recorde{ad}, we have that 
$${\cal C}_{\ge 2}(\{M_1|C,K\})\bigtriangleup{\cal C}_{\ge 2}(\{N|C\}_n)=\{C\}.\eqno{\bf (1)}$$
Thus, 
$$\hbox{${\cal C}_{\ge 2}(K)={\cal C}_{\ge 2}(\{N\}_n)$.}\eqno{\bf (2)}$$
Observe that $K$ has a connected component $H$ such that $r(H)>0$ and $H$ is not a connected component of $N_i$, for every $i\in\{1,2,\dots,n\}$, otherwise 
${\cal A}(\{K',K\})={\cal A}(\{N|C\}_n)$, where $K'$ is a matroid over $C$ such that $r(K')=|C|$.

In this paragraph, we shall prove that $H$ has circuits $C'$ and $C''$ such that $$\{C',C''\}\not\subseteq{\cal C}(N_i),\hbox{ for every $i\in\{1,2,\dots,n\}$}.\eqno{\bf (3)}$$
If {\bf (3)} does not occur, then ${\cal C}(H)\subseteq{\cal C}(N_i)$, for some $i\in\{1,2,\dots,n\}$, since every element of ${\cal C}_{\ge 2}(\{N\}_n)$ has multiplicity one, by {\bf (2)}. So, $N_i$ has a connected component $H'$ such that ${\cal C}(H)\subseteq{\cal C}(H')$. By {\bf (2)}, ${\cal C}(H')\subseteq{\cal C}(K)$. Hence ${\cal C}(H')={\cal C}(H)$ and $H=H'$. We arrive at a contradiction and {\bf (3)} follows. Now, we shall prove that:
\vspace{0.3cm}

\noindent {\bf Step 1:} There are circuits $C_1$ and $C_2$ of $H$ such that $C_1\cup C_2$ is a connected Tutte-line of $H$, $C_1$ is a circuit of $N_i$ and $C_2$ is a circuit of $N_j$, for $j\not=i$, say $i=1$ and $j=2$. 
\vspace{0.3cm}

By {\bf (3)}, there are circuits $C'$ and $C''$ of $H$ such that, for every $i\in\{1,2,\dots,n\}$, $\{C,C''\}\not\subseteq{\cal C}(N_i)$. So, by \recorde{tc}, there is a sequence of circuits of $H$
$$C'=D_0,D_1,D_2,\dots,D_k=C''$$
such that $D_{i-1}\cup D_{i}$ is a connected Tutte-line of $H$, for every $i\in\{1,2,\dots,k\}$. Thus, for some $i\in\{1,2,\dots,k\}$, 
$\{D_{i-1},D_{i}\}\not\subseteq{\cal C}(N_j)$, for every $j\in\{1,2,\dots,n\}$. Take $C_1=D_{i-1}$ and $C_2=D_i$.
\vspace{0.3cm}

Let $\{L_1,L_2,\dots,L_k\}$ be the canonical partition of $L=C_1\cup C_2$. We may suppose that $C_1=L\setminus L_1$ and $C_2=L\setminus L_2$. As $L$ is connected, it follows that $k\ge 3$. By {\bf (1)}, we have that
$${\cal C}_{\ge 2}(\{M|L\}_m)\setminus\{C\}={\cal C}_{\ge 2}(\{N|L\}_n)={\cal C}(H|L)=\{L\setminus L_1,L\setminus L_2,\dots,L\setminus L_k\}.$$
\vspace{0.3cm}

\noindent {\bf Step 2:} For every $i\in\{1,2,\dots,n\}$, $|{\cal C}(H|L)\cap {\cal C}(N_i|L)|\le 1$.
\vspace{0.3cm}

Suppose that step 2 is not true. So, there is an $i\in\{1,2,\dots,n\}$ such that $|{\cal C}(H|L)\cap {\cal C}(N_i|L)|\ge 2$, say $i=2$. Let $C_3$ be a circuit of $N_2|L$ different from $C_2$. Observe that $(C_2\cup C_3)\setminus e$, for $e\in L_1$, does not contains a circuit of $N_2|L$, otherwise this circuit would be $C_1$. We arrive at a contradiction and the result follows.
\vspace{0.3cm}

Without loss of generality, we may suppose that $C_i=L\setminus L_i$ is a circuit of $N_i$, for $i\in\{1,2,\dots,k\}$. As an element $e$ of $E(\{M\}_m)$ cannot be a loop in at most two $N_i$'s, it follows that $k=3$. By \recorde{af}, $L\in{\cal C}_{uni}(\{M\}_m)$. Thus, $L=C$.\qeds

\bteo\rotulo{ag}
Suppose that $m=2$, $L$ is a connected Tutte-line of $M_2$ and $r([M_1|L]^*)\le 1$. If ${\cal C}_{uni}(\{M|L\}_m)=\emptyset$, then $L$ is a Tutte-line of $N_i$ having the same canonical partition as in $M_2$, for some $i\in\{1,2,\dots,n\}$.
\eteo

\noindent {\bf Proof:} By \recorde{ae}, ${\cal C}_{\ge 2}(\{N|L\}_n)\subseteq {\cal C}_{\ge 2}(\{M|L\}_m)$, since $m=2$. Hence, by hypothesis, $${\cal C}_{\ge 2}(\{N|L\}_n)={\cal C}_{\ge 2}(\{M|L\}_m).$$
Suppose that $\{L_1,L_2,\dots,L_k\}$ is the canonical partition of $L$ in $M_2$.
Consider the set $X=\{i:{\cal C}(N_i)\cap {\cal C}(M_2|L)\not=\emptyset\}$.

In this paragraph, we shall prove that $|X|\le 2$. Suppose that $|X|\ge 3$. If $k\ge 4$, then, for $l\in\{1,2,3\}$, there are $i_l$ and $j_l$ such that $|\{i_1,i_2,i_3\}|=|\{j_1,j_2,j_3\}|=3$ and $C_{j_l}\in {\cal C}(N_{i_l})\cap {\cal C}(M_2|L)$. Hence 
$$\sum_{i\in X}r_{N_i}(\{e\})\ge 3,$$
for every $e\in C_{j_1}\cap C_{j_2}\cap C_{j_3}$. We arrive at a contradiction. Thus, $k=3$ and we arrive at a contradiction by \recorde{af}. So, $|X|\le 2$. 

If $|X|=1$, then the result follows. Thus, we may suppose that $|X|=2$. We can order the sets on the canonical partition of $L$ in $M_2$ and the matroids in $\{N\}_n$ such that $C_1,\dots,C_t$ are circuits of $N_1$ and $C_{t+1},\dots,C_k$ are circuits of $N_2$. Without loss of generality, we may suppose that $t\ge 2$. Hence $L$ is a Tutte-line of $N_1$ having canonical partition $\{L_1,\dots,L_t,L\setminus(L_1\cup\cdots\cup L_t)\}$. If $t=k-1$, then $C'=C_k$, where $C'$ is the circuit of $M_1|L$, and the result follows. So, we may suppose that $t\le k-2$. Observe that $C'=L_1\cup\cdots\cup L_t$. Similarly, we have that $C'=L_{t+1}\cup\cdots\cup L_k$ and we arrive at a contradiction.\qed

If $K$ is a matroid and $Z\subseteq E(K)$, a $Z$-{\it arc} of $K$ is a minimal non-empty subset $A\subseteq E(K)\setminus Z$ such that there is a circuit $C$ of $K$ with $C\setminus Z=A$ and $C\cap Z\not=\emptyset$. A {\it fundamental for $A$} is a circuit $C$ such that $C\setminus Z=A$, where $A$ is a $Z$-arc. We denote by $arc_K(Z)$ the set of all $Z$-arcs of $K$. When $A\in arc_K(Z)$ and $P\subseteq Z$, we say that $A\rightarrow P$ in $K$ if there is a $Z$-fundamental for $A$ included in $A\cup P$. For its negation, we write $A\not\rightarrow P$.

\bteo\rotulo{ref}
Suppose that $L\in {\cal C}_{uni}(\{M\}_m)$. If $L$ is a circuit of $M_1$, $L$ is a Tutte-line of $M_2$ and $A$ is a $L$-arc of $M_2$ which does not contain properly a $L$-arc of $M_1$, then ${\cal C}_{uni}(\{M\}_m)\cap {\cal C}(M_1|(L\cup A))\not=\{L\}$.
\eteo

\noindent {\bf Proof:} Suppose that ${\cal C}_{uni}(\{M\}_m)\cap {\cal C}(M_1|(L\cup A))=\{L\}$. Without loss of generality, we may suppose that $E(\{M\}_m)=L\cup A$. Choose $a\in A$. By \recorde{ab}, there is a matroid $K$ such that $\{M/(A\setminus\{a\})\}_m$ is equivalent to $\{M_2/(A\setminus\{a\}),K\}$. Observe that $L\cup\{a\}$ is a connected Tutte-plane of $M_2/(A\setminus\{a\})$, since $A$ is a $L$-arc of $M_2$. As $A$ does not contain properly a $L$-arc of $M_1$, it follows that $L$ is a circuit of $K$ and $r(K^*)\le 2$.  Moreover, $a$ is a loop of $K$ or a coloop of $K$ or $L\cup\{a\}$ is a Tutte-line of $K$.

By \recorde{ae}, we may suppose that ${\cal C}(M_2|L)=\{C_1,C_2,C_3\}$ and that $C_i$ is a circuit of $N_i$, for every $i\in\{1,2,3\}$. By \recorde{ta}, it follows that there is a connected Tutte-line $L_i'$ of $M_2/(A\setminus\{a\})$ such that $C_i\subset L_i'\subset L\cup\{a\}$ and $L_i'\not=L$. Hence $|{\cal C}(M_1/(A\setminus\{a\})|L_i')|\le 1$, since $L_i'\subset L\cup\{a\}$. Thus, applying \recorde{ag} to the pair $(\{M\}_m,\{N\}_n)/(A\setminus\{a\})|L_i'$, we have that $L_i'$ is a connected line of $N_i/(A\setminus\{a\})$. We arrive at a contradiction because $a\in L_i'$, for every $i\in\{1,2,3\}$, and hence $r_{N_1}(\{a\})+r_{N_2}(\{a\})+r_{N_3}(\{a\})=3$.\qeds

\bteo\rotulo{ah}
Suppose that $L$ is a circuit of $M_1$, $L$ is a Tutte-line of $M_2$ having canonical partition $\{L_1,L_2,L_3\}$ and $C_i=L\setminus L_i$ is a circuit of $N_i$, for every $i\in\{1,2,3\}$. Then, $arc_{M_1}(L)=arc_{M_2}(L)$. Moreover, there is a partition $\{{\cal A}_1,{\cal A}_2,{\cal A}_3\}$ of $arc_{M_1}(L)$ such that 
\begin{description}

\item{(i)} $arc_{N_i}(L)=arc_{M_1}(L)\setminus {\cal A}_i$, for every $i\in\{1,2,3\}$;

\item{(ii)} for every $i\in\{1,2,3\}$ and $\{i,j,k\}=\{1,2,3\}$, if $A\in {\cal A}_i$, then $A\rightarrow L_i$ in $K$, for every $K\in\{M_1,M_2,N_j,N_k\}$.

\end{description}
\vspace{0.05cm}

\noindent {\bf Proof:} Observe that $L_i$ is a set of loops of $N_i$, for every $i\in\{1,2,3\}$, and that $L\in{\cal C}_{uni}(\{M\}_m)$. When $A\in arc_{M_i}(L)$, for some $i\in\{1,2\}$, then $M_i|(L\cup A)$ is connected. By \recorde{ab}, $\{M|(L\cup A)\}_m$ is equivalent to $\{M_i|(L\cup A),K_A\}$, for some matroid $K_A$ defined on $L\cup A$.
\vspace{0.3cm}

\noindent {\bf Step 1:} ${\cal C}_{\ge 2}(\{N|(L\cup A)\}_n)\subseteq {\cal C}_{\ge 2}(\{M|(L\cup A)\}_m)$, when $A\in arc_{M_1}(L)\cup arc_{M_2}(L)$.
\vspace{0.3cm}

As $A$ is a $L$-arc of $M_i$, for some $i\in\{1,2\}$, by \recorde{ae}, it follows that 
$${\cal C}_{\ge 2}(\{N|(L\cup A)\}_n)\subseteq {\cal C}_{\ge 2}(\{M_i|(L\cup A),K_A\}),$$ 
since $|\{M_i|(L\cup A),K_A\}|=2$ and the step follows. 
\vspace{0.3cm}

Observe that $arc_{M_1}(L)=arc_{M_2}(L)$ follows from the next step:
\vspace{0.3cm}

\noindent {\bf Step 2:} If $A$ is a minimal set belonging to $arc_{M_1}(L)\cup arc_{M_2}(L)$, then $A$ belongs to $arc_{M_1}(L)\cap arc_{M_2}(L)$.
\vspace{0.3cm}

Suppose that this step fails. Choose a minimal set $A$ of $arc_{M_1}(L)\cup arc_{M_2}(L)$ which does not belong to $arc_{M_1}(L)\cap arc_{M_2}(L)$. We have two cases to deal with: (a) $A\in arc_{M_1}(L)\setminus arc_{M_2}(L)$; or (b) $A\in arc_{M_2}(L)\setminus arc_{M_1}(L)$. In both cases, for $i\in\{1,2\}$, $\{a\}$ is a $L$-arc of $M_i|(L\cup A)/(A\setminus\{a\})$ if and only if $A$ is a $L$-arc in $M_i$. Now, we shall prove that
$${\cal C}_{\ge 2}(\{M|(L\cup A)/(A\setminus\{a\})\}_m)\bigtriangleup{\cal C}_{\ge 2}(\{N|(L\cup A)/(A\setminus\{a\})\}_n)=\{L\}.\eqno{\bf (1)}$$
By setp 1, we need only to prove that ${\cal C}_{uni}(\{M|(L\cup A)/(A\setminus\{a\})\}_m)=\{L\}$. In both cases, $a$ is a loop or coloop of $K_A/(A\setminus\{a\})$. By \recorde{ae}, if $C\in {\cal C}_{uni}(\{M|(L\cup A)/(A\setminus\{a\})\}_m)$, then $a\not\in C$. So, $C=L$ and we have {\bf (1)}. 

Suppose that (a) occurs. Let $L=D_0,D_1,D_2,\dots,D_s$ be the circuits of $M_1|(L\cup A)/(A\setminus\{a\})$ contained in $L\cup\{a\}$. As $D_i\cup D_j$ contains $L$, when $i\not=j$, and $L_i$ is a set of loops of $N_i$, it follows that each $N_i|(L\cup A)/(A\setminus\{a\})$ has at most one $D_j$ as a circuit. Thus, by {\bf (1)}, we may suppose that $s=2$ and $D_i$ is a circuit of $N_i|(L\cup A)/(A\setminus\{a\})$. Hence, by {\bf (1)}, $C_i\cap D_i=\emptyset$, otherwise $C_i\cup D_i$ contains another circuit of $M_1|(L\cup A)/(A\setminus\{a\})$. So, $(D_1\cup D_2)\cap L_3=\emptyset$ and we arrive at a contradiction. When (ii) happens, we arrive at a contradiction by \recorde{ref} and {\bf (1)}. Thus, step 2 follows.
\vspace{0.3cm}

\noindent {\bf Step 3:} If $A\in arc_{M_1}(L)$, then, there is an $i\in\{1,2,3\}$, such that $A\in arc_K(L)$ and $A\rightarrow L_i$ in $K$, for every $K\in\{M_1,M_2,N_j,N_k\}$, where $\{i,j,k\}=\{1,2,3\}$.
\eteo

By step 2, for every $A\in arc_{M_1}(L)$, $P=A\cup L$ is a Tutte-line of $M_1$ and $P$ is a Tutte-plane of $M_2$. By \recorde{ref}, there is $L'\in {\cal C}_{uni}(\{M\}_m)\cap {\cal C}(M_1|P)$ such that $L\not=L'$. By \recorde{ae}, $L'$ is a Tutte-line of $P$. Thus, the circuit of $M_2$ contained in $L\cap L'$, which exists by \recorde{tb}, must be equal to $C_i$, for some $i\in\{1,2,3\}$, say $i=1$. 

First, we shall determine the structure of the Tutte-plane $P$ of $M_2$ using the Tutte-lines $L$ and $L'$ of $M_2$. By \recorde{ae}, we may suppose that the other circuits contained in $L'$ are $C_2'$ and $C_3'$, with $C_i'$ being a circuit of $N_i$, for $i\in\{2,3\}$. By \recorde{ta}, there is a connected Tutte-line $L_i'$ contained in $P$ such that $C_i'\subset L_i'$ and $L_i'\not=L'$. The circuit of $M_2$ contained in $L_i'\cap L$ is $C_2$ or $C_3$. Observe that $L_i'$ does not contains a circuit of $N_1$, since $A$ is a set of loops of $N_1$ and every circuit of $M_2|L_i'$ different of $C_j$ contains $A$. By \recorde{ae}, ${\cal C}_{uni}(\{M\}_m)\cap {\cal C}(M_1|L_i')=\emptyset$. Hence, by \recorde{ag} applied to the pair $(\{M_1|P,M_2|P\},\{N|P\}_n)$, $L_i'$ is a Tutte-line of $N_2$ or $N_3$ having the same canonical partition. Thus, $L_i'=C_i\cup C_i'$, for $i\in\{2,3\}$. 

Let $C$ be the circuit of $M_2$ contained in $L_2'\cap L_3'$, which exists by \recorde{tb}. So, $C$ is a circuit of both $N_2$ and $N_3$. Hence $C$ is a circuit of both $M_1$ and $M_2$. Moreover, $C\cap (L_2\cup L_3)=\emptyset$. Hence $A\rightarrow L_1$ in $H$, for $H\in\{M_1,M_2,N_2,N_3\}$ because $C$ is a circuit of all these matroids and $A\subset C\subseteq A\cup L_1$. We have stablished step 3.
\vspace{0.3cm}

So, we get the partition $\{{\cal A}_1,{\cal A}_2,{\cal A}_3\}$ of $arc_{M_1}(L)$ setting:
$${\cal A}_i=\{A\in arc_{M_1}(L): A\rightarrow L_i\},$$
for $i\in\{1,2,3\}$, and the result follows, from step 1, step 2 and step 3.\qed

For the proof of theorem 1, we need the following result of Seymour:

\bteo\rotulo{seymour}
(3.8 of [7]) Let $K$ be a matroid, let $Z\subseteq E(K)$, and let $\{P_1,P_2\}$ be a partition of $Z$. Then either there is a $Z$-arc $A$ such that $A\not\rightarrow P_1,A\not\rightarrow P_2$, or there is a partition $\{X_1,X_2\}$ of $E(K)$ such that $X_i\cap Z=P_i (i=1,2)$ and 
$$r(X_1)+r(X_2)-r(K)=r(P_1)+r(P_2)-r(Z).\hbox{\qeds}$$
\vspace{0.05cm}

We shall apply Seymour's result only in the case that $Z\not=\emptyset$. Looking in the proof of this result, one concludes that $X_i$ is choosen such that
$$X_i=\bigcup\{P_i\cup A:A\in arc_Z(K)\hbox{ and }A\rightarrow P_i\hbox{ in $K$}\}.$$
So, when we refer to this result, we are supposing that the partitions are taken as described above. 
\vspace{0.3cm}

\noindent {\bf Proof of theorem 1:} As $(\{M\}_m,\{N\}_n)$ is non-preserving, it follows that ${\cal C}_{uni}(\{M\}_m)\not=\emptyset$ or ${\cal C}_{uni}(\{N\}_n)\not=\emptyset$, say ${\cal C}_{uni}(\{M\}_m)\not=\emptyset$. Chose $C\in {\cal C}_{uni}(\{M\}_m)$. Without loss of generality, we may suppose that $C$ is a circuit of $M_1$. By \recorde{ae}, $C$ is a line of $M_i$, for some $i\ge 2$, say $i=2$. Moreover, we may suppose that the canonical partition of $C$ in $M_2$ is $\{L_1,L_2,L_3\}$ and $C_i=L\setminus L_i$ is a circuit of $N_i$, for $i\in\{1,2,3\}$, say. Now, we shall define
$$X_i=\bigcup\{A\cup L_i:A\in arc_{M_1}(C)\hbox{ and }A\rightarrow L_i\hbox{ in $M_1$}\}.$$
By \recorde{ah}, we have that $\{X_1,X_2,X_3\}$ is a partition of $E(\{M\}_m)$, otherwise $\{E(\{M\}_m)\setminus (X_1\cup X_2\cup X_3), X_1\cup X_2\cup X_3\}$ is a 1-separation of $M_i$, for every $i\in\{1,2,\dots,m\}$. By \recorde{seymour}, when $|X_i|\ge 1$, we have that $\{X_i,E(\{M\}_m)\}$ is an exact 2-separation of $M_1,M_2$ and $N_j$, for $j\not=i$. The result follows, since the matroids that are decomposed using these 2-separations and has $X_i$ in its ground set are equal, by \recorde{ag}(ii) and \recorde{lem}.\qeds

\section{The preserving case}

Throughout this section, we shall suppose that $(\{M\}_m,\{N\}_n)$ is a non-trivial preserving pair. We shall denote $E(\{M\}_m)$ by $E$. Let $\rho$ be the rank function of the 2-polimatroid $P$ defined by this pair. 

\bteo\rotulo{ba}
If $\rho(e)=1$, for some $e\in E$, then ${\cal A}(\{M\}_m)\cap {\cal A}(\{N\}_n)\not=\emptyset$.
\eteo

\noindent {\bf Proof:} Let $M_i$ and $N_j$ be matroids such that $r_{M_i}(\{e\})=r_{N_j}(\{e\})=1$. Let $M'$ and $N'$ be the connected component of $M_i$ and $N_j$ that contains $e$, respectively. As $e$ is a loop of $M_k$ and $N_l$, for every $k\not=i$ and $l\not=j$, it follows that $M'$ and $N'$ have the same circuits containing the element $e$, since $(\{M\}_m,\{N\}_n)$ is preserving. Thus, $M'=N'$, by \recorde{lem}.\qeds

\bteo\rotulo{bb}
If $P$ is connected, then ${\cal A}(\{M\}_m)\cap {\cal A}(\{N\}_n)=\emptyset$.
\eteo

\noindent {\bf Proof:} Suppose that ${\cal A}={\cal A}(\{M\}_m)\cap {\cal A}(\{N\}_n)$. Observe that $({\cal A}(\{M\}_m)\setminus{\cal A},{\cal A}(\{N\}_n)\setminus{\cal A})$ is a non-trivial preserving pair which defines a 2-polimatroid $P'$ having rank function $\rho'$. By \recorde{ba}, $\rho'(\{e\})\not=1$, for every $e\in E$. So, $X=\{e\in E:\rho'(\{e\})=0\}$ and $Y=\{e\in E:\rho'(\{e\})=2\}$ is a partition of $E$. As $X$ is a set of loops, for every matroid in $({\cal A}(\{M\}_m)\cup {\cal A}(\{N\}_n))\setminus {\cal A}$, and $Y$ a set of loops, for every matroid in ${\cal A}$, it follows that $\{X,Y\}$ is a 1-separation for the 2-polimatroid $P$, when $X\not=\emptyset$ and $Y\not=\emptyset$. So, $X=\emptyset$ or $Y=\emptyset$. Thus, $X=\emptyset$, otherwise $(\{M\}_m,\{N\}_n)$ is trivial.\qed

For the proof of theorem 2, we need the following result of Lemos:

\bteo\rotulo{ml1}
(2.12 of [2]) If $(M_1,M_2,M_3,M_4)$ is a non-trivial connected quad, then there is at most one $i\in\{1,2,3,4\}$ such that $M_i$ is not a connected matroid. Moreover, when $M_i$ is not connected, it has exactly two connected components.\qeds
\eteo

\noindent {\bf Proof of theorem 2:} For a matroid $H$, we denote by $loop(H)$ the set of its loops. Now, we shall consider a maximal member $S$ of 
$${\cal S}=\{E\setminus loop(H): H\in{\cal A}(\{M\}_m)\cup {\cal A}(\{N\}_n)\}.$$
Without loss of generality, we may suppose that $M_1|S$ is connected. By \recorde{ab}, there is a matroid $K$ such that $\{M|S\}_m$ is equivalent to $\{M_1|S,K\}$. 

In this paragraph, we shall prove that $\{N|S\}_n$ is also equivalent to an admissible multiset containing two matroids. Suppose that this is not the case, and let $\{K\}_p$ be an admissible multiset of matroids equivalent to $\{N|S\}_n$ having $p$ minimum. Thus, $p\ge 3$. For each $i\in\{1,2,\dots,p\}$, set $L_i=E\setminus loop(K_i)$. So, for $\{i,j\}\subset\{1,2,\dots,p\}$, $L_i\cap L_j\not=\emptyset$, otherwise we could replace $K_i$ and $K_j$ by $K_i|L_i\oplus K_j|L_j\oplus K_i|(loop(K_i)\cap loop(K_j))$ in the admissible multiset $\{K\}_p$ and we arrive at a contradiction by the choice of $p$. Choose an element $a_{ij}$ belonging to $L_i\cap L_j$. Observe that $a_{ij}\in loop(K_l)$, for $l\not\in\{i,j\}$. Let $L$ be a connected Tutte-line of $M_1|S$  containing $\{e_{12},e_{13},e_{23}\}$. So, $|{\cal C}(N_i)\cap{\cal C}(M_1|L)|\le 1$ and $L$ contains only three circuits of $M_1$. We arrive at a contradiction, by \recorde{af}. 


%As $\{e_{12},e_{13},e_{23}\}$ cannot be contained in a circuit of $K_i$, it follows that each element of $\{e_{12},e_{13},e_{23}\}$ belongs to a diferent set of the canonical patition $\{L_1,L_2,\dots,L_k\}$ of $L$. Consider the pair
%$(\{M|L\cdot\{e_{12},e_{13},e_{23}\}\}_m,\{N|L\cdot\{e_{12},e_{13},e_{23}\}\}_n)$. As $\{e_{12},e_{13},e_{23}\}$ is a line of %$M_1|L\cdot\{e_{12},e_{13},e_{23}\}$ and %$\{e_{12},e_{13}\},\{e_{12},e_{23}\},\{e_{13},e_{23}\}$ are circuits of diferents $N_i|L\cdot\{e_{12},e_{13},e_{23}\}$, it follows, by \recorde{af}, that $\{e_{12},e_{13},e_{23}\}\in {\cal C}_{uni}(\{M|L\cdot\{e_{12},e_{13},e_{23}\}\}_m)$. We have a contradiction, since $L\in {\cal C}_{uni}(\{M\}_m)$.

Let $(\{M'\}_2,\{N'\}_2)$ be a pair such that $\{M|S\}_m$ and $\{N|S\}_n$ are equivalent to $\{M'\}_2$ and $\{N'\}_2$, respectively. So, $(M_1',M_2',N_1',N_2')$ is a quad. It must be non-trivial, otherwise $M_1'=N_1'$, say, and by the choice of $S$, $M_1'$ and $N_1'$ are connected components of some matroid in $\{M\}_m$ and $\{N\}_n$, recpectively. But this is contrary to \recorde{bb}. By \recorde{ml1}, $M_1'$ and $M_2'$ are connected or $N_1'$ and $N_2'$ are connected, say $M_1'$ and $M_2'$ are connected. Thus, by the choice of $S$, for $i\in\{1,2\}$, $M_i'$ is a connected component of $M_i$. So, $S$ is a separator for both $M_1$ and $M_2$. Hence $\{S,E\setminus S\}$ is a 1-separation for $P$, unless $S=E$. The result follows.\qeds

\section{The proof of the colloraries}

Throughout this section, we shall suppose that $(\{M\}_m,\{N\}_n)$ is a non-trivial pair which defienes a connected 2-polimatroid $P$. We shall denote $E(\{M\}_m)$ by $E$. Let $\rho$ be the rank function of $P$. First, we shall state a result of Lemos [?]:

\bteo\rotulo{ml2}
(theorem 2 of [1]) If $(M_1,M_2,M_3,M_4)$ is a non-trivial connected quad, then 
$$r(M_1)=r(M_2)=r(M_3)=r(M_4).\hbox{\qeds}$$
\vspace{0.05cm}

\bteo\rotulo{ca}
If $(\{M\}_m,\{N\}_n)$ is non-preserving, then it is the unique pair defining $P$.
\eteo

\noindent {\bf Proof:} By theorem 1, we may suppose that $m=2$ and $n=3$. Let $(\{M'\}_m,$ $\{N'\}_n)$ be another non-trivial pair that defines $P$. When $(\{M'\}_{m'},\{N'\}_{n'})$ is non-preseving, we may suppose that $m'=2$ and $n'=3$. By theorem 1, $\{N'\}_{n'}$ is equivalent to $\{N\}_n$. So, $(M_1,M_2,M_1',M_2')$ is a non-trivial connected quad. We arrive at a contaridiction, by \recorde{ml2}, because $r(M_1)\not=r(M_2)$. Thus, $(\{M'\}_{m'},\{N'\}_{n`})$ is preserving. By theorem 2, we may suppose that $m'=n'=2$. In this case, we also arrive at the same contradiction, by \recorde{ml2}, since $(M_1,M_2,M_1',M_2')$ is a non-trivial connected quad.\qed

All the corollaries are imediate consequences of \recorde{ca} and theorem 1 in the case that $(\{M\}_m,\{N\}_n)$ is non-preseving. Thus, we may suppose that every pair of admissible multisets which decomposes $P$ is preserving. By theorem 2, we do not loose generality, if we suppose that $m=n=2$. So, $(M_1,M_2,N_1,N_2)$ is a non-trivial quad. 

\bteo\rotulo{ml3}
(2.2 of [2]) If $(M_1,M_2,M_3,M_4)$ is a non-trivial connected quad, then $M_i$ is loopless, for every $i\in\{1,2,3,4\}$.\qeds
\eteo

Corollary 1 follows \recorde{ml1}. Corollary 2 from \recorde{ml2}. Corollary 3 from \recorde{ml3}. Corollary 4 from {\bf (4.2)} of Lemos [2]. We shall not state the last result of Lemos here, since it envolves some tecnical definitions. The interested reader should consult the original paper. Moreover, the binary connected 2-polimatroid with two non-equivalent decompositions are completely described by Lemos result.

\section{Acknowledgements}

The author would like to thank Professor James Oxley for some discussions on the topic of this paper and all the staff of the Department of Mathematics of Louisiana State University for their hospitality.

\section{References}

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\item{[1]} M.Lemos, {\it Matroids having the same connectivity function}, Discrete Mathematics 131 (1994), 153-161.

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\item{[5]} U.S.R.Murty and I.Simon, {\it A $\beta$-function that is not a sum of rank functions of matroids}, Probl\`emes Combinatoiries et Th\'eorie des Graphes (eds. J.-C.Bermond, J.-C.Fournier, M. Las Vergnas and D. Sotteau), Editions du CNRS, Paris, 1978, 305-306.

\item{[6]} J.Oxley, {\it Matroid Theory}, Oxford University Press, Oxford, 1992.

\item{[7]} P.D.Seymour, {\it Decomposition of regular matroids}, Journal of Combinatorial Theory Series B 28 (1980), 305-359.

\item{[8]} P.D.Seymour, {\it On the connectivity function of a matroid}, Journal of Combinatorial Theory Series B 45 (1988), 25-30.

\item{[9]} W.T.Tutte, {\it Lectures on matroids}, Journal of Research of the National Bureau of Standards Series B 69 (1965), 1-47.

\end{description}

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