\begin{algorithm}
\label{algor2}Let $C=\left(  c_{ij}\right)  $ be an $m\times n$ BPM with
non-null maximal equalizer $E=(e_{ij})$.

\indent For each $c_{ij}=\frac{B_{1}^{ij}|B_{2}^{ij}|\cdots|B_{r}^{ij}}%
{A_{1}^{ij}|A_{2}^{ij}|\cdots|A_{r}^{ij^{\mathstrut}}}$ in $C:$

\indent\indent Let $e$ be the first element in $e_{ij}$.

\indent \indent Let $a^{ij}_{1} | \cdots| a^{ij}_{p} := \mathcal{EP}
_{IS(c_{ij})}(A^{ij}_{e + 1} \cup\cdots\cup A^{ij}_{r})$.

\indent \indent Let $b^{ij}_{1} | \cdots| b^{ij}_{q} := \mathcal{EP}
_{OS(c_{ij})}(B^{ij}_{1} \cup\cdots\cup B^{ij}_{e})$.

\indent\indent For $k$ from $1$ to $p:$

\indent \indent \indent Let $a^{ij}_{k, e + 1} | \cdots| a^{ij}_{k, r} :=
(A^{ij}_{e + 1} \cap a^{ij}_{k}) | \cdots| (A^{ij}_{r} \cap a^{ij}_{k})$.

\indent\indent For $k$ from $1$ to $q:$

\indent\indent\indent Let $b_{k,1}^{ij}|\cdots|b_{k,e}^{ij}:=(B_{1}^{ij}\cap
b_{k}^{ij})|\cdots|(B_{e}^{ij}\cap b_{k}^{ij})$.

\indent\indent Define the formal matrix decomposition
\[
C_{1}^{ij}C_{2}^{ij}=%
\begin{pmatrix}
\frac{b_{1,1}^{ij}|\cdots|b_{1,e}^{ij}}{A_{1}^{ij}|\cdots|A_{e}%
^{ij^{\mathstrut}}}\\
\vdots\\
\frac{b_{q,1}^{ij}|\cdots|b_{q,e}^{ij}}{A_{1}^{ij}|\cdots|A_{e}%
^{ij^{\mathstrut}}}%
\end{pmatrix}%
\begin{pmatrix}
\frac{B_{e+1}^{ij}|\cdots|B_{r}^{ij}}{a_{1,e+1}^{ij}|\cdots|a_{1,r}%
^{ij^{\mathstrut}}} & \cdots & \frac{B_{e+1}^{ij}|\cdots|B_{r}^{ij}}%
{a_{p,e+1}^{ij}|\cdots|a_{p,r}^{ij^{\mathstrut}}}%
\end{pmatrix}
.
\]
The \textbf{transverse decomposition of} $C$ \textbf{with respect to} $E$ is
the formal matrix factorization $C:=C_{1}C_{2}$ where
\[
C_{1}=\left(  C_{1}^{ij}\right)  =%
\begin{pmatrix}
\frac{b_{1,1}^{11}|\cdots|b_{1,e}^{11}}{A_{1}^{11}|\cdots|A_{e}%
^{11^{\mathstrut}}} &  & \frac{b_{1,1}^{1n}|\cdots|b_{1,e}^{1n}}{A_{1}%
^{1n}|\cdots|A_{e}^{1n^{\mathstrut}}}\\
\vdots & \cdots & \vdots\\
\frac{b_{q,1}^{11}|\cdots|b_{q,e}^{11}}{A_{1}^{11}|\cdots|A_{e}%
^{11^{\mathstrut}}} &  & \frac{b_{q,1}^{1n}|\cdots|b_{q,e}^{1n}}{A_{1}%
^{1n}|\cdots|A_{e}^{1n^{\mathstrut}}}\\
\vdots &  & \vdots\\
\frac{b_{1,1}^{m1}|\cdots|b_{1,e}^{m1}}{A_{1}^{m1}|\cdots|A_{e}%
^{m1^{\mathstrut}}} &  & \frac{b_{1,1}^{mn}|\cdots|b_{1,e}^{mn}}{A_{1}%
^{mn}|\cdots|A_{e}^{mn^{\mathstrut}}}\\
\vdots & \cdots & \vdots\\
\frac{b_{q,1}^{m1}|\cdots|b_{q,e}^{m1}}{A_{1}^{m1}|\cdots|A_{e}%
^{m1^{\mathstrut}}} &  & \frac{b_{q,1}^{mn}|\cdots|b_{q,e}^{mn}}{A_{1}%
^{mn}|\cdots|A_{e}^{mn^{\mathstrut}}}%
\end{pmatrix}
\]
and $C_{2}=\left(  C_{2}^{ij}\right)  =$
\[%
\begin{pmatrix}
\frac{B_{e+1}^{11}|\cdots|B_{r}^{11}}{a_{1,e+1}^{11}|\cdots|a_{1,r}%
^{11^{\mathstrut}}} & \cdots & \frac{B_{e+1}^{11}|\cdots|B_{r}^{11}}%
{a_{p,e+1}^{11}|\cdots|a_{p,r}^{11^{\mathstrut}}} & \cdots & \frac
{B_{e+1}^{1n}|\cdots|B_{r}^{1n}}{a_{1,e+1}^{1n}|\cdots|a_{1,r}^{1n^{\mathstrut
}}} & \cdots & \frac{B_{e+1}^{1n}|\cdots|B_{r}^{1n}}{a_{p,e+1}^{1n}%
|\cdots|a_{p,r}^{1n^{\mathstrut}}}\\
& \vdots &  &  &  & \vdots & \\
\frac{B_{e+1}^{m1}|\cdots|B_{r}^{m1}}{a_{1,e+1}^{m1}|\cdots|a_{1,r}%
^{m1^{\mathstrut}}} & \cdots & \frac{B_{e+1}^{m1}|\cdots|B_{r}^{m1}}%
{a_{p,e+1}^{m1}|\cdots|a_{p,r}^{m1^{\mathstrut}}} & \cdots & \frac
{B_{e+1}^{mn}|\cdots|B_{r}^{mn}}{a_{1,e+1}^{mn}|\cdots|a_{1,r}^{mn^{\mathstrut
}}} & \cdots & \frac{B_{e+1}^{mn}|\cdots|B_{r}^{mn}}{a_{p,e+1}^{mn}%
|\cdots|a_{p,r}^{mn^{\mathstrut}}}%
\end{pmatrix}
.
\]
\medskip

\noindent Using first elements of the entries in $E_{1}:=E$ compute the
transverse decomposition $C=C_{1}C_{2};$ then $C_{1}$ is indecomposable. If
$C_{2}$ is decomposable, its maximal equalizer $E_{2}$ is non-null. Using the
first elements of the entries in $E_{2},$ calculate the transverse
decomposition $C_{2}:=C_{2}^{\prime}C_{3};$ then $C_{2}^{\prime}$ is
indecomposable. Continue recursively $s-1$ steps until the maximal equalizer
is null. The \textbf{indecomposable factorization} is $C=C_{1}C_{2}^{\prime
}\cdots C_{s-1}^{\prime}C_{s}$.
\end{algorithm}

\begin{algorithm}
Let $C=\left( c_{ij}\right) $ be an $m\times n$ BPM with
non-null maximal equalizer $E=(e_{ij})$.
\indent For each $c_{ij}=\frac{B_{1}^{ij}|B_{2}^{ij}|\cdots|B_{r}^{ij}}%
{A_{1}^{ij}|A_{2}^{ij}|\cdots|A_{r}^{ij^{\mathstrut}}}$ in $C:$
\indent\indent Let $e$ be the first element in $e_{ij}$.
\indent \indent Let $a^{ij}_{1} | \cdots| a^{ij}_{p} := \mathcal{EP}
_{IS(c_{ij})}(A^{ij}_{e + 1} \cup\cdots\cup A^{ij}_{r})$.
\indent \indent Let $b^{ij}_{1} | \cdots| b^{ij}_{q} := \mathcal{EP}
_{OS(c_{ij})}(B^{ij}_{1} \cup\cdots\cup B^{ij}_{e})$.
\indent\indent For $k$ from $1$ to $p:$
\indent \indent \indent Let $a^{ij}_{k, e + 1} | \cdots| a^{ij}_{k, r} :=
(A^{ij}_{e + 1} \cap a^{ij}_{k}) | \cdots| (A^{ij}_{r} \cap a^{ij}_{k})$.
\indent\indent For $k$ from $1$ to $q:$
\indent\indent\indent Let $b_{k,1}^{ij}|\cdots|b_{k,e}^{ij}:=(B_{1}^{ij}\cap
b_{k}^{ij})|\cdots|(B_{e}^{ij}\cap b_{k}^{ij})$.
\indent\indent Define the formal matrix decomposition
\[
C_{1}^{ij}C_{2}^{ij}=%
\begin{pmatrix}
\frac{b_{1,1}^{ij}|\cdots|b_{1,e}^{ij}}{A_{1}^{ij}|\cdots|A_{e}%
^{ij^{\mathstrut}}}\\
\vdots\\
\frac{b_{q,1}^{ij}|\cdots|b_{q,e}^{ij}}{A_{1}^{ij}|\cdots|A_{e}%
^{ij^{\mathstrut}}}%
\end{pmatrix}%
\begin{pmatrix}
\frac{B_{e+1}^{ij}|\cdots|B_{r}^{ij}}{a_{1,e+1}^{ij}|\cdots|a_{1,r}%
^{ij^{\mathstrut}}} & \cdots & \frac{B_{e+1}^{ij}|\cdots|B_{r}^{ij}}%
{a_{p,e+1}^{ij}|\cdots|a_{p,r}^{ij^{\mathstrut}}}%
\end{pmatrix}
.
\]
The \textbf{transverse decomposition of} $C$ \textbf{with respect to} $E$ is
the formal matrix factorization $C:=C_{1}C_{2}$ where
\[
C_{1}=\left( C_{1}^{ij}\right) =%
\begin{pmatrix}
\frac{b_{1,1}^{11}|\cdots|b_{1,e}^{11}}{A_{1}^{11}|\cdots|A_{e}%
^{11^{\mathstrut}}} & & \frac{b_{1,1}^{1n}|\cdots|b_{1,e}^{1n}}{A_{1}%
^{1n}|\cdots|A_{e}^{1n^{\mathstrut}}}\\
\vdots & \cdots & \vdots\\
\frac{b_{q,1}^{11}|\cdots|b_{q,e}^{11}}{A_{1}^{11}|\cdots|A_{e}%
^{11^{\mathstrut}}} & & \frac{b_{q,1}^{1n}|\cdots|b_{q,e}^{1n}}{A_{1}%
^{1n}|\cdots|A_{e}^{1n^{\mathstrut}}}\\
\vdots & & \vdots\\
\frac{b_{1,1}^{m1}|\cdots|b_{1,e}^{m1}}{A_{1}^{m1}|\cdots|A_{e}%
^{m1^{\mathstrut}}} & & \frac{b_{1,1}^{mn}|\cdots|b_{1,e}^{mn}}{A_{1}%
^{mn}|\cdots|A_{e}^{mn^{\mathstrut}}}\\
\vdots & \cdots & \vdots\\
\frac{b_{q,1}^{m1}|\cdots|b_{q,e}^{m1}}{A_{1}^{m1}|\cdots|A_{e}%
^{m1^{\mathstrut}}} & & \frac{b_{q,1}^{mn}|\cdots|b_{q,e}^{mn}}{A_{1}%
^{mn}|\cdots|A_{e}^{mn^{\mathstrut}}}%
\end{pmatrix}
\]
and $C_{2}=\left( C_{2}^{ij}\right) =$
\[%
\begin{pmatrix}
\frac{B_{e+1}^{11}|\cdots|B_{r}^{11}}{a_{1,e+1}^{11}|\cdots|a_{1,r}%
^{11^{\mathstrut}}} & \cdots & \frac{B_{e+1}^{11}|\cdots|B_{r}^{11}}%
{a_{p,e+1}^{11}|\cdots|a_{p,r}^{11^{\mathstrut}}} & \cdots & \frac
{B_{e+1}^{1n}|\cdots|B_{r}^{1n}}{a_{1,e+1}^{1n}|\cdots|a_{1,r}^{1n^{\mathstrut
}}} & \cdots & \frac{B_{e+1}^{1n}|\cdots|B_{r}^{1n}}{a_{p,e+1}^{1n}%
|\cdots|a_{p,r}^{1n^{\mathstrut}}}\\
& \vdots & & & & \vdots & \\
\frac{B_{e+1}^{m1}|\cdots|B_{r}^{m1}}{a_{1,e+1}^{m1}|\cdots|a_{1,r}%
^{m1^{\mathstrut}}} & \cdots & \frac{B_{e+1}^{m1}|\cdots|B_{r}^{m1}}%
{a_{p,e+1}^{m1}|\cdots|a_{p,r}^{m1^{\mathstrut}}} & \cdots & \frac
{B_{e+1}^{mn}|\cdots|B_{r}^{mn}}{a_{1,e+1}^{mn}|\cdots|a_{1,r}^{mn^{\mathstrut
}}} & \cdots & \frac{B_{e+1}^{mn}|\cdots|B_{r}^{mn}}{a_{p,e+1}^{mn}%
|\cdots|a_{p,r}^{mn^{\mathstrut}}}%
\end{pmatrix}
.
\]
\medskip
\noindent Using first elements of the entries in $E_{1}:=E$ compute the
transverse decomposition $C=C_{1}C_{2};$ then $C_{1}$ is indecomposable. If
$C_{2}$ is decomposable, its maximal equalizer $E_{2}$ is non-null. Using the
first elements of the entries in $E_{2},$ calculate the transverse
decomposition $C_{2}:=C_{2}^{\prime}C_{3};$ then $C_{2}^{\prime}$ is
indecomposable. Continue recursively $s-1$ steps until the maximal equalizer
is null. The \textbf{indecomposable factorization} is $C=C_{1}C_{2}^{\prime
}\cdots C_{s-1}^{\prime}C_{s}$.
\end{algorithm}
{
    "title": "computing the dimension of a bipartition matrix",
    "id": "2111.05799",
    "abstract": "this article presents a computer program that computes the dimension of a bipartition matrix. its dimension has three independent components: row dimension, column dimension, and entry dimension. the program applies four routines of independent interest: (1) a routine that factors a bipartition as a formal product of indecomposables, (2) a routine that computes the inverse and recovers the bipartition, (3) a routine that factors a bipartition matrix as a formal product of indecomposables, and (4) a routine that calculates the \"transpose-rotation\" of a bipartition matrix (the column dimension is the row dimension of the transpose-rotation).",
    "categories": "math.co math.at",
    "doi": "",
    "created": "2021-11-09",
    "updated": "2024-01-29",
    "authors": [
        "dawson freeman",
        "ronald umble"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2111.05799"
}
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