Factorizations of bisymmetric matrices with relatively prime entries in the semigroup of non-negative integer matrices

The file "Bisymmetric Factorization.py" implements a factorization algorithm of a two-by-two bisymmetric matrix with relatively prime entries, to search for factorizations into non-negative integer matrices. Details can be found the be paper "Atoms in the Semigroup of Non-Negative Integer Matrices" by Dever, Goedhart, Heilbrunn, and Wong. Given a range of minimum entries, the algorithm will search all two-by-two bisymmetric matrices with relatively prime entries and find all factorable matrices. All matrices with relatively prime entries that are not found by the algorithm are necessarily atoms. By the main theorem of our paper, the factors will necessarily be bisymmetric. The files labeled "BisymDatamtoM.csv" contain the factorizations of such matrices with minimum values ranging from m to M. The columns listed are x, y, a, b, c, d where (x y; y x) = (a b; b a)(c d; d c) is the factorization into two-by-two bisymmetric matrices. Rows are sorted in ascending order by x, y values. Only matrices with x<y are stored, although there will be multiple versions of the same factorization, since associates of the factors or commuted matrices are also stored. The total number of factorizations from each file is: Minimums up to 1000: 5,461,852 factorizations Minimums from 1001 to 2000: 21,278,318 factorizations Minimums from 2001 to 3000: 40,244,970 factorizations Minimums from 3001 to 4000: 61,115,352 factorizations文件`Bisymmetric Factorization.py`实现了一种针对元素互质的二阶双对称矩阵（bisymmetric matrix）的分解算法，用于搜索其到非负整数矩阵的分解方式。详细内容可参阅Dever、Goedhart、Heilbrunn及Wong发表的论文《Atoms in the Semigroup of Non-Negative Integer Matrices》。 给定元素最小值的范围，该算法将遍历所有元素互质的二阶双对称矩阵，并找出所有可分解的矩阵。所有经该算法未找到分解方式的元素互质矩阵，必然为半群原子。根据我们论文中的核心定理，分解得到的因子矩阵必然为双对称矩阵。 命名为`BisymDatamtoM.csv`的文件存储了元素最小值介于m到M之间的此类矩阵的分解结果。文件列依次为x、y、a、b、c、d，对应分解式$(x y; y x) = (a b; b a)(c d; d c)$，即分解为两个二阶双对称矩阵的乘积。 数据行按x、y值升序排列，且仅存储x<y的矩阵。由于同时存储了因子的相伴矩阵与交换后的矩阵，因此同一分解可能存在多个版本。 各文件包含的分解结果总数如下： - 元素最小值不超过1000：5,461,852组分解结果 - 元素最小值介于1001至2000之间：21,278,318组分解结果 - 元素最小值介于2001至3000之间：40,244,970组分解结果 - 元素最小值介于3001至4000之间：61,115,352组分解结果