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

Description

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

Steps to reproduce

The csv file of matrix factorizations can be replicated by running "Bisymmetric Factorization.py" in Python. The variables "m" and "M" give the range of minimum values of the bisymmetric matrix to be factored.

Institutions

• Millersville University

  Pennsylvania, Millersville
• Bryn Mawr College

  Pennsylvania, Bryn Mawr
• Kutztown University

  Pennsylvania, Kutztown

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