Atoms in the semigroup of non-negative integer matrices

Abstract: In the semigroup of $2\times 2$ matrices with non-negative integer entries and non-zero determinant, we study the factorization of matrices into atoms, or irreducible matrices. In 2022, Baeth et al. discovered classes of atoms in this semigroup; however, the factorability of most matrices remains unknown. As the result of joint work with Eva Goedhart, Gregory Heilbrunn, and Tony W. H. Wong, I will discuss additional classes of atoms: a class of atoms with determinant $p$, $2p$, or $4p$, where $p$ is prime, and a class of atoms where the main diagonal is much ``larger'' than the off-diagonal (or vice-versa). In addition, we find that bisymmetric matrices with relatively prime entries are a divisor-closed subset and use a factor-search algorithm to classify bisymmetric atoms with minimum entry up to 4000.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)