Invertible and nilpotent matrices over antirings

TL;DR

This paper characterizes invertible matrices over commutative antirings, analyzes nilpotent matrices, and explores their decomposition into sums of square-zero matrices, extending matrix theory to antiring structures.

Contribution

It provides a comprehensive characterization of invertible matrices over commutative antirings and establishes decomposition results for nilpotent matrices into square-zero matrices.

Findings

Characterization of invertible matrices over commutative antirings

Number of nilpotent matrices over finite antirings

Decomposition of nilpotent matrices into sums of square-zero matrices

Abstract

In this paper we characterize invertible matrices over an arbitrary commutative antiring S and find the structure of GL_n (S). We find the number of nilpotent matrices over an entire commutative finite antiring. We prove that every nilpotent $n \times n$ matrix over an entire antiring can be written as a sum of $\lceil \log_2 n \rceil$ square-zero matrices and also find the necessary number of square-zero summands for an arbitrary trace-zero matrix to be expressible as their sum.