Invertible and nilpotent matrices over antirings

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.

9 pages, 1 figure, minor changes