Idempotent matrices over antirings

Invertible and Nilpotent Matrices over Antirings

Abstract. In this paper we characterize invertible matrices over an arbitrary commutative antiring S with 1 and find the structure of GLn(S). We find the number of nilpotent matrices over an entire commutative finite antiring. We prove that every nilpotent n×n matrix over an entire antiring can be written as a sum of ⌈log2 n⌉ square-zero matrices and also find the necessary number of square-zer...

On nilpotency of matrices over antirings

Article history: Received 13 May 2010 Accepted 1 June 2010 Available online 4 July 2010 Submitted by R.A. Brualdi AMS classification: 15A15 15A18

On idempotent matrices over semirings

Idempotent matrices play a significant role while dealing with different questions in matrix theory and its applications. It is easy to see that over a field any idempotent matrix is similar to a diagonal matrix with 0 and 1 on the main diagonal. Over a semiring the situation is quite different. For example, the matrix J of all ones is idempotent over Boolean semiring. The first characterizatio...

Spectral Lattices of Reducible Matrices over Completed Idempotent Semifields

Previous work has shown a relation between L-valued extensions of FCA and the spectra of some matrices related to L-valued contexts. We investigate the spectra of reducible matrices over completed idempotent semifields in the framework of dioids, naturally-ordered semirings, that encompass several of those extensions. Considering special sets of eigenvectors also brings out complete lattices in...

Sheaves over Right Sided Idempotent Quantales