Regular matrices and their strong preservers over semirings
Regular elements in a ring as generalized inverse elements were originated by E. H. Moore in 1935. Let \(R\) be a ring. An element \(a\) in this ring is called regular if there is an element \(x\in R\) such that \(axa=a\). When \(x\) exists, it is called a generalized inverse of \(a\). This concept has been considered and studied in rings of matrices. Some authors characterized regular matrices over semirings (R. B. Bapat, J. Denes, K. H. Kim,\dots). The authors of the paper under review consider characterization of linear operators, \(T\) say, on the matrices over semirings, that preserve regularity, i.e. those linear operators, \(T\) for which a matrix \(A\) over a semiring is regular if and only if \(T(A)\) is regular. Their results include the Boolean algebra, the nonnegative reals, the nonnegative integers and the fuzzy scalars.
scientific article; zbMATH DE number 5285770