Regular matrices and their strong preservers over semirings

An m×n matrix A over a semiring S is called regular if there is an n×m matrix G over S such that AGA=A. We study the problem of characterizing those linear operators T on the matrices over a semiring such that T(X) is regular if and only if X is. Complete characterizations are obtained for many semirings including the Boolean algebra, the nonnegative reals, the nonnegative integers and the fuzzy scalars.