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 characterization of idempotent matrices over semirings dates back to 1963, see [3], where idempotent Boolean relation matrices of finite order were characterized from the graph theoretical point of view. In parallel, in 1969 structural characterization of idempotent matrices over the semiring of the nonnegative reals was obtained in [1] in terms of a special rectangle block structure of matrices. Later, in [4] the characterization of Boolean idempotent matrices was given in terms of quasi-orders and in [2] it was provided in terms of limit-dominating matrices. However it was an open problem to obtain a structural characterization of Boolean idempotent matrices, analogous to the result obtained in [1]. We solve this problem for matrices over the binary Boolean semiring. Some related results and applications will be presented. In particular, we construct minimal idempotent envelope of a given matrix and describe all matrices that are majorized by a given idempotent matrix with respect to a minus order. We will also address some applications and open problems.