Abstract:

The authors proved that a simple inverse semigroup of a matrix is actually a group of a matrix by the method of matrix diagonalization;  a 0-simple inverse semigroup of a matrix is actually a 0-group when the zero is a prime element;   and  a semigroup of matrix is a completely 0simple inverse semigroup if and only if it is isomorphic to the brandt semigroup B_n which the trivial group corresponds to by characterizing complete 0-simple inverse semigroups of matrix via Rees construction.

Key words: semigroups of matrix, inverse semigroups, simple semigroups, 0-simple semigroups