On strongly invertible matrices over semirings.

A square matrix over a semiring is called strongly invertible if all of its leading principal submatrices are invertible. In this paper, the strongly invertible matrices over a semiring are discussed and an equivalent condition for a square matrix over a semiring to be strongly invertible is given. Also, some equivalent descriptions are obtained for a semiring over which the product of any two strongly invertible matrices with the same size is strongly invertible. Some of the results obtained in this paper generalize and develop previous results for matrices over the field of complex numbers and matrices over commutative rings. [ABSTRACT FROM AUTHOR]