GONDRAN-MINOUX ENVELOPING RANK OF MATRICES ON SEMIRINGS

Abstract

In semiring theory, rank of matrices and its characteristic properties have played an important role in the semirings structure analysis and have achieved many interesting results on the class of commutative semirings, including Gondran-Minoux rank and Gondran-Minoux enveloping rank of matrices. These rank functions have been considered on the class of entire zerosumfree semirings such as max-plus semiring, extensions of the max-plus semiring, quasi-selective semiring without zero divisors, etc. However, there are not many research results about Gondran-Minoux enveloping rank of matrices over general semirings now. In this paper, we review definitions which relate to Gondran-Minoux enveloping rank of matrices, considering several characteristic inequalities of Gondran-Minoux enveloping column rank of matrices on class of commutative semirings, comparing with factor rank of matrices, indicating the necessary and sufficient conditions for Gondran-Minoux enveloping column rank and factor rank of all matrices to coincide, indicate several cases of Gondran-Minoux enveloping column rank and Gondran-Minoux enveloping row rank equals.