| The invertible matrix over a semiring is one of important topics in algebra.It has been widely used in cryptography,optimization theory and networking,etc.As a generalization of invertible matrix,we introduce the concept of e-invertible matrix over a semiring,where e is a non-zero multiplicative idempotent element.The e-invertible matrix over a commutative semiring is studied,and the main results are as follows:First,the structure of semigroup that e-invertible matrices with respect to matrix multiplication is studied.By a congruence relation,we considered the maximal subgroups of this matrix semigroup,proved the existence of its maximal subgroups,and gave a characterization about these maximal subgroups.Also,the decomposition theorem of this matrix semigroup is obtained.Second,e-invertible matrix over a commutative semiring is discussed.The characterizations of e-invertible matrices over general commutative semirings and the commutative semirings with ε-function are given,respectively.Third,e-invertible matrix over a extensive semiring is investigated.According to the relationship about,c-invertible matrix between a commutative semiring and its extensive semiring,and the characterization of e-invertible ma-trix over the commutative semiring with ε-function,another characterization of e-invertible matrix over a commutative semiring is obtained. |
