Some Topics On Permanents Of Matrices Over Semirings

In recent years, as the development of research on semirings, permanents of matrices over semirings have become a research focus. An algebraic system (R,+,·) is called a semiring if (R,+) is commutative monoid with identity 0 and (R,·) is another monoid with identity 1,0 is not equal to 1, connected by ring-like distributivity, and x·0= 0·x= 0 for any x in R. Semirings not only include ring and field but also include Boolean algebra, fuzzy algebra, distributive lattice and incline. And matrices over rings, Boolean matrices, fuzzy matrices, lattice matrices, incline matrices and nonnegative matrices are the special examples of matrices over an semiring. In previous research for permanents of matrices over some special semirings, one used different methods to discuss the permanents of matrices for different types of semirings. These methods have no generality. This paper will use unified idea and method to study permanents of matrices and adjoint matrices over commutative semirings and generalize some previous results for permanents of matrices over some special semirings to general commutative semiring (or difference ordered semiring).The paper is organized as follows:In section 1, some necessary concepts and basic lemmas are introduced. In section 2, permanents of matrices over a commutative semiring are discussed and some identities for permanents of the matrices are given. Especially, some inequalities for permanents of the matrices over a commutative difference ordered semiring are obtained. Also, An upper bound and a lower bound for the permanent of any matrix over a commutative total ordered semiring are given and some equivalent descriptions for a matrix over an commutative integral difference ordered semiring to satisfy that its permanent equals zero are obtained. In section 3, the adjoint matrix of a matrix over a commutative difference ordered semiring is studied and some inequalities for the adjoint matrix are obtained. The main results obtained in this paper generalize the corresponding results for fuzzy matrices, for lattice matrices, for incline matrices and for nonnegative matrices.