Chapter 1
Introduction
It is known that most of practical problems within the fields of economics， engineering， medical sciences， environmental sciences mvolve data that contain uncertainties.For this reason we can not successfully use traditional mathematical tools. In order to solve these problems， many scientists have put forth some special tools suchas probability theory， fuzzy set theory[354]， rough set theory and soft set theory .
1.1Hemirings
A serni''ri''ng is an algebraic system S， +， . consisting of a non-empty set S together ith two binary operations on S called addition and multiplication denoted in he usual manner such that S，+ and S， . are semigroups and the following distributive laws
a b + c = a . b+ b . c and a + b . c= a-c+b-c
are satisfied for all a， b， c e S.
By zero of a semiring we mean an element 0E S such that O.x - x d O+x - x+0 - x for all x∈ S. A semiring with zero is called a hemiring if S， + is commutative. For the sake of simplicity， we shall omit the symbol :e . "，writing ab for a . b a， b∈ S. If A and B are non-empty subsets in S， let AB denote he set of all finite sums {albl+a2b2+-+anb7.~n∈ N， ai E A， bi E B}. Throughout this book， S is always a hemiring.
A non-empty subset A of S is called a teft resp.， ''right ideat of S if A is closed under addition and SA至 A resp.， AS∈ A. Further， A is called an ideal of S if it is both a left ideal and a right ideal of S. A non-empty subset B of S is called a bi-ideal of S if B is closed under addition and multiplication satisfying BSB∈ B. A non-empty subset Q of S is called a qZLasi-ideal of S if Q is closed under addition and SQ n oS∈ Q. A subset A of S is called an interior ideal of S if A is closed under addition and multiplication such that SAS c A.
The h-closure A ofA in S is defined as
b+z for some a，be A， z∈ S}
A subhemiring A of S is called a stro''ng h-s''ubhemiring if and +a+z - y+b+z implies x c y+A. SDong h-ideals are defined similarly. Clearly，every strong h-subhemiring h-ideal is an h-subhemiring h-ideal. A quasi-ideal Q f S is called an h-quasi-ideal of S if SQ n SQ∈ Q and x + a + z = b + z impliesx E Q for all x，z∈ Sja，b∈ Q.
Lernma l.1.1 For a hemiring S， we have
1 A∈ A， where {0}∈ A c S;
2 If A∈ B C S， then A C B;
3 A - A， VA c S;
4 AB - AB;
5
6
For any left right h-ideal， h-bi-ideal， h-quasi-ideal or h-interior ideal A of
Proof We only show 4， the other properties can be easily proved. Since and ， then AB c AB， and so .
To show the converse inclusion， let x E A and y e B. Then there exist ai， a2EA， bi， b2 E B and zi，such that and Then we have
imply aiy， a2y E AB. Thus that that is， AB∈ AB. This completes the proof.
A left ideal resp.， right ideal， ideal， bi-ideal， quasi-ideal， interior ideal A of S is called a teft h-ideal resp.， right h-ideal， h-ideal， h-bi-ideal， h-quasi-ideal， h-interio-rideal if x， z∈ S， a，b∈ A， and x + a + z = b+ z implies x E A.
Remark l.1.2 1 Every h-ideal h-bi-ideal， h-quasi-ideal of S is an ideal bi-ideal， quasi-ideal ， respectively;
2 Every h-ideal of S is an h-quasi-ideal of S;
3 Every h-quasi-ideal of S is an h-bi-ideal of S;
4 Every h-ideal of S is an h-interior ideal of S.
However， the converse of the above remark does not hold in general as shown in the following examples.
Example l.1.3 Let S = {0， a， b} be a set with an addition operation + and a multiplication operation . as follows:
Then S is a hemiring. Let A一 {0， b}. Evidently A is an ideal of S and it is not an h-ideal of S， since a+ 0 + b = 0+ b while ag A.
Example l.1.4 Let No be the set of all non-negative integers and S be the set of all 2 x 2 matrices. Then S is a heiniring with respect to the usual addition and multiplication of matrices. Consider the set Q of all matrices of the form.Evidently Q is an h-quasi-ideal of S and it is not a left right h-ideal of S.
Example l.1.5 We denote by N and p the sets of all positive integers and positive real numbers， respectively. The set S of all matrices of the form together with is a hemiring with respect to the usual addition and multiplication of matrices. Let R and L be the sets of all matrices b 0 together with t and together with ， respe，tively. It is easy to show that R and L are a right h-ideal and a left h-ideal of S， respectively. Now the product RL is an h-bi-ideal of S and it is not an h-quasi-ideal of S. Indeed， theelement Chapter I Introduction belongs to the intersection SRL Cl RS， but it is not an element of R. Hence
SRL -l RLS垡 RL
Example l.1.6 Let S = and a multiplication operation be a set with an addition operation + follows:
Then S is a hemiring. Let I = {0， a} is not an h-ideal of S， since a e i， b Evidently I is an h-interior ideal of S and itS while a . b .
Definition l.1.7 A hemiring S is said to be h-he''miregula''r if for each x∈ S there exist a， a， z E S such that
Lernrna l.1.8 If A and B are， respectively， a right and a left h-ideal of S then AB c A -l B.
Proof and Since
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