(∈γ,∈γ∨qδ)-intuitionistic Fuzzy(Soft)Filter of BL-algebras

§1. Introduction

Since filter theory plays an important role in studying logic algebras,many researchers combine it with mathematical approaches to discuss its generalized properties.Liu and Li[1]applied the concept of fuzzy sets to filter theory and proposed the notions of fuzzy filters and fuzzy prime filters in BL-algebras.Using the notions of membership and quasicoincidence of fuzzy points related with fuzzy sets,Ma and Zhan[2]presented(∈,∈∨q)-fuzzy filters in BL-algebras.Farther more,Yin and Zhan[3]introduced(α,β)-fuzzy filters in BL-algebras where α,β∈(∈γ,qδ,∈γ∧qδ,∈γ∨qδ)andα/=∈γ∧qδ.Zhan and Jun[4]applied the notion of soft setsto the theory of BL-algebras and introduced the notions of soft BL-algebras based on fuzzy sets,then they investigated some characterizations of filteristic soft BL-algebras.

In this paper,in order to study filter theory in BL-algebras more comprehensively,we introduce the notion of(∈γ,∈γ∨qδ)-intuitionistic fuzzy filters in BL-algebras,then investigate some oftheir properties.Based on soft sets,we give the definition of(∈γ,∈γ∨qδ)-intuitionistic fuzzy filters in BL-algebras and study the intuitionistic fuzzy soft image and intuitionistic fuzzy soft inverse image of(∈γ,∈γ∨qδ)-intuitionistic fuzzy filters of BL-algebras.

§2. Preliminaries

In this section,we review some basic notions and results which will be needed in the sequel.

Recall that an algebra L=(L,∧,∨,⊙,→,0,1)is a BL-algebra[5]if it is a bounded lattice such that

(1)(L,⊙,1)is a commutative monoid;

(2)⊙a(bǔ)nd→form an adjoin pair,i.e.,x≤y→z if and only if x⊙y≤z for all x,y,z∈L;

(3)x∧y=x⊙(x→y);

(4)(x→y)∨(y→x)=1.

A non-empty subset A of L is called a filter of L if it satisfies the conditions

(1)?x,y∈A?,x⊙y∈A;

(2)?x∈A,x≤y?y∈A.

A fuzzy set f of X is a function f:X→[0,1](see[6]).

Defi nition 2.1[1]Let f be a fuzzy set in BL-algebra L.f is called a fuzzy filter if f satisfies the following conditions

(1)?x,y∈A,f(x⊙y)≥min{f(x),f(y)};

(2)f is order-preserving,that is,if x≤y,then f(y)≥f(x).

Defi nition 2.2[7]Let X be a non-empty fixed set.An intuitionistic fuzzy set A is an object having the form

whereμA:X → [0,1]andλA:X → [0,1]denote the degrees of membership and nonmembership of x∈X to the set A,respectively and 0≤μA(x)+λA(x)≤1 for each x∈X.

The set ofallintuitionistic fuzzy sets in X is denoted by I F(X).For the sake ofsimplicity, we use A=〈μA,λA〉for A={〈x,μ(x),λ(x)〉|x∈X}.

Defi nition 2.3[8]Let A,B∈I F(X).Then the intersection A∧B={〈x,min{μA(x), μB(x)},max{λA(x),λB(x)〉|x∈X},the union A∨B={〈x,max{μA(x),μB(x)},min{λA(x),λB(x)〉|x∈X}.

Defi nition 2.4[9]Let r,t∈[0,1)be two real numbers such that 0≤r+t≤1.An intuitionistic fuzzy set A in a set X of the form

is called an intuitionistic fuzzy point with the support x and value〈r,t〉,denoted by x(r,t).

Letγ,δ∈[0,1]be such thatγ＜δ.For an intuitionistic fuzzy point x(r,t)and A∈I F(X), we say

(1)x(r,t)∈γA ifμA(x)≥r＞γandλA(x)≤t＜1?γ;

(2)x(r,t)qδA ifμA(x)+r＞2δandλA(x)+t＜2?2δ;

(3)x(r,t)∈γ∨qδA if x(r,t)∈γA or x(r,t)qδA;

Based on the concept ofsoft sets[10],Gunduz and Bayramov gave the concept ofintuitionistic fuzzy soft sets and their operations as follows.

Defi nition 2.5[11]Let U be an initial universe and E be a set of parameters,A?E. Then a pair(F,A)is called an intuitionistic fuzzy soft set over U,where F is a mapping given by F:A→I F(U).

Defi nition 2.6[11]The intersectionand the unionoftwo intuitionistic fuzzy soft sets(F,A)and(G,B)over U are intuitionistic fuzzy soft sets,defined as

respectively,for allε∈C(C′),where C=C′=A∪B.

§3. (∈γ,∈γ∨qδ)-intuitionistic Fuzzy(Soft) Filters of BL-algebras

In what follows,let L be a BL-algebra and E be a parameter set related to objects in L unless other statements.

Defi nition 3.1 An intuitionistic fuzzy set A= 〈μA,λA〉of L is called(∈γ,∈γ∨qδ}-intuitionistic fuzzy filter of L if for all r1,r2∈(0,1],t1,t2∈[0,1)and x,y∈L,

(F1)x(r1,t1)∈γA and y(r2,t2)∈γA?x⊙y(min{r1,r2},max{t1,t2})∈γ∨qδA;

(F2)x(r1,t1)∈γA?y(r1,t1)∈γ∨qδA with x≤y.

Example 3.2 Let L={0,a,b,c,1}be a chain where 0＜a＜b＜c＜1.For all x,y∈L,we define x∧y=min{x,y},x∨y=max{x,y}and⊙a(bǔ)nd→as follows

then(L,∧,∨,⊙,→)is a BL-algebra.Define an intuitionistic fuzzy set A in L as

It is easy to show A=〈μA,λA〉is an(∈0.3,∈0.3∨q0.4)-intuitionistic fuzzy filter of L.

Proposition 3.3 Let 2δ=1+γand A be an(∈γ,∈γ∨qδ)-intuitionistic fuzzy filter of L, then?Aγ={x∈L|μA(x)＞γandλA(x)＜1?γ}/=?is a filter of L.

Proof Let A is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy filter of L.Let x,y∈?Aγ,thenμA(x)＞γ,λA(x)＜1?γandμA(y)＞γ,λA(y)＜1?γ.Suppose thatμA(x⊙y)≤γ,λA(x⊙y)≥1?γ, then x(μA(x),λA(x))∈γA,y(μA(y),λA(y))∈γA and

Theorem 3.4 If I is a non-empty set of L and 2δ=1+γ.Then I is a filter of L if and only if the intuitionistic fuzzy set A=〈μA,λA〉of L such that

(1)μA(x)≥δandλA(x)≤1?δfor all x∈I;

(2)μA(x)=γandλA(x)=1?γotherwise

is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy filter of L.

Proof Assume that I is a filter of L.Let x,y∈I and r1,r2∈(γ,1],t1,t1∈[0,1?γ)be such that x(r1,t1),y(r2,t2)∈γA.ThenμA(x)≥r1＞γ,λA(x)≤t1＜1?γand so x∈I. Similarly,it can be proved that y∈I.

Since I is a filter of L,then x⊙y∈I,that isμA(x⊙y)≥δandλA(x⊙y)≤1?δ.If min{r1,r2}≤δand max{t1,t2}≥1?δ,thenμA(x⊙y)≥δ≥min{r1,r2}＞γandλA(x⊙y)≤1?δ≤max{t1,t2}＜1?γ,i.e.,x⊙y(min{r1,r2},max{t1,t2})∈γA.Ifmin{r1,r2}＞δand max{t1,t2}＜1?δ,thenμA(x⊙y)+min{r1,r2}＞2δandλA(x⊙y)+max{t1,t2}＜2?2δ. Thus x⊙y(min{r1,r2},max{t1,t2})qδA.Hence x⊙y(min{r1,r2},max{t1,t2})∈γ∨qδA.

Now let x,y∈L and r∈(γ,1],t∈[0,1?γ)be such that x≤y and x(r,t)∈γA.we can prove that x∈I in the above way.Since I is a filter of L,then y∈I,which impliesμA(y)≥δ andλA(y)≤1?δ.If r≤δand t≥1?δ,thenμA(y)≥δ≥r＞γandλA(y)≤1?δ≤t＜1?γ, i.e.,y(r,t)∈γA.If r＞δand t＜1?δ,thenμA(y)+r＞2δandλA(y)+t＜2?2δ,i.e., y(r,t)qδA.Hence y(r,t)∈γ∨qδA.

Therefore,A=〈μA,λA〉is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy filter of L.

Conversely,suppose that A is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy filter of L.It is easy to see thatHence,I is a filter of L.

Theorem 3.5 Let A=〈μA,λA〉be an intuitionistic fuzzy set of L.Then A is an (∈γ,∈γ∨qδ)-intuitionistic fuzzy filter of L if and only if the following conditions hold:for all x,y∈L,

(P1) max{μA(x⊙y),γ}≥min{μA(x),μA(y),δ}and min{λA(x⊙y),1?γ}≤max{λA(x), λA(y),1?δ};

(P2) max{μA(y),γ}≥min{μA(x),δ}and min{λA(y),1?γ}≤max{λA(x),1?δ}with x≤y.

Proof Let A be an(∈γ,∈γ∨qδ)-intuitionistic fuzzy filter of L.Suppose that(P1)does not hold,then there exist x,y∈L such that max{μA(x⊙y),γ}＜r=min{μA(x),μA(y),δ}, min{λA(x⊙y),1?γ}＞t=max{λA(x),λA(y),1?δ}.Hence,μA(x)≥r＞γ,μA(y)≥t＞γ, μA(x⊙y)＜r andμA(x⊙y)+r＜2r≤2δ,λA(x)≤t＜1?γ,λA(y)≤t＜1?γ,λA(x⊙y)＞t andλA(x⊙y)+t＞2t≥2?2δ.That is,x(r,t)∈γA,y(r,t)∈γA,buta contradiction.Thus,(P1)holds.

Let x≤y.Suppose that max{μA(y),γ}＜r=min{μA(x),δ}and min{λA(y),1?γ}＞t= max{λA(x),1?δ}.ThenμA(x)≥r＞γ,μA(y)＜r,μA(y)+r＜2r≤2δandλA(x)≤t＜1?γ, λA(y)＞t,μA(y)+t＞2t≥2?2δ,i.e.,x(r,t)∈γA buta contradiction. Hence,(P2)is valid.

Conversely,suppose that(P1)and(P2)hold.Let x,y∈L,r1,r2∈(γ,1]and t1,t2∈[0,1?γ)be such that x(r1,t1)∈γA,y(r2,t2)∈γA.Then we getμA(x)≥ r1＞ γ, λA(x)≤t1＜1?γ,μA(y)≥r2＞γ,λA(y)≤t2＜1?γ,

Now,if min{r1,r2}≤δand max{t1,t2}≥1?δ,thenμA(x⊙y)≥min{r1,r2}andλA(x⊙y)≤ max{t1,t2},i.e.,x⊙y(min{r1,r2},max{t1,t2})∈γA.If min{r1,r2}＞ δandmax{t1,t2}＜1?δ,thenμA(x⊙y)+min{r1,r2}≥δ+min{r1,r2}＞2δandλA(x⊙y)+ max{t1,t2}≤1?δ+max{t1,t2}＜2?2δ,i.e.,x⊙y(min{r1,r2},max{t1,t2})qδA.Hence, x⊙y(min{r1,r2},max{t1,t2})∈γ∨qδA and so(F1)is satisfied.

Let x≤y such that x(r,t)∈γA,where x,y∈L,r∈(γ,1]and t∈[0,1?γ).Then we haveμA(x)≥r＞γ,λA(x)≤t＜1?γ,max{μA(y),γ}≥min{μA(x),δ}≥max{r,δ}, min{λA(y),1?γ}≤max{λA(x),1?δ}≤{t,1?δ}.

Now,if r≤δand t≥1?δ,thenμA(y)≥r andλA(y)≤t,i.e.,y(r,t)∈γA.If r＞δand t＜1?δ,thenμA(y)+r≥δ+r＞2δandλA(y)+t≤1?δ+t＜2?2δ,i.e.,y(r,t)qδA. Hence,y(r,t)∈γ∨qδA and so(F2)holds.

For any intuitionistic fuzzy set A=〈μA,λA〉of L and r∈(γ,1]and t∈[0,1?γ),we denote A(r,t)={x∈L|x(r,t)∈γA}as∈γ-levelset,as qδ-levelset and

The following theorem and corollary presentthe relationshipsbetween(∈γ,∈γ∨qδ)-intuitionistic fuzzy filters and crisp filters of L.

Theorem 3.6 Let 2δ=1+γand A=〈μA,λA〉be an intuitionistic fuzzy set of L.Then A is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy filter of L if and only ifis a filter of L for each r∈(γ,1],t∈[0,1?γ).

Proof Assume that A is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy filter of L.Letfor some r∈(γ,1],t∈[0,1?γ),then x(r,t)∈γA or x(r,t)qδA.We haveμA(x)≥r＞γ, λA(x)≤t＜1?γorμA(x)＞2δ?r≥2δ?1=γ,λA(x)＜2?2δ?t≤2?2δ=1?γ.Since A is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy filter of L,thusμA(x⊙y)≥min{μA(x),μA(y),δ} andλA(x⊙y)≤max{λA(x),λA(y),1?δ}.We consider two cases.

Case 1 r∈(γ,δ],t∈[1?δ,1?γ).Then 2δ?r≥δ＞r,2?2δ?t≤1?δ＜t＜1?γand soμA(x)≥r,μA(y)≥r,λA(x)≤t,λA(y)≤t.Thus,μA(x⊙y)≥min{μA(x),μA(y),δ}≥r andλA(x⊙y)≤max{λA(x),λA(y),1?δ}≤t and so x⊙y(r,t)∈γA.

Case 2 r∈(δ,1],t∈[0,1?δ).Then 2δ?r＜δ＜r,2?2δ?t＞1?δ＞t and μA(x)＞2δ?r,μA(y)＞2δ?r,λA(x)＜2?2δ?t,λA(y)＜2?2δ?t.ThusμA(x⊙y)≥min{μA(x),μA(y),δ}＞2δ?r andλA(x⊙y)≤max{λA(x),λA(y),1?δ}＜2?2δ?t and so x⊙y(r,t)qδA.

Therefore,x⊙y(r,t)∈γ∨qδA,i.e.,It can be showed thatand x≤y impliesin the similar way.Hence,is a filter of L.

Therefore,(P1)holds.(P2)can be proved in the similar way.Thus,A is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy filter of L.

Corollary 3.7 Let A=〈μA,λA〉be an intuitionistic fuzzy set of L.

(1)A is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy filter of L if and only if A(r,t)(/=?)is a filter of L for each r∈(γ,δ],t∈[1?δ,1?γ).

(2)If 2δ=1+γ,then A is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy filter of L if and only ifis a filter of L for each r∈(δ,1],t∈[0,1?δ).

In the following,we apply the concept of intuitionistic fuzzy soft sets to filter theory and propose the notion of(∈γ,∈γ∨qδ)-intuitionistic fuzzy soft filters of BL-algebras.

Defi nition 3.8 Let(F,A)be an intuitionistic fuzzy soft set of L,where A?E.Then (F,A)is called an(∈γ,∈γ∨qδ)-intuitionistic fuzzy soft filter of L if F(ε)is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy filter of L for eachε∈A.

Proposition 3.9 Let(F,A)and(G,B)be two(∈γ,∈γ∨qδ)-intuitionistic fuzzy soft filters of L,where A,B?E.If C=A∪B,thenis an(∈γ,∈γ∨qδ)-intuitionistic fuzzy soft filter of L.

Proof Since(F,A)and(G,B)are(∈γ,∈γ∨qδ)-intuitionistic fuzzy soft filters of L and for eachε∈C,we consider three cases.

Case 1 Ifε∈A,then H(ε)=F(ε)is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy soft filter of L.

Case 2 Ifε∈B,then H(ε)=G(ε)is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy soft fi lter of L.

Case 3 Ifε∈A∩B,then F(ε)and G(ε)are(∈γ,∈γ∨qδ)-intuitionistic fuzzy filters of L.Let x,y∈L,then

and

Let x,y∈L be such that x≤y.Similarly,we can prove that max{(μF(ε)∧μG(ε))(y),γ}≥min{(μF(ε)∧μG(ε))(x),δ}and min{(λF(ε)∨λG(ε))(y),1?γ}≤max{(λF(ε)∨λG(ε))(x),1?δ}.

Hence,F(ε)∧G(ε)is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy filter of L.

Therefore,in any case,H(ε)is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy soft filter of L.Asεis arbitrary,thusis an(∈γ,∈γ∨qδ)-intuitionistic fuzzy soft filter of L.

Proposition 3.10 Let(F,A)and(G,B)be two(∈γ,∈γ∨qδ)-intuitionistic fuzzy soft filters of L,where A,B?E.If A∩B=?,thenis an(∈γ,∈γ∨qδ)-intuitionistic fuzzy soft filter of L.

Proof Since A∩B=?,for eachε∈C,thenε∈A?B orε∈B?A.Since(F,A)and (G,B)are(∈γ,∈γ∨qδ)-intuitionistic fuzzy soft fi lters of L,ifε∈A?B,then H(ε)=F(ε)is an (∈γ,∈γ∨qδ)-intuitionistic fuzzy filter of L.Ifε∈B?A,then H(ε)=G(ε)is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy filter of L.Hence,in any case,H(ε)is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy filter of L.Asεis arbitrary,(H,C)is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy soft filter of L.

In the above theorem,if A∩B/=?,(H,C)maybe not an(∈γ,∈γ∨qδ)-intuitionistic fuzzy soft filter of L.To show this,we present the following example.

Example 3.11 Let L={0,a,b,c,d,1}be a chain,where 0＜b＜a＜1,0＜d＜a＜1 and 0＜d＜c＜1.Define x∧y=min{x,y},x∨y=max{x,y}and⊙a(bǔ)nd→as follows

Then(L,∧,∨,⊙,→,0,1)is a BL-algebra(see[3]).Define two intuitionistic fuzzy sets A and B in L byμA(0)=0.2, μA(a)=0.7,μA(b)=0.6,μA(c)=0.2,μA(d)=0.3,μA(1)=0.7, λA(0)=0.5,λA(a)=0.2,λA(b)=0.3,λA(c)=0.5,λA(d)=0.5,λA(1)=0.1 andμB(0)=0.1, μB(a)=0.3,μB(b)=0.25,μA(c)=0.4,μB(d)=0.2,μB(1)=0.6,λB(0)=0.4,λB(a)=0.2, λB(b)=0.4,λB(c)=0.1,λB(d)=0.3,λB(1)=0.2.

Defi ne intuitionistic fuzzy soft sets(F,A′)and(G,B′),where A′={ε1,ε2}and B′= {ε1,ε3}as

Then F(ε1)and G(ε1)are two(∈0.3,∈0.3∨q0.6)-intuitionistic fuzzy filters of L,F(ε1)∨G(ε1)is not an(∈0.3,∈0.3∨q0.6)-intuitionistic fuzzy filter since b(0.6,0.3)∈0.3F(ε1)∨G(ε1), c(0.4,0.1)∈0.3F(ε1)∨G(ε1),but.Hence,(F,A)■(G,B)= (H,C)is not an(∈0.3,∈0.3∨q0.6)-intuitionistic fuzzy soft filter of L.

§4.The Image and Inverse Image of (∈γ,∈γ∨qδ)-intuitionistic Fuzzy Soft Filters of BL-algebras

In what follows,we denote C(L,E),C(S,E′)and C(T,E′′)as the classes of intuitionistic fuzzy soft sets of L,S and T with parameters from E,E′and E′′respectively,where L,S and T are BL-algebras.

Defi nition 4.4 Let f=(u,p):C(L,E)→C(S,E′)be a mapping,where u:L→S and p: E→E′are two mappings.For(G,B)∈C(S,E′),the inverse image of(G,B)under f,denoted by f?1(G,B),is an intuitionistic fuzzy soft set of L defined by f?1(G,B)=(u?1(G),p?1(B)), whereμu?1(G)(ε)(x)=μG(p(ε))(u(x))andλu?1(G)(ε)(x)=λG(p(ε))(u(x)),for allε∈p?1(B)and x∈L.

Defi nition 4.2 Let f=(u,p):C(L,E)→ C(S,E′)be a mapping,where u:L→ S and p:E→E′are mappings.For(F,A)∈C(L,E),the image of(F,A)under f,denoted by f(F,A),is an intuitionistic fuzzy soft set of S defined by f(F,A)=(u(F),p(A)),where

for allη∈p(A)and y∈S.

Theorem 4.3 Let f(u,p):C(L,E)→ C(S,E′)be a mapping,where p:E→ E′is a mapping and u is a homomorphic mapping from L to S.If(G,B)is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy soft fi lter of S,then f?1(G,B)is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy soft filter of L.

Proof For eachε∈p?1(B),i.e.,p(ε)∈B,then G(p(ε))is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy filter of S.For all x,y∈L,

Let x,y∈L be such that x≤y,then

Thus,u?1(G)(ε)is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy filter of L. Asεis arbitrary, f?1(G,B)is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy soft filter of L.

Theorem 4.4 Let f:C(L,E)→C(S,E′)be a mapping,p:E→E′be a mapping and u be a surjective homomorphism from L to S.If(F,A)is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy soft filter of L,then f(F,A)is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy soft filter of S.

Proof For eachη∈B and y1,y2∈S,then

and

If y1≤y2,similarly,we can prove that max{μu(F)(η)(y2),γ}≥min{μu(F)(η)(y1),δ}and min{λu(F)(η)(y2),1?γ}≤max(λu(F)(η)(y1),1?δ}.

Thus u(F)(η)is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy filter of S.Asηis arbitrary,f(F,A)is an(∈γ,∈γ∨qδ)-intuitionistic fuzzy soft filter of S.

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YANG Yong-wei,XIN Xiao-long,HE Peng-fei

(Department of Mathematics,Northwest University,Xi’an 710127,China)

In the paper,in order to further study the properties of filters of BL-algebras, we propose the concepts of the(∈γ,∈γ∨qδ)-intuitionistic fuzzy filters and(∈γ,∈γ∨qδ)-intuitionistic fuzzy soft fi lters of BL-algebras and derive some related results.Finally,we discuss the properties of images and inverse images of(∈γ,∈γ∨qδ)-intuitionistic fuzzy soft filters of BL-algebras.

BL-algebras;intuitionistic fuzzy sets;(∈γ,∈γ∨qδ)-intuitionistic fuzzy filters; soft sets

tion:03E72,08A72

1002–0462(2014)01–0065–11

date:2012-09-11

Supported by the Graduate Independent Innovation Foundation of Northwest University(YZZ12061)

Chin.Quart.J.of Math. 2014,29(1):65—75

Biographies:YANG Yong-wei(1984-),male,native of Shangqiu,Henan,a Ph.D.candidate of Northwest University,engages in fuzzy algebras,logic algebras;XIN Xiao-long(1955-),male,native of Xi’an,Shaanxi,a professor of Northwest University,Ph.D.,engages in logic algebras;HE Peng-fei(1986-),male,native of Xi’an, Shaanxi,a Ph.D.candidate of Northwest University,M.S.D.,engages in fuzzy algebras,rough set theory.

CLC number:O159 Document code:A