Generalized TL fuzzy rough rings via TL fuzzy relational morphisms

R E S E A R C H

Open Access

Generalized

T

L-fuzzy rough rings via

T

L-fuzzy relational morphisms

Canan Ekiz

_{, Yıldıray Çelik}

_{and Sultan Yamak}

*_{Correspondence:} [email protected] 1_{Department of Mathematics,} Giresun University, Giresun, 28100, Turkey

2_{Department of Mathematics,} Karadeniz Technical University, Trabzon, 61080, Turkey

Abstract

In this paper, we introduce the notion ofTL-fuzzy (full) relational morphisms of rings and investigate some properties ofI-lower andT-upper fuzzy rough approximations on rings with respect to them.

MSC: 20M99; 03E72; 16W20

Keywords: rough sets;I-lowerL-fuzzy rough approximations;T-upperL-fuzzy rough approximations;TL-fuzzy (full) relational morphisms; rings;TL-fuzzy rough rings

1 Introduction

The theory of rough sets proposed by Pawlak [] is a new mathematical approach for the study of incomplete or imprecise information. The usefulness of rough sets has been demonstrated by some successful applications in many areas such as knowledge discovery, machine learning, data analysis, approximate classification, conflict analysis, and so on. Equivalence relation is a key notion in Pawlak’s rough set model. The equivalence classes are employed to construct lower and upper approximations. Some constructive and alge-braic properties of rough sets are investigated by using equivalence relations. However, the requirement of an equivalence relation in Pawlak’s rough set models seems to be a very restrictive condition that may limit the applications. Thus one of the aims of many re-searchers working on rough set theory has been to generalize this theory. Pawlak’s model has been generalized by use of arbitrary relations in place of equivalence relations, or by use of covering or neighborhood systems or set-valued mappings in place of equivalence classes (see [–]).

Fuzzy set theory which was introduced by Zadeh [] in  and was generalized by Goguen [] in  is another mathematical tool to cope with the vague concepts via grading the turbidity. Dubois and Prade [] introduced the problem communicating with the fuzzy sets and the rough sets. Many researchers have dealt with the integration of rough sets and fuzzy sets which are two distinct and complementary theories. For a deep study of fuzzy rough sets, the reader is referred to [–].

Biswas and Nanda [] applied the notion of rough sets to algebra and introduced the notion of rough subgroups. Kuroki [] introduced the notion of a rough ideal in a semi-group. In [, ], Davvaz analyzed a relationship between rough sets and ring theory considering a ring as a universal set and introduced the notion of rough ideals and rough subrings with respect to an ideal of a ring. By considering a ring as a universal set, Liet al.

[] studied (ν,T)-fuzzy rough approximation operators with respect to aTL-fuzzy ideal of the ring. Li and Yin [] generalized lower and upper approximations toν-lower and T-upper fuzzy rough approximations with respect toT-congruenceL-fuzzy relation on a semigroup. In order to have a more flexible tool for analysis of an information system, recently, Davvaz has studied the concept of generalized rough sets called by himT-rough sets []. This is another generalization of rough sets. In this type of generalized rough sets, instead of equivalence relations, we require set-valued maps. This technique is use-ful, where it is difficult to find an equivalence relation among the elements of the universe set. In this generalized rough sets, a set-valued map gives rise to lower and upper gener-alized approximation operators. Aliet al.[] studied some topological properties of the sets which are fixed by these operators. They also studied the degree of accuracy (DAG) for generalized rough sets and some properties of fuzzy sets which are induced by DAG. Davvaz, in [], also introduced the concept of set-valued homomorphisms for groups, which is a generalization of an ordinary homomorphism. Yamaket al.[, ] investigated some properties of rough approximations with respect to set-valued homomorphisms of rings and modules in the perspective of set-valued homomorphisms. In [], Aliet al.

initiated the study of roughness in hemirings with respect to the Pawlak approximation space and also with respect to the generalized approximation space.

Recently, Ignjatovićet al.[] have introduced the notion of a (relational morphism) fuzzy relational morphism which is more general than a (congruence relation) fuzzy congruence relation. Set-valued homomorphisms and relational morphisms are related closely because a set-valued homomorphism defines a relational morphism andvice versa. This paper, in one respect, is an attendance in the sense of fuzzy generalization of the ideas presented in references []. This study offers a wider perspective than [] in terms of usingTL-fuzzy relational morphisms which areL-fuzzy relations from any ring to any other ring instead ofT-congruenceL-fuzzy relation on a semigroup. In this paper, we investigate some properties of theTL-fuzzy relational morphisms looking from the side of homomorphisms of rings and designate them to construct anL-fuzzy approximation space and investigate some properties of thisL-fuzzy approximation space with respect to rings.

2 Preliminaries

The following definitions and preliminaries are required in the sequel of our work and hence they are presented in brief.

Let (L,∧,∨, , ) be a complete lattice with the least element  and the greatest element . A triangular norm [], ort-norm in short, is an increasing, associative and commutative mappingT :L×L→Lthat satisfies the boundary condition: for allα∈L,T(α, ) =α. A t-normT onL is called∨-distributive if T(α,β∨β) =T(α,β)∨T(α,β) for all α,β,β∈L.T is also called infinitely∨-distributive ifT(α,

i∈βi) =

i∈T(α,βi) for

allα,βi∈L, whereis an index set. An implicator is a functionI:L×L→L

satisfy-ing the conditionsI(, ) =  andI(, ) =I(, ) =I(, ) =  (see []). The minimum

t-normTMand the drastic productt-normTDonLare defined as follows:

TM(α,β) =α∧β, TD(α,β) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

β ifα= ;

α ifβ= ;

 otherwise

An implicatorIdefined as

I(x,y) = T(x,α)≤y

α, α∈L

for allx,y∈Lis called anR-implicator (residual implicator) based on thet-normT. For anR-implicatorIbased on thet-normT, the following statements hold:∀a,b,c,d,ai∈L

(i∈K) [, ],

() aTb≤c⇔a≤I(b,c), () I(a, ) = andI(,a) =a, () a≤b⇒I(a,b) = ,

() a≤b⇒I(a,c)≥I(b,c)andI(c,a)≤I(c,b), () I(a,b)T I(c,d)≤T(a,c)IT(b,d),

() I(T(a,b),c)≤I(a,I(b,c)), () T(I(a,b),c)≤I(a,T(b,c)), () I(_i∈Kai,b) =

i∈KI(ai,b),

() I(b,_i∈Kai) =

i∈KI(ai,b).

Throughout this paper, unless otherwise indicated,Lis referred to as any lattice, andT andIare referred to as anyt-norm and any implicator onL, respectively.

2.1 L-fuzzy subsets

In this subsection, we give some basic notions and results (see [, , , , –]). LetXbe a non-empty set called the universe of discourse. AnL-fuzzy subset ofXis any function fromXinto L(see []). The class of all subsets andL-fuzzy subsets ofX

will be denoted byP(X) andF(X,L), respectively. In particular, ifL= [, ] (where [, ] is the unit interval), then it is appropriate to replace a fuzzy subset with anL-fuzzy subset. In this case the set of all fuzzy subsets ofXis denoted byF(X). For anyμ∈F(X,L), the α-cut (or level) set ofμwill be denoted byμα, that is,μα={x∈X|μ(x)≥a}, whereα∈L.

In what follows,αywill denote the fuzzy singleton with valueαatyand  elsewhere. For

A⊆Xandα∈L,αA∈F(X,L) is defined byαA(x) =αifx∈AandαA(x) =  otherwise.

Ais called a characteristic function of the setA⊆X. Letμandνbe any twoL-fuzzy

subsets ofX. The symbolsμ∨ν,μ∧νandμTνwill mean the followingL-fuzzy subsets ofX, for allxinX,

(μ∨ν)(x) =μ(x)∨ν(x),

(μ∧ν)(x) =μ(x)∧ν(x),

(μTν)(x) =T μ(x),ν(x).

2.2 L-fuzzy relations

LetX,Y andZbe non-empty sets.

L-fuzzy relation if(x,z)≥_y∈X((x,y)T(y,z)) for allx,z∈X. Let∈F(X×Y,L) and –∈F(Y×X,L) beL-fuzzy relations which satisfy the condition–(y,x) =(x,y) for allx∈Xandy∈Y. Then–_{is called the inverseL-fuzzy relation of.}

TheT-compositions of theL-fuzzy relations∈F(X×Y,L) and∈F(Y×Z,L) is anL-fuzzy relation◦T:X×Z→Ldefined by (◦T)(x,z) =_y∈Y((x,y)T(y,z)) for all (x,z)∈(X,Z) (see []).

2.3 GeneralizedL-fuzzy rough approximation operators

LetXandYbe two non-empty sets andbe anL-fuzzy relation fromXtoY. The triple (X,Y,) is called a generalizedL-fuzzy approximation space. Ifis anL-fuzzy relation on

X, then it is denoted by the pair (X,), and especially ifL= [, ], then the triple (X,Y,) is called a generalized fuzzy approximation space. LetT be at-norm andIbe an implicator onL. For anyL-fuzzy subsetμofY, theT-upper andI-lowerL-fuzzy rough approxi-mations ofμ, denoted asT(μ) and_I(μ) respectively, are twoL-fuzzy sets ofXwhose membership functions are defined respectively by

T(μ)(x) =

y∈Y

T (x,y),μ(y), _I(μ)(x) =

y∈Y

I (x,y),μ(y) ∀x∈X.

The operators T and _I from F(Y,L) to F(X,L) are referred to as T-upper and I-lowerL-fuzzy rough approximation operators of (X,Y,) respectively, and the pair (_I(μ),T(μ)) is called the (I,T)-L-fuzzy rough set ofμwith respect to (X,Y,).

In [], Wuet al.studied some properties ofT-upper andI-lower fuzzy rough approxi-mation operators of the generalized approxiapproxi-mation space (X,Y,), whereT is a continu-oust-norm andIis a hybrid monotonic implicator on [, ]. The same properties for the T-upper and theI-lowerL-fuzzy rough approximation operators can be obtained.

Example . LetL={,α,β,γ, }be a lattice whose Hasse diagram is depicted in Figure .

Let:Z×Z→Lbe defined by

(x,y) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

 ifx= ;

 ifx= ;

β ifx= ;

 ifx= 

for allx∈Zandy∈Z, and letμ,ν,ηbeL-fuzzy subsets ofZindicated in the following table:

x   

μ(x)    ν(x)  α α η(x) γ β β

Then T_D-upper andI-lowerL-fuzzy rough approximations of them are obtained as follows, whereIis anR-implicator based on thet-normTD:

x    

T(μ)(x)   β 

T(ν)(x)   β 

T(η)(x) δ   

x    

_I(μ)(x)     _I(ν)(x) α  δ  _I(η)(x) β  δ 

2.4 TL-fuzzy subrings

Let (R, +,·) be a ring andIbe a non-empty subset ofR. Then we say thatIis a subring whenevera,b∈Ithena–b,ab∈I. A subringIofRwhich satisfies the conditionra∈I

andar∈Ifor allr∈R,a∈Iis said to be an ideal ofR. LetRandSbe rings. A function

f :R→Sis a homomorphism providedf(a+b) =f(a) +f(b) andf(ab) =f(a)f(b) for all

a,b∈R.

Definition . LetRbe a ring andμ∈F(R,L). If, for allx,y∈R,μsatisfies the following conditions:

(i) μ(x)≤μ(–x), (ii) μ(x)Tμ(y)≤μ(x+y), (iii) μ(x)Tμ(y)≤μ(xy),

thenμ is called aTL-fuzzy subring ofR. Moreover, a TL-fuzzy subringμis called a TL-fuzzy ideal ofRifμ(x)∨μ(y)≤μ(xy).

Definition . LetRbe a ring andμ,ν∈F(R,L). Defineμ+_T ν,μ·_T νandμ_T νas follows:

(μ+_T ν)(x) =

x=a+b

μ(a)Tν(b);

(μ·T ν)(x) =

x=ab

μ(a)Tν(b);

(μT ν)(x) =

x=n_i=aibi,n∈N

Tn

i=μ(ai)Tν(bi).

The L-fuzzy sets μ+_T ν, μ·T ν andμT ν are called the T-sum, ·T-product and T-product, respectively.

3 TL-fuzzy relational morphism of rings

min{(a,b),(c,d)} ≤(ac,bd) for alla,b,c,d∈A(see [, ]). We will give a more

general definition for rings in the light of the reference [].

Definition . LetRandSbe two rings and∈F(R×S,L). If, for all (x,y), (a,b)∈R×S,

satisfies the following conditions:

(i) (x,y)T(a,b)≤(x+a,y+b), (ii) (x,y)≤(–x, –y),

(iii) (x,y)T(a,b)≤(xa,yb),

thenis called aTL-fuzzy relational morphism fromRtoS. Moreover, aTL-fuzzy re-lational morphismis said to be full if(x,y)∨(a,b)≤(xa,yb). The set of all the TL-fuzzy (full) relational morphisms fromRtoSis denoted byHom(R×S,T,L) (respec-tively, Hom[R×S,T,L]). It is obvious thatHom[R×S,T,L]⊆Hom(R×S,T,L) and if ∈Hom(R×S,T,L), then–∈Hom(S×R,T,L).

Example . LetR,S,R,R,SandSbe rings.

() LetμandνbeTL-fuzzy subrings ofRandS, respectively. Let aTL-fuzzy relation

be defined by(a,b) =μ(a)Tν(–b)for all(a,b)∈R×S. Thenis aTL-fuzzy relational morphism fromRtoS. Moreover, ifT is a∨-distributivet-norm andμ

andνareTL-fuzzy ideals ofRandS, respectively, thenis full.

() Let∈Hom(R×S,T,L)andf :R→Randg:S→Sbe homomorphisms of

rings. Then:R×S→Ldefined by(a,b) =(f(a),g(b))for all(a,b)∈R×S

is aTL-fuzzy relational morphism fromRtoS. In particular, ifis full, thenis

full.

() Letf :R→Rbe a homomorphism of rings andμbe aTL-fuzzy subring ofR.

Then:R×R→Ldefined by(a,b) =μ(f(a))Tμ(b)for all(a,b)∈R×Ris a TL-fuzzy relational morphism fromRtoR.

() Letf :R→Rbe a homomorphism of rings andα,β∈Lsuch thatα≤β. Then :R×R→Ldefined by

(a,b) =

⎧ ⎨ ⎩

β iff(a) =b;

α iff(a)=b

for all(a,b)∈R×Ris aTL-fuzzy relational morphism fromRtoR.

() Zis a ring under the usual operations of addition and multiplication. LetLbe any lattice with the least elementand the greatest element.:Z×Z→Ldefined by

(a,b) =

⎧ ⎨ ⎩

 ifa=b;

 ifa=b

is aTL-fuzzy relational morphism onZbut it is not full since

Figure 2 LatticeL.

Let:Z×Z→Lbe defined by

(x,y) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

 ifxis odd;

α ifxis even andx= ;

 ifx= 

is aTL-fuzzy full relational morphism.

Proposition . LetT be an infinitely∨-distributive t-norm.

(i) If∈Hom(R×S,T,L),then for allx,a∈Randy,b∈S,

(x,y)T(a,b)≤(x–a,y–b).

(ii) Let∈Hom(R×S,T,L)andT≤T.Then∈Hom(R×S,T,L). (iii) Let,∈Hom(R×S,T,L).ThenT∈Hom(R×S,T,L).

Proof It is straightforward.

By Proposition .(iii), it is clear thatHom(R×S,T,L) is a monoid.

Theorem . LetT be an infinitely∨-distributive t-norm.If∈Hom(R×S,T,L)and

∈Hom(S×K,T,L),then◦_T ∈Hom(R×K,T,L).

Proof Let (r,k), (r,k)∈R×K. Thus

(◦T )(r,k)T(◦T )(r,k)

=

a∈S

(r,a)T(a,k)

T

b∈S

(r,b)T(b,k)

=

a∈S

b∈S

(r,a)T(r,b)T(a,k)T(b,k)

≤

a∈S

b∈S

(r+r,a+b)T(a+b,k+k)

=

s∈S

(r+r,s)T(s,k+k)

= (◦T )(r+r,k+k),

(◦_T )(r,k)T(◦_T )(r,k)

=

a∈S

(r,a)T(a,k)

T

b∈S

(r,b)T(b,k)

=

a∈S

b∈S

≤

a∈S

b∈S

(rr,ab)T(ab,kk)

≤

s∈S

(rr,s)T(s,kk)

= (◦T )(rr,kk),

(◦T )(r,k) =

s∈S

(r,s)T(s,k)

≤

s∈S

(–r, –s)T(–s, –k)

= (◦T )(–r, –k).

So,◦_Tis aTL-fuzzy relational morphism of rings. It is clear to see that if∈Hom[R× S,T,L] and∈Hom[S×K,T,L], then◦_T ∈Hom[R×K,T,L].

Definition . Let∈Hom(R×S,T,L). Then the kernel and image of(denoted by

KerandIm, respectively) are defined to be theL-fuzzy subsets ofRandS, respectively that satisfyKer(r) =(r, ) andIm(s) =_r∈R(r,s) for allr∈R,s∈S.

Remark Letf:R→Sbe a homomorphism of rings and let∈F(R×S,L) be defined by

(a,b) =

⎧ ⎨ ⎩

 iff(a) =b;

 iff(a)=b.

Then∈Hom(R×S,T,L) andKer= Kerf andImR= Imf.

Proposition . Let∈Hom(R×S,T,L).Then:

(i) Keris aTL-fuzzy subring ofR.

(ii) If∈Hom[R×S,T,L],thenKeris aTL-fuzzy ideal ofR.

(iii) Imis aTL-fuzzy subring ofSifT is an infinitely∨-distributivet-norm onL.

Proof

(i) Letx,y∈R. Then

Ker(x)T Ker(y) =(x, )T(y, )

≤(x+y, ) ≤Ker(x+y),

Ker(x)T Ker(y) =(x, )T(y, )

=(xy, )

=Ker(xy),

Ker(x) =(x, )

≤(–x, )

(ii) If∈Hom[R×S,T,L], then it follows immediately from (i). (iii) Letx,y∈S. Then

Im(x)TIm(y) =

a∈R

(a,x)

T

b∈R

(b,y)

=

a∈R

b∈R

(a,x)T(b,y)

≤

a∈R

b∈R

(a+b,x+y)

=

a,b∈R

(a+b,x+y)

≤

r∈R

(r,x+y)

=Im(x+y),

Im(x)TIm(y) =

a∈R

(a,x)

T

b∈R

(b,y)

=

a∈R

b∈R

(a,x)T(b,y)

≤

a∈R

b∈R

(ab,xy)

=

a,b∈R

(ab,xy)

≤

r∈R

(r,xy)

=Im(xy),

Im(x) =

a∈R

(a,x)

≤

a∈R

(–a, –x)

=

–a∈R

(–a, –x)

=

r∈R

(r, –x)

=Im(–x).

ThusImis aTL-fuzzy subring ofS.

Let∈F(R×S,L) andT be anyt-norm onL. For anα∈L, let someα-level sets be defined as follows:

(S)α=

s∈S| ∃r∈R:(r,s)Tα≥α,

(R)α=

∗_Sα=s∈S|(,s)Tα≥α, ∗_Rα=r∈R|(r, )Tα≥α.

Theorem . Let∈Hom(R×S,T,L).Then:

(i) If(S)α=∅,then(S)αis a subring ofS.

(ii) If(R)α=∅,then(R)αis a subring ofR.

(iii) If(∗_S)α=∅,then(∗_S)αis a subring of(S)α.

(iv) If(∗_R)α=∅,then(∗_R)αis a subring of(R)α.

Proof Leta,b∈(S)α. Then there existx,y∈Rsuch that(x,a)Tα≥αand(y,b)T ×

α≥α. So, we have(x,a)T(y,b)Tα≥α. Since∈Hom(R×S,T,L), then(x–y,a–

b)Tα≥αby Proposition .(i) and(xy,ab)Tα≥α. Thereforea–b,ab∈(S)α. Hence

(S)αis a subring ofS. For (ii), the proof is similar to (i). For (iii), it is similar to (i) that

(∗_S)αis a subring ofS. Since (∗_S)α⊆(S)α, then (∗_S)αis a subring of (S)α. For (iv), the

proof is similar to (iii).

Definition . Letμbe aTL-fuzzy ideal ofR. TheL-fuzzy subsety+μofRdefined by (y+μ)(x) =μ(x–y) is called a coset of theTL-fuzzy idealμ(see []).

Proposition . Letμbe aTL-fuzzy ideal of R which satisfiesμ() = and let x,y,u,v be any elements in R.If x+μ=u+μand y+μ=v+μ,then

(i) (x+y) +μ= (u+v) +μand

(ii) (xy) +μ= (uv) +μ.

Proof It is similar to the proof of Proposition . in [].

Letμbe aTL-fuzzy ideal ofRwhich satisfiesμ() = . Then Proposition . allows us to define two binary operations ‘+’ and ‘·’ on the setR/μof all cosets ofμas follows:

(x+μ) + (y+μ) = (x+y) +μ,

(x+μ)·(y+μ) = (xy) +μ.

It is straightforward to see thatR/μis a ring under these binary operations with additive identityμ, multiplicative identity  +μand –(x+μ) = (–x) +μ.

Theorem . Letμbe aTL-fuzzy ideal of R which satisfiesμ() = and:R×R/μ→L be defined by(x,y+μ) =μ(x–y)for all x,y∈R.Then∈Hom(R×R/μ,T,L).

Proof Letx,a∈Randy+μ,b+μ∈R/μ. Thus

(x,y+μ)T(a,b+μ) =μ(x–y)Tμ(a–b)

≤μ(x–y+a–b)

=μ x+a– (y+b)

=(x+a,y+b+μ)

(x,y+μ) =μ(x–y)

≤μ –(x–y)

=μ(–x+y)

=μ –x– (–y)

=(–x, –y+μ),

(x,y+μ)T(a,b+μ) =μ(x–y)Tμ(a–b)

≤μ (x–y)aTμ y(a–b)

=μ(xa–ya)Tμ(ya–yb)

≤μ(xa–ya+ya–yb)

=μ(xa–yb)

=(xa,yb+μ)

= xa, (y+μ)(b+μ).

So,is aTL-fuzzy relational morphism of rings.

Let∈F(R×S,L). For anyr∈Rands∈S,(r,S)∈F(S,L) and(R,s)∈F(R,L) is defined as follows:

(r,S)(s) =(r,s) ∀s∈S, (R,s)(r) =(r,s) ∀r∈R.

(r,S)and(R,s)are called the (r,S)- and (R,s)-restriction of, respectively (see []).

Theorem . LetT be an infinitely∨-distributive t-norm.Ifis aTL-fuzzy(full) rela-tional morphism from R to S,then(R,)and(,S)areTL-fuzzy subrings(ideals)of S and

R,respectively.

Proof Since(R,)=Ker, then it is obvious that(R,)is aTL-fuzzy subgroup ofRby Proposition .. Similarly,(,S)isTL-fuzzy subgroup ofSsince–∈Hom(S×R,T,L) and(,S)=Ker–. Thus if∈Hom[S×R,T,L], then it is easy to see(R,) and(,S)

areTL-fuzzy ideals ofSandR, respectively.

4 I-lower andT-upper fuzzy rough approximations on a ring

In this section, we study the properties ofI-lower andT-upper fuzzy rough approxima-tion operators with respect to aTL-fuzzy (full) relational morphism of rings.

Theorem . Let μ,ν ∈F(S,L), ∈Hom(R×S,T,L) andT be an infinitely∨

-dis-tributive t-norm.Then:

Proof

(i) Letx∈R. Then

T(μ) +T T(ν)(x) =

x=a+b

T(μ)(a)TT(ν)(b)

=

x=a+b

k∈S

(a,k)Tμ(k)

T

t∈S

(b,t)Tν(t)

=

x=a+b

k,t∈S

(a,k)T(b,t)Tμ(k)Tν(t)

≤

x=a+b

s=k+t

(a+b,k+t)Tμ(k)Tν(t)

=

x=a+b s=k+t

(a+b,k+t)

T

s=k+t

μ(k)Tν(t)

=

x=a+b

s∈S

(a+b,s)T(μ+T ν)(s)

=

s∈S

(x,s)T(μ+_T ν)(s)

=T(μ+_T ν)(x).

So, we haveT(μ) +_T T(ν)≤T(μ◦_T ν). (ii) It is similar to the proof of (i).

(iii) Letx∈R. Then

T(μ)_T T(ν)(x)

=

x=ni=aibi,n∈N

Tn i=

T

(μ)(ai)TT(ν)(bi)

=

x=n_i=aibi,n∈N

Tn i=

ki∈S

(ai,ki)Tμ(ki)

T

ti∈S

(bi,ti)Tν(ti)

=

x=ni=aibi,n∈N

Tn i=

ki,ti∈S

(ai,ki)Tμ(ki)T(bi,ti)Tν(ti)

≤

x=n_i=aibi,n∈N

Tn i=

ki,ti∈S

(aibi,kiti)Tμ(ki)Tν(ti)

=

x=ni=aibi,n∈N

Tn i=

ki,ti∈S

(aibi,kiti)

T Tn i=

ki,ti∈S

μ(ki)Tν(ti)

≤

x=ni=aibi,n∈N

ki,ti∈S

_n

i=

aibi, n

i=

kiti

T Tn i=

ki,ti∈S

μ(ki)Tν(ti)

≤

pi=kiti∈S

x, n i= pi T Tn i=

pi=kiti∈S

μ(ki)Tν(ti)

≤

pi∈S

x, n i= pi

T(μT ν)

_n

i=

pi

≤

s∈S

(x,s)T(μT ν)(s)

=T(μT ν)(x).

The following example shows that replacing ‘=’ by ‘≤’ is not true in general in Theo-rem ..

Example . LetLbe the lattice which is given in Example .() andT =TM. Let:

Z×Z→Land:Z×Z→Lbe defined by

(x,y) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

 ifx= ;

 ifx= ;

β ifx= ;

 ifx= ;

(x,y) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

 ifx=y= ;

α ifx= ,y= ;

 ifx= .

Then∈Hom(Z×Z,T,L) and∈Hom(Z×Z,T,L). Letωbe anL-fuzzy subset of Zdefined by

ω(x) =

⎧ ⎨ ⎩

α ifx= ;

 ifx= .

Then (T(ω) +_T T(ω))()≤α andT(ω+_T ω)() = . Thus, T(ω) +_T T(ω)= T(μ+_T μ). Letμ,νbeL-fuzzy subsets ofZdefined by

x   

μ(x)  α α ν(x) β  

Then (T(μ) ·T T(ν))() =  and T(μ ·T ν)() = β. Thus, T(μ) ·T T(ν) = T(μ·T ν).

Definition . Let (R,S,) be aTL-fuzzy approximation space andμbe anL-fuzzy sub-set ofS.μis called aI-lowerTL-fuzzy rough ring ofSif_I(μ) is aTL-fuzzy subring of

Randμis called aT-upperTL-fuzzy rough ring ofSif_T(μ) is aTL-fuzzy subring ofR. The (I,T)-L-fuzzy rough set (_I(μ),_T(μ)) is called aTL-fuzzy rough ring ofμif both _I(μ) and_T(μ) areTL-fuzzy subrings ofR.

Theorem . LetT be an infinitely∨-distributive t-norm,∈Hom(R×S,T,L)andμ

be aTL-fuzzy subring of S.Thenμis aT-upperTL-fuzzy rough ring of S.

Proof Letx,y∈R. Then

T(μ)(x)TT(μ)(y) =

k∈S

(x,k)Tμ(k)

T

t∈S

(y,t)Tμ(t)

=

k∈S

t∈S

≤

k∈S

t∈S

(x+y,k+t)Tμ(k+t)

≤

p∈S

(x+y,p)Tμ(p)

=T(μ)(x+y),

T(μ)(x)TT(μ)(y) =

k∈S

(x,k)Tμ(k)

T

t∈S

(y,t)Tμ(t)

=

k∈S

t∈S

(x,k)Tμ(k)T(y,t)Tμ(t)

≤

k∈S

t∈S

(xy,kt)Tμ(kt)

≤

p∈S

(xy,p)Tμ(p)

=T(μ)(xy),

T(μ)(x) =

k∈S

(x,k)Tμ(k)

≤

k∈S

(–x, –k)Tμ(–k)

=T(μ)(–x).

Hence we obtainT(μ) is aTL-fuzzy subring ofR. Thusμis aT-upperTL-fuzzy rough

ring ofS.

Theorem . LetT be an infinitely∨-distributive t-norm,∈Hom[R×S,T,L]andμ

be aTL-fuzzy ideal of S.ThenT(μ)is aTL-fuzzy ideal of R.

Proof Letx,y∈R. Then

T(μ)(x)∨T(μ)(y) =

k∈S

(x,k)Tμ(k)

∨

t∈S

(y,t)Tμ(t)

=

k,t∈S

(x,k)Tμ(k)∨ (y,t)Tμ(t)

≤

k,t∈S

(x,k)∨(y,t)T μ(k)∨μ(t)

≤

k,t∈S

(xy,kt)Tμ(kt)

≤

p∈S

(xy,p)Tμ(p)

=T(μ)(xy).

The following example shows thatμis not aI-lowerTL-fuzzy rough ring ofSin general under the condition of Theorem ., and Theorem . may not be true for theI-lower

L-fuzzy rough approximation ofμeven ifis full.

Example . ConsiderLandas in Example .(). LetT =TM. Thenis aTML-fuzzy

full relational morphism. Let aL-fuzzy subsetμofZbe defined by

x  

μ(x)  α

Then μis a TML-fuzzy ideal ofZ and we obtain the TML-fuzzy lower approximation

operator ofμas follows:

_I(μ)(x) =

⎧ ⎨ ⎩

α ifx= ;

 ifx= .

Since_I(μ)()TM_I(μ)(–) = TM = α=_I(μ)() =_I(μ)( + (–)), then_I(μ)

is not aTML-fuzzy subring ofZ.

5 Conclusions

The generalized rough sets on algebraic sets such as group, ring and module were mainly studied by a set-valued homomorphism [, , ] which is a generalization of a congru-ence relation. In this paper, a definition ofTL-fuzzy (full) relational morphism is consid-ered as a generalization of fuzzy congruence relations (orT-congruenceL-fuzzy relations) for rings. Then we obtained some new properties of aTL-fuzzy (full) relational morphism to provide opportunity of putting reasonable interpretations and explored the features of generalizedI-lower andT-upper fuzzy rough approximations of anL-fuzzy subset on rings. From these points of view, taking a fresh look at the generalized (I,T)-L-fuzzy rough sets on rings is an interesting research topic. Our further work on this topic will focus on the properties of generalized (I,T)-L-fuzzy rough sets on modules with respect to theTL-fuzzy (full) relational morphism.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The present study was proposed by CE, YC and SY. All authors read and approved the final manuscript.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

Received: 14 December 2012 Accepted: 20 May 2013 Published: 3 June 2013 References

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doi:10.1186/1029-242X-2013-279