Roughness in Quantum B-algebras

ZHAN Qiuyan,WANG Jing,YANG Yichuan

(School of Mathematical Sciences,Beihang University,Shahe Campus,Beijing 102206,China)

Abstract:Uncertainty theories have the widespread applications and significant influence.In this paper,rough set theory of uncer-tainty theory is applied to quantum B-algebras.The rough subalgebras of(linearly ordered)quantum B-algebras are considered,and it is proved that the subalgebras of quantum B-algebras are rough subalgebras.Rough normal q-filters of lattice-order quantum B-al-gebras are studied.It is proved that normal q-filters on quantum B-algebras are rough normal q-filters.In order to study lattice-or-der quantum B-algebras,quantum B-algebras are divided into three types by means of identity element,and the concrete forms of union and intersection operations are given.Homomorphic images of rough normal q-filters on lattice-order quantum B-algebras are studied.In addition,rough set theory is applied to a special class of quantum B-algebra—KL-algebras,and rough set algebras are proved to be CKL-algebras by selecting suitable implication operators.Finally,rough soft set theory is applied to quantum B-alge-bras,and a decision making method is given on quantum B-algebras.

Key words:quantum B-algebra;rough subalgebra;rough normal q-filter;CKL-algebra;decision making

1 Introduction

Quantum B-algebras,the partially ordered implicational algebras arising as subreducts of quantales,were in-troduced by Rump and Yang[1].It provided a unified semantics for non-commutative algebraic logic including pseduo-BCK algebras,po-groups,BL-algebras,MV-algebras,GPE-algebras and residuated lattices[1].Botur and Paseka[2]studied filters on integral quantum B-algebras,they established an embedding of a cartesian product of polars of a pseudo-hoop.Zhang et al.[3]established the quotient structures by using normal q-filters in perfect quantum B-algebras and investigated the relation between basic implication algebras and quantum B-algebras.In[4],dual quantum B-algebras(Girard quantum B-algebras)were investigated.And they proved that every dual quantum B-algebra is a residuated poset and that complete dual quantum B-algebras and dual quantales are equiv-alent to each other.Han et al.[5]considered untial quantum B-algebras.Then,in[6],they considered three class-es of quantum B-algebras such as locally unital quantum B-algebras,two-sided quantum B-algebras and implica-tion-commutative quantum B-algebras.Han et al.gave the injective hulls of quantum B-algebras in terms of the injective hulls of posemigroups in[7].Following the idea of Fleischer who represented BCK-algebras by means of residuable elements of commutative integral po-monoids,in[8],Kuhr and Paseka described quantum B-alge-bras as subsets of residuable elements of po-semigroups.Moreover,they showed that quantum B-algebras corre-spond one-to-one to what we call Fleischer posemigroups.In[9],Ciungu defined and studied the involutive and weakly involutive quantum B-algebras.And they proved that any weakly involutive quantum B-algebra is a re-siduated poset.Then the result of any pair of quantifiers is a monadic operator on weakly involutive quantum B-algebras obtained.Moreover,in[10],Ciungu defined the monadic quantum B-algebras and investigated their properties.

Rough set theory was introduced by Pawlak[11].Apart from many applications in information processing,at-tribute reduction,medical diagnosis and so on,rough set theory has also been applied to many logical algebras,such as BCI-algebras[12],BCK-algebras[13],MV-algebras[14],Lattice-ordered effect algebras[15].Yang and Xu[16]in-vestigated homomorphic images of rough ideals and rough prime ideals in quantales.Luo and Wang[17]applied rough set theory to fuzzy ideals of quantales,and defined rough fuzzy(prime,semi-prime,primary)ideals of quantales,which generalized Yang and Xu's work on quantales in fuzzy environment.Since quantum B-algebras is arising as subreducts of quantales,it is natural to apply rough set theory to quantum B-algebras.On the other hand,soft set theory was put forward by Molodstsov[18]as a new mathematic tool for dealing with uncertainty in-formation,which was free from the inadequacy of the parametrization limitation.Moreover,as a generalized models of rough set theory,the rough soft set theory came forth rapidly,especially with regard to their applica-tions in decision making.

Based on above discussions,we know that most of non-commutative logical algebras can be regarded as a special case of quantum B-algebras.So it is an essential topic to consider these rough set theory on quantum B-al-gebras,whose results will unify and enrich previous works related to uncertainty theories on logical algebras.

The article is organized as follows:Section 2 provides some necessary preliminaries.The rough subalgebras of quantum B-algebras are investigated in Section 3.The rough normal q-filters of quantum B-algebras are stud-ied,a result of a normal q-filter is a rough normal q-filter of quantum B-algebras is given(Theorem 1),and homo-morphic images of rough normal q-filters of lattice-ordered quantum B-algebras(Proposition 9)are also consid-ered in Section 4.In Section 5,we apply rough set theory to CKL-algebras,a special class of quantum B-alge-bras,and prove that the rough set algebra is a CKL-algebra(Theorem 2).Finally,we apply rough soft set theory to quantum B-algebras in decision making and design a decision making algorithm for rough soft quantum B-al-gebras in Section 6.

2 Preliminaries

We first review some basic knowledge about quantum B-algebras.

Definition 1[1]A quantum B-algebra is a poset X with two binary operations→and⇝satisfying the follow-ing conditions:

for all x,y,z∈X.

A quantum B-algebra X is called unital if it admits an element u,the unit element,which satisfies u→x=u⇝x=x for all x∈X.Notice that the unit element u is always unique.In fact,if u and u' are unit elements,then u≤u'→u,which implies that u'≤u⇝u=u.Similarly,we have u≤u'⇝u'=u'.Sou=u'.

A quantum B-algebra X is called commutative if x→y=x⇝y for all x,y∈X.A subset Y of a quantum B-alge-bra X is called a subalgebra if Y is closed with respect to→and⇝.

Proposition 1[1]Let X be a quantum B-algebra.Then for all x,y,z∈X,we have:

Definition 2[3]A nonempty subset F of a quantum B-algebra X is called a normal q-filter of X if it satisfies the following conditions for all x,y∈X:

Definition 3[3]Let X be a quantum B-algebra and F a normal q-filter of X.For all x,y∈X,define the binary relation θFas follows:

Then θFis called a congruence relation on X.

Similar to the result gave in[19,Example 1.25]that there is difference between two cardinalities of congru-ence relations and ideals in a semimodule over an incline,we construct following two examples to show that there is difference between two cardinalities of normal q-filters F and the congruence relations induced by F,too.

Example 1 Let X=(Z×Z,+,≤,(0,0))with trivial partial order,and define a→b:=a⇝b:=-a+b,then one could check that X is a commutative quantum B-algebra.Consider a normal q-filter F1={(a,b)|a≥0,b∈Z}of X,then for a1,a2,b1,b2∈Z,(a1,b1)θF1(a2,b2)⇔(a1,b1)→(a2,b2)∈F1,(a2,b2)→ (a1,b1)∈F1⇔(a2-a1,b2-b1)∈F1,(a1-a2,b1-b2)∈F1⇔a1=a2,that is,(a1,b1)θF1(a2,b2)⇔a1=a2.Consider another normal q-filter F2={(a,b)|a=0,b∈Z}of X,then(a1,b1)θF2(a2,b2)⇔a1=a2.So we have:two normal q-filters F1and F2correspond to the same congruence relation.

Example 2 Let X={0,a,b,1}with the operation→and the order in Table 1 and in Fig.1,respectively.One can check that(X,→,≤)is a commutative quantum B-algebra.Table 2 shows that three normal q-filters F1,F2,and F3correspond to the same congruence relation θ1.

Table 1 →on X of Example 2

Table 2 Normal q-filters and congruence relations on X of Example 2

Remark 1 As shown in following Example 3,a congruence relation given by Definition 3 can not guaran-tee[x→y]θF=[x]θF→[y]θF,[x⇝y]θF=[x]θF⇝ [y]θFin quantum B-algebras in general.If the equations hold for all x,y∈X,we call θFa complete congruence relation.

Example 3 Let X={0,x,y,z,1}with0≤x≤y≤1,0≤x≤z≤1.Define→,and⇝in Table 3 and Table 4.

Table 3 →on X of Example 3

Table 4 ⇝on X of Example 3

Table 5 →on X of Example 4

3 Rough subalgebras

Example 4 Let X={x,y,z,r,t,1}be the poset with x≤y≤z≤1,r≤t≤1.Define a binary relation → as Ta-ble 5.

Now we study rough approximation on linearly ordered quantum B-algebras.Before that,we give the defini-tion of convex set on quantum B-algebras.

Definition 6 A subset S of a quantum B-algebra X is called convex,if for every x,y∈Sand x≤z≤y implies that z∈S.

Proof We first prove that[x]θFis convex.Lett,s∈[x]θF,t≤z≤s,z∈X,we have to prove z∈[x]θF.Since t≤z,z≤s,so we have x→t≤x→z,s→x≤z→x for any x∈X.Note that θFis a congruence relation and F is a normal q-filter,then we have x→t∈F,s→x∈F,and so x→z∈F,z→x∈F,that is,z∈[x]θF,hence[x]θFis convex.

Assume that there exists t∈[x]θFands∈[y]θFsuch thats≤t.If t≤y,thens≤t≤y,t∈[y]θF,it is a contradic-tion;if y≤t,then x≤y≤t,y∈[x]θF,it is a contradiction.So for each t∈[x]θFands∈[y]θF,t≤s. □

However,the following example shows that Proposition 2 is not true if X is not linearly ordered.

Example 5 Let X={0,a,b,c,1}satisfy Table 6 and its Hasse diagram is Fig.2.

Fig.2 The order "≤" on X of Example 5

Table 6 →on X of Example 5

Next we prove that a subalgebra on a quantum B-algebra is also a rough subalgebra.

Proposition 4 Let θFbe a complete congruence relation on a quantum B-algebra X.If A is a subalgebra of X,then A is a rough subalgebra of X.

According to(i)and(ii),we complete the proof. □

Now we give an example to illustrate that the condition θFis a complete congruence relation on X in Proposi-tion 4 and could not be dropped.

4 Rough normal q-filters

4.1 Rough normal q-filters of quantum B-algebras

Similar to subalgebras,normal q-filters are another significant tool to study substructures of quantum B-alge-bras.In what follows we investigate some properties about rough normal q-filters of quantum B-algebras.We first give the definition of rough normal q-filters of quantum B-algebras.

Proposition 5 Let X be a quantum B-algebra.Then for an approximation space(X,θF)and A,B⊆X,the following properties hold for x∈X:

Fig.3 Hasse digram on X of Example 7

Table 7 →on X of Example 7

Table 8 ⇝on X of Example 7

Analogous to the result about rough subalgebras which present in Section 3,next we prove that a normal q-filter is a rough normal q-filter of quantum B-algebras.

We now show that two conditions(1)A is a normal q-filter and(2)F ⊆A in Theorem 1 could not be re-moved.Naturally,we will analyse it in three cases.

4.2 Homomorphic images of rough normal q-filters of lattice-ordered quantum B-algebras

Observed that a quantum B-algebra X is a partially ordered set with two binary operations→ and⇝,if it is a lattice-ordered,then we call X a lattice-ordered quantum B-algebra.Naturally,how to define join and meet opera-tions on X becomes an essential problem.Motivated by[1],in the following we give the concrete form of join and meet operations on X by three cases,and the unit element helps us to analyse them.

Case 1 If the unit element is the greatest one,then it consists of pseudo-BCK algebras[21],MV-algebras[22],the algebras of Lukasiewicz' infinite-valued logic,and their non-commutative extensions[23].

Recall that a quantum B-algebra X is called bounded if X admits a smallest element 0[1].That is,if X is a bounded quantum B-algebra,where the unit element is the greatest one,then

hold for all x∈X.And by[1,(14)],we have x≤y⇔1≤x→y⇔1≤x⇝y,which follows that x≤x⇔1≤x→x⇔1≤x⇝x,so

Notice that[1,Prop.12]shows that every unital quantum B-algebra has a pseudo-BCK subalgebra,and then there is a natural partial order

Based on above facts,next we begin to show the concrete form of join and meet operations in Case 1.First,we have:

Lemma 1 Let X be a bounded quantum B-algebra,where the unit element is the greatest one,for all x,y∈X,we have

is an upper bound for{x,y}.

Proof In fact,by(8),we have x≤(x→y)⇝y.Using(15),which implies that x→y≤y→y=1,that is,y≤(x→y)⇝y.So x≤(x→y)⇝y and y≤(x→y)⇝y hold.Thus(x→y)⇝y is an upper bound for{x,y}.Similarly,we have x≤(y→x)⇝x,y≤(y→x)⇝x,so(y→x)⇝xis an upper bound for{x,y}. □

Remark 2 In general,if X is a bounded quantum B-algebra with the unit element is the greatest one,then we have(x→y)⇝y≠(y→x)⇝x.For instance,see Example 3,(x→y)⇝y=y≠1=(y→x)⇝x.If for all x,y∈X,(x→y)⇝y=(y→x)⇝x,then X is called ∨-semi-lattice-ordered.

Proposition 6 Let X be a bounded quantum B-algebra with the unit element is the greatest one,if X is∨-semi-lattice-ordered,then(x→y)⇝y is the least upper bound of{x,y}for all x,y∈X.

Proof Note that(x→y)⇝y is an upper bound of{x,y}.Let z be another upper bound of{x,y},then x≤z,y≤z,so x→z=1,y→z=1,(y→z)⇝z=z.We have to prove(x→y)⇝y≤z,i.e.((x→y)⇝y)→z=1.Since x≤z,by(7),we have z→y≤x→y,which implies that(x→y)⇝y≤(z→y)⇝y,so((x→y)⇝y)→z=((x→y)⇝y)→((y→z)⇝z)=((x→y)⇝y)→((z→y)⇝y)=1.Therefore,(x→y)⇝y is the least upper bound of{x,y}. □

According to Lemma 1 and Proposition 6,we could write x∨y=(x→y)⇝y(y∨x=(y→x)⇝x)for x,y∈X in Case 1.

Dually,in what follows we investigate ∧-semi-lattice-ordered quantum B-algebra in Case 1.First,we define two negations on a bounded quantum B-algebra X which are denoted by x-=x→0,x～=x⇝0.Then we have:

Proposition 7 Let X be a bounded quantum B-algebra,where the unit element is the greatest one,then the following conditions hold for all x,y∈X.

Proof According to the definition of negations and(18),we have0-=0→0=1,0～=0⇝0=1.Using(8),we obtain1≤(1→0)⇝0.And by(15),which implies that(1→0)⇝0=1,that is,1→0≤0.Note that0≤1→0,so 1→0=0,which follows that1-=0.Similarly,we have1～=0.So(19)holds.

Notice that x≤(x→y)⇝y,take y=0,then we obtain x≤(x→0)⇝0=(x-)～.Analogously,we have x≤(x⇝0)→0=(x～)-.Hence(20)holds.

Using(1),put z=0,we have(y→0)≤(x→y)→(x→0)⇔x→y≤(y→0)⇝(x→0)=y～⇝x～.Similarly,take z=0,we obtain x⇝y≤y～→x～.Therefore,(21)holds.

Since y≤x,by(21),we have y→x≤x～→y～,so x～→y～=1,x～≤y～.Analogously,we get x-≤y-.So,(22)holds. □

Naturally,we obtain the result that is dual to Lemma 1.

Lemma 2 Let X be a ∨-semi-lattice-ordered quantum B-algebra with the unit element is the greatest one,then(x-∨y-)～((y-∨x-)～)is a lower bound for{x,y}.

Proof In fact,since x-,y-≤x-∨y-,by(22),we have(x-∨y-)～≤(x-)～,(y-)～,so(x-∨y-)～is a lower bound of{(x-)～,(y-)～}.Note that x=(x→0)⇝0=(x-)～,similarly,we obtain y=(y-)～,hence(x-∨y-)～is a lower bound of{x,y}.Analogously,we have(y-∨x-)～is a lower bound of{x,y}. □

Proposition 8 Let X be a ∨-semi-lattice-ordered quantum B-algebra,where the unit element is the greatest one,then(x-∨y-)～((y-∨x-)～)is the greatest lower bound of{x,y}and(x-∨y-)～=(y-∨x-)～for all x,y∈X.

Proof Observed that(x-∨y-)～is a lower bound of{x,y}.Assume that s is another lower bound of{x,y},thens≤x,s≤y,so x-≤s-,y-≤s-,that is,s-is an upper bound of{x-,y-}.Note that X is∨-semi-lattice-ordered,so according to Proposition 6,it implies that x-∨y-is the least upper bound of{x-,y-},hence x-∨y-≤s-,which implies that(x-∨y-)～≥(s-)～=s.Thus(x-∨y-)～is the greatest lower bound of{x,y}.Similarly,we can check that(y-∨x-)～is the greatest lower bound of{x,y}.Note that(x→y)⇝y=(y→x)⇝x hold for all x,y∈X,which implies that(x-∨y-)～=(y-∨x-)～. □

From Lemma 2 and Proposition 8,we could write x∧y=(x-∨y-)～(y∧x=(y-∨x-)～)for all x,y∈X.And so we complete the discussion of lattice-ordered quantum B-algebras in Case 1.After that,we continue to study the other two cases.

Case 2 If the unit element is placed in an intermediate position.Note that any residuated poset[24]is a quan-tum B-algebra,if it is a residuated lattice[25],then the join and meet operations are given by the lattice structure,that is,x∨y=sup{x,y},x∧y=inf{x,y}.

Case 3 If the unit element is the smallest one,then it consists of effect algebras[26]and their non-commuta-tive versions.

Lattice-ordered effect algebras are considered as a class of L-algebras[27-28],and we could consider them as L-algebras which are also quantum B-algebras.So we can give the join and meet form by x∨y=(x′→y′)→x,x∧y=((x→y)→x′)′on the basis of[28,Prop 3.4],where x'=x→0.

By now,we complete the discussion of lattice-ordered quantum B-algebras by three cases and present the concrete form of join and meet operations.After that we investigate homomorphic images of rough normal q-fil-ters of lattice-ordered quantum B-algebras.Motivated by the definition of morphism on quantum B-algebras in[1],we first give the concept of homomorphism of lattice-ordered quantum B-algebras.

Definition 8 Let X1and X2be two lattice-ordered quantum B-algebras.A mapping f:X1→X2is called a ho-momorphism if it satisfies the following equalities for all x,y∈X:

Obviously,f is order preserving.Observed that the quotient structures of quantum B-algebras are constructed by normal q-filters in[3].And the following result shows the case of homomorphic images of rough normal q-filters of lattice-ordered quantum B-algebras.

Proposition 9 Let X1and X2be two lattice-ordered quantum B-algebra,f:X1→ X2a surjective homomor-phism from X1to X2and θ2a congruence relation on X2.If θ1={(x1,x2)∈X1×X1|(f(x1),f(x2))∈θ2},then for a subset A⊆X1,we have:

(I) θ1is a congruence relation on X1,→ .

Proof (I) It is obvious that θ1is an equivalence relation on X1.Now let x,y,w,z∈X1such that xθ1y,wθ1z,which follows that f(x)θ2f(y),f(w)θ2f(z),then we have

(f(x)→f(w))θ2(f(y)→f(z)).

Note that f is a homomorphism,so f(x)→f(w)=f(x→w),f(y)→f(z)=f(y)→f(z),which implies that(f(x→w))θ2(f(y→z)),that is,(x→w)θ1(y→z).Similarly,we can prove that(x⇝w)θ1(y⇝z).There-fore,θ1is a congruence relation on X1.

5 An example:roughness in CKL-algebras

Definition 9[27]Let(L,→)be a set L with a binary operation→.An element 1 of L is called a logical unit if it satisfies

for all x∈L.If(L,→)with a logical unit1∈L satisfies

for all x,y,z∈L,then(L,→,1)is said to be an L-algebra.

Definition 10[27,29]An L-algebra L which satisfies

for all x,y∈L is called a KL-algebra.ACKL-algebra is an L-algebra L which satisfies

for all x,y,z∈L.

ACKL-algebra L is indeed a KL-algebra since x→(y→x)=y→(x→x)=y→1=1for any x,y∈L.

Lemma 3[30]For a non-empty set U,assume that X,Y ⊆U,then(X,Y)∈N(Apr)if and only if X ⊆Y and(Y -X)∩S=∅,where S={x∈U,|[x]θ|=1}and Y-X is the difference of sets Y and X.

It is widely known that with the help of congruence relations in a nonclassical logical algebra,one could con-struct the rough set algebra from a nonclassical logic algebra.Conversely,after proper selection of implication operators,one could express the rough set algebra as a nonclassical logical algebra.In what follows we present an appropriate way to express the implication operator:

Next we express the rough set algebra as a nonclassical logical algebra applying the above definition of→.

Theorem 2 (N(Apr),→,(U,U))is a CKL-algebra.

Proof It suffices to check that(C),(L),(26)and(27)hold.

For any(X1,X2),(Y1,Y2),(Z1,Z2)∈N(Apr),using the definition of→,we obtain

Thus(C)holds.

Note that

Therefore,(26)holds.

Therefore,(N(Apr),→,(U,U))is a CKL-algebra. □

6 Rough soft quantum B-algebras in decision making

Definition 11[18]Let U be a non-empty set and E a set of parameters.A pair S=(s,A)is called a soft set over U,where s is a mapping given by s:A→P(U),P(U)denotes the powerset of U and A⊆E.

Remark 3 From the definition of soft set,we obtain that a soft set over U is a parameterized family of sub-sets of U rather than a set.For a parameter ε∈A,sA(ε)can be considered as the set of ε-approxiamtion elements of soft set(sA,A).

Example 8 Consider Example 7.F={c,d,1}is a normal q-filter and the congruence classes are:E1={c,d,1},E2={a},E3={b},E4={0}.

Define a soft set S=(s,A)over X by s(0)={0,a,b},s(a)={0,a,b,c},s(b)={a,b,c},s(c)={c,d},s(d)={0,1},s(1)={a,b,c,d},where A=X.

To find the farthest(closest)accurate quantum B-algebra on S w.r.t.a normal q-filter of a quantum B-alge-bra,in the following,we present a decision making method in quantum B-algebras.That is,we have to find the worst(best)parameter for a given soft set S=(s,A).

Decision making method I:

Step 1 Input the original description quantum B-algebra X,soft set S and Pawlak approximation space(X,F),where F is a normal q-filter of X.

Step 3 Compute the values of‖s(ei)‖,in which

Step 4 Find the maximal value ‖s(em)‖ of‖s(ei)‖ (Find the minimum value ‖s(en)‖ of‖s(ei)‖.

Step 5 Outputs(em)(s(em)).

Remark 4 By the decision making method,we know that the final optimal object is decided by the deci-sion maker.In real life,the decision maker can give different weighted selection values according to the actual situation.As we know,many decision-making problems are human in nature and therefore subjective,so there is actually no uniform standard for decision-making.Hence the algorithm is more suitable for practical application.

Now consider Example 7.We give a soft set S which is represented by Table 9,where parameters setA={e1,e2,e3,e4}.

Table 9 A soft set of Example 7

Table 10 of Example 7

Table 11 of Example 7

So we have‖s(e1)‖=0,‖s(e2)‖=0.3,‖s(e3)‖=0.2,‖s(e4)‖=0.2.Then it is easy to find the maximal(mini-mum)value is‖s(em)‖=‖s(e2)‖(‖s(en)‖=‖s(e1)‖).That is,s(e2)={a,b,c}(s(e1)={0,a,b})is the farthest(closest)accurate quantum B-algebra on S.

7 Conclusion

Quantum B-algebras,a class of partially ordered algebras with two binary operations(→ and⇝,featuring non-commutative implication),can be characterized by their embeddability into quantales.They cover the major-ity of implicational algebras and provide a unified semantics for a wide class of algebraic logics.In this paper,we introduce rough approximation of quantum B-algebras.We study rough subalgebras and rough normal q-fil-ters of quantum B-algebras.We give a result of a normal q-filter is a rough normal q-filter of quantum B-alge-bras.Moreover,motivated by[1],we consider lattice-ordered quantum B-algebras by three cases.And we inves-tigate homomorphic images of rough normal q-filters of lattice-ordered quantum B-algebras.We apply rough set theory to CKL-algebras,a special class of quantum B-algebras,and prove that the rough set algebra is a CKL-al-gebra.Finally,we apply rough soft set theory to quantum B-algebras in decision making and design a decision making algorithm for rough soft quantum B-algebras.

The relation between quantum B-algebras and rough set theory is established.We hope these connections will lead to some new research.With rough normal q-filters and their algebraic structures,one can show approxi-mate classification for the elements of an information system,as well as quantum B-algebras.It may be possible to establish some approximate reasoning theories relating to quantum rough normal q-filters.The next task of our research is to find some concrete applications of the results.