Fuzzy ideals of ordered semigroups with fuzzy orderings

Letf,g∈F(X).f⊆∨qkgifandonlyifg(x)⩾f(x)∧1−k2forallx∈X.

Let f⊆∨qkg and x be an element of X. If possible, let g(x)<f(x)∧1−k2. Then there exists a real number α such that g(x)<α<f(x)∧1−k2 which yields that xα∈fbutxα∈∨qk¯g, a contradiction. Thus g(x)⩾f(x)∧1−k2forallx∈X.

Conversely, assume that g(x)⩾f(x)∧1−k2forallx∈X.Letxα be any fuzzy point such that xα∈f. Then g(x)⩾f(x)∧1−k2⩾α∧1−k2.Ifα⩽1−k2,theng(x)⩾α,i.e.,xα∈g.Ifα⩾1−k2,theng(x)⩾1−k2, which implies g(x)+α>1−k2+1−k2=1−k, and hence xαqkg. Thus, in both cases we always have xα∈∨qkg. Therefore, f⊆∨qkg.

The following lemma can be obtained directly from Lemma 2.1.

Letf,g,h∈F(X).Then

(1) f⊆∨qkf.

(2) f⊆∨qkgandg⊆∨qkhimplyf⊆∨qkh.

It is natural to ask wether f⊆∨qkgandg⊆∨qkfimplyf=g. The following example gives a negative answer to this question.

Let f and g be twofuzzy subsets of a setX=a,b,csuchthatfa=0.6,fb=0.6,fc=0.5andga=0.5,gb=0.5,gc=0.6.Then, for anyk∈[0,1),wehavef⊆∨qkgandg⊆∨qkf,butf≠g.

Let f, g ∈ F(X). We define a relation on F(X) as follows. If f⊆∨qkgandg⊆∨qkf,then we writef≡kg. It follows from Lemma 2.2 that ≡ k is an equivalence relation on F(X).

([24, 34 Assume thatXis a nonempty set. A fuzzy relatione : X×X→[0,1]onXis called an fuzzy partial order iffor any x, y, z ∈ X,

(1) (Reflexivity) e(x, x) = 1;

(2) (Transitivity) e(x,y)∧e(y,z)⩽e(x,z);

(3) (Anti-symmetry) e(x,y)=e(y,x)=1impliesx=y.

A nonempty set X equipped with a fuzzy partial order is called afuzzy partially ordered set, or shortly afuzzy poset.

Definition 2.5 ([31, 32]). A fuzzy ordered semigroup is a triple (S, e ⋅)consisting of a nonempty setStogether witha fuzzy relationeand a binary operations ⋅ onSsuch that

(1) (S, e) is a fuzzy poset;

(2) (S, ·)is a semigroups;

(3) e(x,y)⩽e(z⋅x,z⋅y)ande(x,y)⩽e(x⋅z,y⋅z)forallx,y,z∈S.

A fuzzy ordered semigroup (S, e, ⋅) is said to be commutative if x ⋅ y = y ⋅ x for all x, y ∈ S. By the identity of S we mean an element I ∈ S such that x ⋅ I = I ⋅ x = x for all x ∈ S. In what follows, for the sake of simplicity, we shall write xy instead of x ⋅ y, for any x, y ∈ S.

LetSbe a nonempty set, ⩽ bea binary relation onS and ⋅ be a binary operation on S. Forα∈[0,1),we define a fuzzy relatione⩽ on Sas follows: ∀(x,y)∈S×S,

In the sequel, the symbol θ denotes the Godel implication on [0, 1], i.e., θ(r,t)=1,r⩽tt,r>t,∀r,t∈0,1 (see [33]). Then, a binary operation on F(S), which is induced directly by θ, can be defined by θ(f,g)(x)=θ(f(x),g(x))foranyf,g∈F(S)andx∈S.

Define a fuzzy relatione : [0,1]×[0,1]→[0,1]by(∀x,y∈[0,1],e(x,y)=θ(x,y))and a binaryoperation . : [0,1]×[0,1]→[0,1]by(∀x,y∈[0,1],x⋅y=x∧y).Then([0, 1], e, ⋅)is a commutative fuzzyordered semigroup with the identity 1.

([31]). Let(S, e·)be a fuzzy ordered semigroup andf∈F(S).Thenfis called a fuzzy left (resp. right) ideal ofSif the following conditions hold: (1) f(y)⩾f(x)∧e(y,x)foranyx,y∈S; (2) f(xy)⩾f(y)(resp.f(xy)⩾f(x))foranyx,y∈S.

By a fuzzy ideal, we mean the one which is both a fuzzy left and fuzzy right ideal.

Let (S,e,୵) be a fuzzy ordered semigroup. We define a unary operation and a binary operation (·] on F(S), respectively, as follows: ∀f,g∈F(S),∀x∈S,

(1) (f](x)=∨y∈X⁡e(x,y)∧f(y).

(2) (f∘eg)(x)=∨(y,z)∈S×S⁡f(y)∧e(x,yz)∧g(z)

Particularly, for an element x of S, we shall write (x] instead of f_{x}]. It is clear that

In addition, if ( f] = f, then f is called afuzzy lower set (See [24, 34]).

Let (S,e,୵) be a fuzzy ordered semigroup. Then

(1) ∘_eis associative onF(S), that is, (foeg)oeh=foe(goeh)for anyf,g,h∈F(S).

(2) IfSis a commutative fuzzy ordered semigroup, then ∘_eis commutative onF(S), that is, foeg=goefforanyf,g, ∈ F(S).

Finally, we develop some basic properties of the multiplication ∘_e. We will omit the proofs because they are trivial.

Let (S,e,୵) be a fuzzy ordered semigroup andf,g,h_i ∈ F(S), i ∈ I. Then

(1) (foeg]=foeg.

(2) foe(⋃ihi)=⋃ifoehi,(⋃ihi)oef=⋃ihioef.

(3) foe(⋂ihi)⊆⋂ifoehi,(⋂ihi)oef⊆⋂ihioef.

The item (1) in above proposition indicates that for any f;g∈F(S),foeg is a fuzzy lower set, and the item (2) shows that ∘>_e is distributive over arbitrary unions.

Let (S,e,୵) be a fuzzy ordered semigroup andfi;gi∈F(S)(i=1,2)such thatf1⊆∨qkf2andg1⊆∨qkg2.

(1) f1∪g1⊆∨qkf2∪g2.

(2) f1∩g1⊆∨qkf2∩g2.

(3) f1oeg1⊆∨qkf2oeg2.

Let (S,e,୵) be a fuzzy ordered semigroup andA,B⊆S.Then

(1) A⊆BifandonlyiffA⊆∨qkfB.

(2) fA∪fB=fA∪BandfA∩fB=fA∩B.

(3) fA∘efB=(fAB].

A fuzzy subset $f$ of a fuzzy ordered semigroupSis called an (∈, ∈ ∨ q_k)-fuzzy left (resp. right) idealofSiffor anyx, y ∈ Sand α ∈ (0,1)], the following conditions hold.

(F1a) xα∈f⇒yα∈∨qkθ(x],f.

(F2a) y∈Sandxα∈f⇒yxα∈∨qkfresp.xyα∈∨qkf.

A fuzzy subset $f$ of $S$ is called an (∈, ∈ ∨q_k)-fuzzy ideal if it is both an (∈, ∈ ∨q_k)-fuzzy left ideal and (∈, ∈ ∨ q_k)-fuzzy right ideal of $S$. We note that, whenever a statement is made about (∈, ∈ ∨q_k)-fuzzy left ideals, analogous statement holds for (∈, ∈ ∨q_k)-fuzzy right ideals.

In what follows, we develop some characterizations of (∈, ∈ ∨q_k)-fuzzy left ideals of a fuzzy ordered semigroup.

LetSbe a fuzzy ordered semigroup andf ∈ F(S). Thenfis an (∈, ∈ ∨q_k-fuzzy left ideal ofSif and only if

(Flb) f(y)⩾f(x)∧e(y,x)∧1−k2for anyx,y∈S;

(F2b) f(xy)⩾f(y)∧1−k2for anyx,y∈S.

(⟹) Let f be an (∈, ∈ ∨q_k-fuzzy left ideal of S. If f(y)<f(x)∧e(y,x)∧1−k2for somex,y∈S, then there exists α∈[0,1]such thatf(y)<α<f(x)∧e(y,x)∧1−k2. Then it follows that xα∈f. Meanwhile, since f(y)<e(y,x),we haveθ((x],f)(y)=θ(e(y,x),f(y))=f(y). This yields the following two assertions.

(i) yα∈¯θ((x],f).Indeed,sinceθ((x],f)(y)=f(y)<α,we haveyα∈¯θ((x]cf).

(ii) yαqk¯θ((x],f). In fact, a routine calculation implies that θ((x],f)(y)+α+k=f(y)+α+k<1−k2+1−k2+k=1, thus we have yαqk¯θ((x],f).

Hence, yα∈∨qk¯θ((x],f), which contradicts to (Fla). Therefore f(y)⩾f(x)∧e(y,x)∧1−k2 for any x,y ∈ S. In a similar way, we can prove that f(x,y)⩾f(y)∧1−k2 for any x,y∈S.

(⟸) Suppose that (Flb) and (F2b) hold. Let x,y ∈ Sx,y ∈ S and α ∈ (0,1] be such that x_α ∈ f. Then f(x)⩾α. By condition (Flb), we have f(y)⩾α∧e(y,x)∧1−k2.Ifα∧e(y,x)>1−k2,thenf(y)⩾1−k2, which implies θ((x],f)(y)+α+k=θ(e(y,x),f(y))+α+k⩾f(y)+α+k>1,and thenyαqkθ((x],f). If α∧e(y,x)⩽1−k2,thenf(y)⩾f(x)∧e(y,x),it derivesf(y)⩾f(x)orf(y)⩾e(y,x).Sincef(y)⩾f(x) implies θ((x],f)(y)=θ(e(y,x),f(y))⩾f(y)⩾f(x)⩾αandf(y)⩾e(y,x)impliesθ((x],f)(y)=1⩾α, we always have yα∈θ((x],f).Henceyα∈∨qkθ((x],f). This means that (Fla) holds. Similarly, we can prove that (F2a) holds.

Note that every fuzzy left (resp. right) ideal of S according to Definition 2.8 is an (∈, ∈ ∨q_k-fuzzy left (resp. right) ideal of S. However, the following example reveals that an (∈, ∈ ∨q_k-fuzzy left (resp. right) ideal is not necessarily a fuzzy left (resp. right) ideal.

Consider the fuzzy ordered semigroupS = {a,b,c}, where fuzzy ordereand multiplication · are defined respectively as follows:

Letfbe a fuzzy subsets ofSsuch that

Put $k=0.4$. Then it is easy to veri thatfis an (∈, ∈ ∨q_k-fuzzy left ideal ofS, but not a fuzzy left ideal of S because f(ab) = f(a) = 0.6 < 0.8 = f(b).

Let(S,e,·)be a fuzzy ordered semigroup andf ∈ F(S) . Thenfis an (∈, ∈ ∨q_k)-fuzzy left idealofSif and only iffor eachα∈[0,1−k2],f_αis either empty or a left ideal of the ordered semigroup(S,⩽α;⋅)wherefα={x∈S:f(x)⩾α}and⩽α={(x,y)∈S×S:e(x,y)⩾α}.

(⟹) Let α∈[0,1−k2] Then it is easy to check that (S,⩽α;⋅) is an ordered semigroup. Now, if f_α ≠ ⊘, we show that f_α is aleft ideal of (S,⩽α;⋅). Let x, y ∈ S be such that y⩽αxandx∈fα,thene(y,x)⩾αandf(x)⩾α. Since f is an (∈, ∈ ∨q_k-fuzzy left ideal, it follows from Proposition 3.2 that f(y)⩾f(x)∧e(y,x)∧1−k2⩾α, which implies y∈fα.Next, letx∈Sandy∈fα. Then f(xy)⩾f(y)∧1−k2⩾α is a left ideal of (S,⩽α;⋅).

(⟸) For x,y ∈ S, set α=f(x)∧e(y,x)∧1−k2,thenα∈[0,1−k2] and f_α is nonempty. By the assumption, f_α is a left ideal of (S,⩽α;⋅). Since y⩽αxandx∈fα,We have y ∈ f__α, which implies that f(y)⩾α=f(x)∧e(y,x)∧1−k2. Similarly, we have f(xy)⩾f(y)∧1−k2by settingα=f(y)∧1−k2. Therefore, by Proposition 3.2, f is an (∈, ∈ ∨q_k-fuzzy left ideal of (S, e, ·).

LetSbe a fuzzy ordered semigroup andf ∈ F(S) . Thenfis an (∈, ∈ ∨q_k)-fuzzy left ideal ofSif and only if

(F1C)(f]⊆∨qkf;

(F2(C)) fS∘ef⊆∨qkf.

(⟹) Let f be an (∈, ∈ ∨q_k-fuzzy left ideal of S and x∈S. Then, by Proposition 3.2, we have (f](x)∧1−k2=∨y∈Se(x,y)∧f(y)∧1−k2⩽f(x)and(fS∘ef)(x)∧1−k2=∨(y,Z)∈S×S⁡ex,yz∧f(z)∧1−k2⩽∨(y,z)∈S×S⁡ex,yz∧f(yz)∧1−k2⩽fx. It follows from Lemma 2.1 that (f]⊆∨qkf; and fS∘ef⊆∨qkf, i.e., (Flc) and (F2c) hold.

(⟸) Assume that (Flc) and (F2c) hold. If possible, let f(y)<f(x)∧e(y,x)∧1−k2 for some x,y ∈ S. Then there exists a real number α such that f(y)<α<f(x)∧e(y,x)∧1−k2, which implies yα∈(f],yα∈¯f and yαqk¯f, a contradiction to (Flc). Therefore, the condition (Flb) holds. Similarly, we can prove that (F2b) is valid.

Let (S,e,#183;) be a fuzzy ordered semigroup and x∈S. Recall from Section 2 that the fuzzy subset (x] is defined by (x](y)=e(y,x) for all y ∈ S. Now we generalize it as follows. For any x∈S and α ∈ (0,1], U(x;α) is a fuzzy subset of S defined by U(x;α)(y)=α∧e(y,x) for all y ∈ S. One may easily observe that U(x;1) = (x] and xα∈U(x;α).

LetSbe a fuzzy ordered semigroup andf ∈ F(S) . Thenfis an (∈, ∈ ∨q_k)-fuzzy left ideal ofSif and only iffor anyx,y ∈ Sand α ∈ (0,1], the following conditions hold.

(F1d)xα∈f⇒U(x;α)⊆∨qkf;

(F2d) y∈S,xα∈f⇒U(yx;α)⊆∨qkf.

(⟸) Let f be an (∈, ∈ ∨q_k-fuzzy left ideal of S and let x, y ∈ S and α∈(0,1] be such that x_α ∈ f. Then f(x) ⩾ α. By Proposition 3.2, we have f(y)⩾f(x)∧e(y,x)∧1−k2⩾α∧e(y,x)∧1−k2=U(x;α)(y)∧1−k2. Thus Lemma 2.1 implies that U(x;α)⊆∨qkf, and hence (Fld) holds. Moreover, for any z ∈ S, we have f(z)⩾f(yx)∧e(z,yx)∧1−k2⩾f(x)∧e(z,yx)∧1−k2⩾α∧e(z,yx)∧1−k2=U(yx;α)(z)∧1−k2. Thus U(yx;α)⊆∨qkf. So (F2d)$ holds.

(⟸) Assume that (F1d) and (F2d) hold. Let x,y ∈ S and set α=f(x)∧e(y,x)∧1−k2 Then x_α ∈ f. By the assumption, we have U(x;α)⊆∨qkf, which implies f(y)⩾U(x;α)(y)∧1−k2=α∧e(y,x)∧1−k2=f(x)∧e(y,x)∧1−k2 Next, we set α=f(x)∧1−k2 Then (F2d) implies U(yx;α)⊆∨qkf. Thus we have f(yx)⩾U(yx;α)(yx)∧1−k2=α∧e(yx,yx)∧1−k2=α∧1−k2=f(x)∧1−k2. Therefore, by Proposition 3.2, f is an (∈, ∈ ∨q_k-fuzzy left ideal of S.

In the following, we intend to construct an (∈, ∈ ∨q_k-fuzzy left ideal from an arbtrary fuzzy subset of a fuzzy ordered semigroup.

LetSbe a fuzzy ordered semigroup andf ∈ F(S). Then(1−k2)s∘efis an (∈, ∈ ∨q_k-fuzzyleft ideal ofS.

Straightforward by Propositions 2.10 and 3.5.

LetSbe a fuzzy ordered semigroup andf ∈ F(S) . Put

We will complete the proof by three steps.

Step 1: 〈f〉 is an (∈, ∈ ∨q_k-fuzzy left ideal of S. For any x,y ∈ S, define two sets

Let γ=α∧e(y,x)∧(1−k2)∈A1 Then α⩽1−k2;x∈[fα]αorα>1−k2;x∈fα.Ifα⩽1−k2,x∈[fα]α, then γ∈[0,1−k2],andγ⩽αyieldsx∈[fα]α⊆[fγ]γ. If α>1−k2;x∈fα,thenγ∈[0,1−k2]andx∈fα⊆fγ⊆[fγ]γ. So, in both cases, we always have γ∈[0,1−k2]andx∈[fγ]γ. Since e(y,x)⩾γ, we have y∈[fγ]γ Thus γ∈A2. This proves that A1⊆A2. Thus, 〈f〉(y)⩾∨A2⩾∨A1=〈f〉(x)∧e(y,x)∧1−k2. In a similar way, we can prove that 〈f〉(xy)⩾〈f〉(y)∧1−k2 for any x,y ∈ S. Therefore, 〈f〉 is an (∈, ∈ ∨q_k-fuzzy left ideal of S.

Step 2: f⊆{f}. For any x∈S, since fα⊆[fα]α for every α ∈[0,1], we have

This implies f ⊆ 〈f〉, as required.

Step 3: 〈f〉 ⊆ g for any (∈,∈ ∨ q_k-fuzzy left ideal g of S with f ⊆ g. For any α∈[0,1−k2] it follows from Proposition 3.4 that g_α is either empty or a left ideal of the ordered semigroup (S,⩽_α,· In both cases, we always have g_α = [g_α]_α. Thus, for any x∈S, we have

This implies that 〈f〉 ⊆ g.

Naturally, we can consider the greatest (∈, ∈ ∨q_k-fuzzy left ideal of a fuzzy ordered semigroup contained in a fuzzy set. For this, we have the following result.

Letfbe any fuzzy subset of S. Then

For x,y ∈ S, define

Now, we split the proof into three parts as follows.

(i) i(f) is an (∈, ∈ ∨q_k-fuzzy left ideal of S. For any γ=α∧e(y,x)∧(1−k2)∈B1, we have U(x;α)⊆∨qkf,U(ax;α)⊆∨qkf(∀a∈S). Thus, for any z∈S,f(z)⩾U(x;α)(z)∧(1−k2)=α∧e(z,x)∧(1−k2)⩾α∧e(z,y)∧e(y,x)∧(1−k2)=U(y;α∧e(y,x)∧(1−k2))(z)∧(1−k2)=U(y;γ)(z)∧(1−k2)andf(z)⩾Uax;αz∧1−k2=α∧ez,ax∧1−k2⩾α∧ez,ay∧eay,ax∧1−k2⩾α∧ez,ayey,x∧(1−k2)=U(ay,α∧e(y,x)∧(1−k2)(z)∧(1−k2)=U(ay,γ)(z)∧(1−k2),∀a∈S, which implies Uy;γ⊆∨qkf,Uay,y⊆∨qkf,∀a∈S. Thus γ∈B2. This proves that B1⊆B2 and hence i(f)(y)=∨B2⩾∨B1=i(f)(x)∧e(y,x)∧1−k2. In a similar way, we can prove i(f)(xy)⩾i(f)(y)∧1−k2. Therefore, i(f) is an (∈, ∈ ∨q_k-fuzzy left ideal of S.

(ii) i(f)⊆∨qkf For any γ=α∧(1−k2)∈B3, we have U(x;α)⊆∨qkf. Combining Lemma 2.1, we get that f(x)⩾U(x;α)(x)∧(1−k2)=α∧(1−k2)=γ. Since γ is an arbitrary element of B₃, we have f(x)⩾∨B3=i(f)(x)∧(1−k2), i.e.,i(f)⊆∨qkf.

(iii) g ⊆ i(f) for any (∈, ∈ ∨q_k-fuzzy left ideal g that is contained (under ⊆ ∨q_k) in f. For any fuzzy point x_α ∈ g, since g is an (∈, ∈ ∨q_k)-fuzzy left ideal, it follows from Proposition 3.6 that U(x;α)⊆∨qkg and U(ax;α)⊆∨qkgfor anya∈S. Thus, for any x∈S, we have

This implies g ⊆ i(f), as required.

The following two propositions are easy to prove.

Let {f_i : i ∈ Ibe a family of (∈, ∈ ∨q_k)-fuzzy left ideals of a fuzzy ordered semigroup S. Then⋃ifiand⋂ifiare both (∈, ∈ ∨q_k)-fuzzy left ideals ofS.

Letfandgbe two (∈, ∈ ∨q_k)-fuzzy left ideals of afuzzy ordered semigroup S. Then so isfoeg.

Let Fidl (S) be the set of all (∈, ∈ ∨q_k-fuzzy ideals of S. Then it follows from Theorem 3.8, 3.9 and Proposition 3.10 that

Let us recall that a quantale is a triple (Q,*,⩽) such that (Q, ⩽) is a complete lattice, (Q,*) is a semigroup and for any x ∈ Q and {y_i}_i∈I ⊆ Q,

Combing Proposition 2.10, 3.10 and 3.11, we obtain the following theorem.

LetSbe a fuzzy ordered semigroup. Then (Fidl(S), ⊆, ∘_e) is a quantale.

LetSbe a fuzzy ordered semigroup andf ∈ F(S) . Thenfis called an (∈, ∈ ∨q_k-fuzzy interior ideal ofSif it satisfies(Fla)and for anyx,y,z ∈ Sand α, β ∈ (0,1], the following conditions hold. (F3a)xα;YB∈f⇒(xy)α∧B∈∨qkf;(F4a)yα∈f⇒(xyz)α∈∨qkf.

LetSbe a fuzzy ordered semigroup andf ∈ F(S) . Thenfis an (∈, ∈ ∨q_k-fuzzy interior idealofSif and only if(F1b)f(y)⩾f(x)∧e(y,x)∧1−k2foranyx,y∈S;(F3b)f(xy)⩾f(x)∧f(y)∧1−k2foranyx,y∈S;(F4b)f(xyz)⩾f(y)∧1−k2foranyx,y,z∈S.

The proof is similar to that of Proposition 3.2. □

Let(S, e, ⋅)be a fuzzy ordered semigroup andf ∈ F(S). Thenf is an (∈, ∈ ∨q_k)-fuzzy interior ideal of S if and only if for eachα∈[0,1−k2],fαis either empty or an interior ideal of the ordered semigroupS,⩽α,⋅.