The notions of double-framed soft subfields, double-framed soft algebras over double-framedsoft subfields, and double-framed soft hypervector spaces are introduced, and theirproperties and characterizations are considered. The concept of soft set has been introduced by Molodtsov in 1999 [g˘man  studied ginog˘lu  introducginog˘lu  considerginog˘lu –10).The hyperstructure theory was introduced by Marty  at the 8In this paper, using the notion of DFS sets which is introduced in , we intrH × H → P∗(H) is called a hyperoperation or join operation, where P∗(H) is the set of all nonempty subsets of H. The join operation is extended to subsets of H in natural way, so that A∘B is given bya∘A and A∘a are used for {a}∘A and A∘{a}, respectively. Generally, the singleton {a} is identified by its element a.A map ∘:F be a field and  an abelian group. A hypervector space over F is defined to be the quadruplet , where “∘” is a mappinga, b ∈ F and x, y ∈ V the following conditions hold:(H1)a∘(x + y)⊆a∘x + a∘y,(H2)a + b)∘x⊆a∘x + b∘x,((H3)a∘(b∘x) = (ab)∘x,(H4)a∘(−x) = (−a)∘x = −(a∘x),(H5)x ∈ 1∘x. Let V, +, ∘, F) over a field F is said to be strongly left distributive (see [A hypervector space (ive (see ) if it sU be an initial universe set and let E be a set of parameters. We say that the pair  is a soft universe. Let P(U) denote the power set of U and A, B, C,…⊆E.Molodtsov  defined  soft set over U, where A pair U is a parameterized family of subsets of the universe U. For ε ∈ A, ε-approximate elements of the soft set In other words, a soft set over mples in .E be a set of parameters, and A, B, C,… are subsets of E.Let  double-framed soft set  over U where α are mappings from A to P(U).A double-framed pair U and two subsets γ and δ of U, the γ-inclusive set and the δ-exclusive set of eA, respectively, are defined as follows: double-framed including set of For a DFS-set F be a field unless otherwise specified.In what follows, let U is called a double-framed soft subfield of F if the following conditions are satisfied: A double-framed soft set F, then  and α(1)⊇α(0). If a ∈ F, we have(1) For all a ∈ F be such that a ≠ 0. Then(2) Let (3) It is by (1).It is easy to show that the following theorem holds.U is a double-framed soft subfield of F if and only if the nonempty γ-inclusive setδ-exclusive setF for all γ, δ ∈ P(U).A soft set V be algebra over F and let U. A double-framed soft set  double-framed soft algebra over Let V be an algebra over F and let U. If α(0)⊆β(x) for all x ∈ V.Let x ∈ V, we have α(0)⊆α(1)⊆β(x).For any V be an algebra over F and let U. Then, a double-framed soft set Let a, b ∈ F and x, y ∈ V. Also conditions (2) and (3) are hold by Assume that α(0)⊆α(1)⊆β(x) for all x ∈ V. Thus,a ∈ F and x ∈ V. Therefore, Conversely, suppose that V be a hypervector space over F and F. A soft set V is called a double-framed soft hypervector space of V related to α(1)⊆β(θ) where θ is the zero of .Let V be a hypervector space over F and F. If V related to  and β(θ)⊇α(0),Let It is an immediate consequence of V be a hypervector space over F. If V related to a double-framed soft subfield F, thenLet x ∈ V. Since x ∈ 1∘x by (H5), we have β(x)⊆⋂y∈1∘xβ(y).Let β(x) = ⋃y∈1∘xβ(y) for all x ∈ V.By using V over F is strongly left distributive. Let F. Then, a DFS-set V is a double-framed soft hypervector space of V related to  and ⋃z∈a∘x+b∘yβ(z)⊆(α(a) ∪ β(x)) ∪ (α(b) ∪ β(y)), and β(x)⊇α(1),for all a, b ∈ F and all x, y ∈ V.Assume that a hypervector space V related to a, b ∈ F and x, y ∈ V. Then,Assume that x, y ∈ V, we haveF, we have β(a)⊇α(1)⊇α(0), and β(a)⊇α(1)⊇α(−1). Note that 0 ∈ 0∘x for all x ∈ V. It follows thatx ∈ V. Let a ∈ F and x ∈ V. Then,α(1)⊆β(θ). Therefore, V related to Conversely suppose the conditions (1) and (2) are true. For all V be a hypervector space over F and F. If a DFS-set V is a double-framed soft hypervector space of V related to γ-inclusive set,δ-exclusive set,V over the fields eF, respectively, for all γ, δ ∈ P(U).Let F. Let y ∈ a∘x. Then,γ ∈ P(U). Let x, y ∈ eV. Then, β(x)⊆δ and β(y)⊆δ. It follows thatx − y ∈ iV. Note that eF is a subfield of F. Let a ∈ eF, x ∈ iV, and y ∈ a∘x. Then,y ∈ iV which shows that a∘x⊆iV. Therefore, iV is a hypervector space over the field eF for all δ ∈ P(U).Let V be a hypervector space over F and F. If a DFS-set V is a double-framed soft hypervector space of V related to V over the field γ, δ ∈ P(U).Let


We give some results concerning the existence oftripled fixed points for a class of condensing operators in Banach spaces. Further, as an application, we study the existence of solutions for a generalsystem of nonlinear integral equations. Then, the intersection set A∞ = ⋂n=1∞An is nonempty and A∞ is precompact.Let  → 1 implies tn → 0. Then, T has at least one fixed point.Let The following result is a corollary of the previous theorem.C  be a nonempty closed, bounded, and convex subset of X and T : C → C a continuous mapping such that for any subset A of Cφ : ℛ+ → ℛ+ is a nondecreasing and upper semicontinuous functions; that is, for every t > 0, φ(t) < t. Then, T has at least one fixed point.Let G : X × X → X is an element  ∈ X × X such that G = x and G = y.A coupled fixed point of a mapping μ1, μ2,…, μn be measures of noncompactness in Banach spaces E1, E2,…, En, (respectively).Let E1 × E2 × ⋯×En, where Xi is the natural projection of X on Ei, for i = 1,2,…, n, and F is a convex function defined byThen, the functionF be as follows:E1 × E2, where, for i = 1,2, μi are measure of noncompactness in Ei  and Xi, i = 1,2 are the natural projections of X on Ei.Aghajani and Sabzali  illustraΩ be a nonempty, bounded, closed, and convex subset of a Banach space E and let φ : ℛ+ → ℛ+. Assume that φ is a nondecreasing and upper semicontinuous function. Let G : Ω × Ω → Ω be a continuous operator satisfyingμ. Then, G has at least a coupled fixed point.Let x, y, z) of a mapping G : X × X × X → X is called a tripled fixed point ifA tripled , we have thatE × E × E where Xi, i = 1,2, 3 are the natural projections of X on Ei.We can notice that by takingSo, we obtain the following theorem.Ω be a nonempty, bounded, closed, and convex subset of a Banach space E and let φ : ℛ+ → ℛ+ be a nondecreasing and upper semicontinuous function such that φ(t) < t for all t > 0. Then, for any measure of noncompactness μ, the continuous operator G : Ω × Ω × Ω → Ω satifyingLet G has at least a tripled fixed point.To prove this theorem, let us define the measure of noncompactness ℛ+) endowed with the normℛ+) for a positive fixed t on ℬℛ+)BC( is defined as follows:ω0(X), we need first to introduce the modulus of continuity.We can see an application of x ∈ X and ϵ > 0;x on  and letLet (i)ξ, η, q : ℛ+ → ℛ+ are continuous and ξ(t) → ∞ as t → ∞;(ii)ψ : ℛ → ℛ is continuous and there exist positive δ, α such thatt1, t2 ∈ ℛ+;the function (iii)f : ℛ+ × ℛ × ℛ × ℛ × ℛ × ℛ → ℛ is continuous and there exists a nondecareasing continuous function Φ : ℛ → ℛ with Φ(0) = 0, so that(iv)f| is bounded on ℛ+; that is,the function defined by |(v)h : ℛ+ × ℛ+ × ℛ × ℛ → ℛ is a continuous function and there exists a positive solution r0 of the inequalityD is positive constant defined by the equalityx, y, z, u, v, w ∈ BC(ℛ+).Assume thatBC(ℛ+) × BC(ℛ+) × BC(ℛ+).Suppose that (i)–(v) hold; then the system of integral equationsG : BC(ℛ+) × BC(ℛ+) × BC(ℛ+) → BC(ℛ+) be an operator defined byx, y, z) ∈ BC(ℛ+) × BC(ℛ+) × BC(ℛ+), letℛ+) × BC(ℛ+) × BC(ℛ+) is equivalent to the tripled fixed point of G.Let ution of  in BC(t) is continuous for any  ∈ BC(ℛ+) × BC(ℛ+) × BC(ℛ+). Hence, we havex, y, z), ɛ > 0,T > 0 such that if t > T, thenx,y, z ∈ BC(ℛ+). We notice two cases.Obviously, Then, by  and 30)G and , there eCase 1. If t > T, then from (hen from  and 40)Case 1. ICase 2. Similarly, for t ∈ , we haveqT = sup{q(t) : t ∈ }, andβ is continuous on  ×  ×  × , we have β(ϵ) → 0 and ϵ → 0. Thus, using (iii), we getG is a continuous function from  we haveGx,y,zt−Gly, from , we concG satisfies all the conditions of T > 0 and ϵ > 0, assume that X1, X2, and X3 are nonempty chosen subsets of t1, t2 ∈ , with |t2 − t1| ⩽ ϵ. Without loss of generality, letx, y, z) ∈ X1 × X2 × X3,f and h on the compact sets   ×    ×     ×    ×   and  ×  ×  ×  × , respectively, we get ωr0,D1T → 0 and ωr0T → 0 as ϵ → 0.Now, we show that the map T → ∞ in ,  ∈ X1 × X2 × X3, and t ∈ ℛ+, we havex, y, z), , and t are arbitrary in  × BC(ℛ+) × BC(ℛ+).Moreover, Φ is a nondecreasing continuous function with Φ(0) = 0 and (iii), and we obtain,ϵ⟶0.By , we getϵ⟶0.By –(v).ξ, η, q : ℛ+ → ℛ+ are continuous and ξ → ∞ as t → ∞. Further, the function ψ : ℛ → ℛ is continuous for δ = α = 1, and we havet1, t2 ∈ ℛ+. Conditions (i) and (ii) hold.Obviously, Now, letMoreover,x, y, z ∈ BC(ℛ+) × BC(ℛ+) × BC(ℛ+),r > 0, we have thatLet us verify the last condition (v). First,ℛ+) × BC(ℛ+) × BC(ℛ+).Consequently, the system has at least one solution in BC(


After publication of the original article the authors realised that Douglas. R. Call’s name was displayed incorrectly. The correct spelling of his name is included in this erratum; the correct spelling has also been updated in the original article. The authors realised that in Line 347—“AAT” should have been “ATA”, and “OAA” should have been “AAO”. In Line 348—“OIO” should have been “IOO”, “CRD” should have been “DRC” and “AAT” should have been “ATA”. In Line 349—“OIO” should have been “IOO” and “CRD” should have been “DRC”. In Line 325—“47” should have been “41”. In Line 329—“48” should have been “47”. In Line 340—“49” should have been “48”. In Line 503 reference ‘47’ should be removed. In Line 506—‘48’ should have been ‘47’ and in Line 510—reference ‘49’ should have been ‘48’.


This paper studies the existence of solutions for a system of coupled hybridfractional differential equations with Dirichlet boundary conditions. We makeuse of the standard tools of the fixed point theory to establish the main results. The existence and uniqueness result is elaborated with the aid of an example. Fractional calculus is the study of theory and applications of integrals and derivatives of an arbitrary (noninteger) order. This branch of mathematical analysis, extensively investigated in the recent years, has emerged as an effective and powerful tool for the mathematical modeling of several engineering and scientific phenomena. One of the key factors for the popularity of the subject is the nonlocal nature of fractional-order operators. Due to this reason, fractional-order operators are used for describing the hereditary properties of many materials and processes. It clearly reflects from the related literature that the focus of investigation has shifted from classical integer-order models to fractional-order models. For applications in applied and biomedical sciences and engineering, we refer the reader to the books –4. For sHybrid fractional differential equations have also been studied by several researchers. This class of equations involves the fractional derivative of an unknown function hybrid with the nonlinearity depending on it. Some recent results on hybrid differential equations can be found in a series of papers see –37)..37]).cDδ,  cDω denote the Caputo fractional derivative of orders δ, ω, respectively, fi ∈ C and hi ∈ C, i = 1,2.Motivated by some recent studies on hybrid fractional differential equations, we consider the following Dirichlet boundary value problem of coupled hybrid fractional differential equations:The aim of this paper is to obtain some existence results for the given problem. Our first theorem describes the uniqueness of solutions for the problem  by meansIn this section, some basic definitions on fractional calculus and an auxiliary lemma are presented , 2.q for a continuous function g is defined asThe Riemann-Liouville fractional integral of order n-times continuously differentiable function g :  denotes the integer part of the real number q.For at least ϕ ∈ C, the integral solution of the problemGiven c0, c1 ∈ R are arbitrary constants. Alternatively, we havex(0) = 0 = x(1) in =f=f∣w(t) ∈ C1} denote a Banach space equipped with the norm ||w|| = max{|w(t)|, t ∈ }, where W = U, V. Notice that the product space ||) with the norm |||| = ||x|| + ||y||,  ∈ U × V is also a Banach space.Let U × V → U × V byA1fi   are continuous and bounded; that is, there exist positive numbers μfi such that |fi| ≤ μfi, ∀ ∈  × R × R.The functions A2) such that |h1| ≤ ρ0 + ρ1|x| + ρ2|y| and |h2| ≤ σ0 + σ1|x| + σ2|y|, ∀x, y ∈ R, i = 1,2.There exist real constants For brevity, let us setIn view of Now we are in a position to present our first result that deals with the existence and uniqueness of solutions for the problem . This reA1) holds and that h1, h2 :  × R2 → R are continuous functions. In addition, there exist positive constants ηi, ζi, i = 1,2 such thatν1(η1 + η2) + ν2(ζ1 + ζ2) < 1ν1 and ν2 are given by  = κ1 < ∞, supt∈h2 = κ2 < ∞ and define a closed ball: x, y)|| ≤ r.Let us set sup].Hence||Θ1ind that||Θ2x1, y1),  ∈ U × V and for any t ∈ , we haveν1(η1 + η2) + ν2(ζ1 + ζ2) < 1, it follows that Θ is a contraction. So Banach's fixed point theorem applies and hence the operator Θ has a unique fixed point. This, in turn, implies that the problem  = (1/2)(|sinu(t)| + 1), f2 = (1/2)(|cosu(t)| + 1), h1 = (1/4(t+2)2)(|x|/(1 + |x|)) + 1 + (1/32)  sin2y  , and h2 = (1/32π)sin(2πx) + |y|/16(1 + |y|) + 1/2. Note thatConsider the following coupled system of hybrid fractional differential equations:Dc3/2. Let P(F) = {x ∈ G : x = λFx forsome 0 < λ < 1}. Then either the set P(F) is unbounded or F has at least one fixed point.Let A1) and (A2) hold. Furthermore, it is assumed that ν1ρ1 + ν2σ1 < 1 and ν1ρ2 + ν2σ2 < 1, where ν1 and ν2 are given by how that||Θ2τ1, τ2 ∈  with τ1 < τ2 and obtainx, y). This implies that the operator Θ is equicontinuous. Thus, by the above findings, the operator Θ is completely continuous.Now we show that the operator Θ is equicontinuous. For that, we take P = { ∈ U × V∣ = λΘ, 0 ≤ λ ≤ 1} is bounded. Let  ∈ P; then we have  = λΘ. Thus, for any t ∈ , we can writeP is bounded. Hence all the conditions of In the next step, it will be established that the set  view of , can be


P-orders and R-orders are introduced and some related properties are investigated. An example is presented to show that the R-union of two internal cubic soft sets might not be internal. A sufficient condition is provided, which ensure that the R-union of two internal cubic soft sets is also internal. Moreover, some properties of cubic soft subalgebras of BCK/BCI-algebras based on a given parameter are discussed.Operations of cubic soft sets including “AND” operation and “OR” operation based on  P-union, P-intersection, R-union, and R-intersection of cubic sets and investigated several related properties. Later on, Jun et al.  the set of all interval numbers. Let us define what is known as refined minimum and refined maximum  of two elements in [I]. We also define the symbols “⪰,” “⪯,” and “ = ” in case of two elements in [I]. Consider two interval numbers i ∈ Λ. We define complement, denoted by By anX be a nonempty set. A function A : X → [I] is called an interval-valued fuzzy set  in X. Let [I]X stand for the set of all IVF sets in X. For every A ∈ [I]X and x ∈ X, A(x) =  is called the degree of membership of an element x to A, where A− : X → I and A+ : X → I are fuzzy sets in X which are called a lower fuzzy set and an upper fuzzy set in X, respectively. For simplicity, we denote A = . For every A, B ∈ [I]X, we defineAc of A ∈ [I]X is defined as follows: Ac(x) = A(x)c for all x ∈ X; that is,Ai | i ∈ Λ} of IVF sets in X where Λ is an index set, the union G = ⋃i∈ΛAi and the intersection F = ⋂i∈ΛAi are defined as follows:x ∈ X, respectively.Let U be an initial universe set and let E be a set of parameters. Let P(U) denote the power set of U and A ⊂ E.Molodtsov  defined F, A) is called a soft set over U, where F is a mapping given byA pair  may be considered as the set of ɛ-approximate elements of the soft set . Clearly, a soft set is not a set. For illustration, Molodtsov considered several examples in ∈[I] and t ∈ .Given a parameter F, A) over U is an ɛ-cubic soft subalgebra over U and let x, y ∈ U. If δ1, δ2]∈[I], then x, y ∈ λF(ɛ)→(t) for all t ∈ , then λF(ɛ)(x) ≤ t and λF(ɛ)(y) ≤ t. Using (λF(ɛ)(x∗y) ≤ max{λF(ɛ)(x), λF(ɛ)(y)} ≤ t, and so x∗y ∈ λF(ɛ)→(t). Therefore λF(ɛ)→(t) are subalgebras of U.Assume that a cubic soft set →(t) are subalgebras of U for all ∈[I] and t ∈ . Assume that there exists a, b ∈ U such thatδ1 < min{γ1, γ3} and δ2 < min{γ2, γ4}. Takingx, y ∈ U. Now, assume that λF(ɛ)(a∗b) > max{λF(ɛ)(a), λF(ɛ)(b)} for some a, b ∈ U. Then there exists t0 ∈  such thata, b ∈ λF(ɛ)→(t0) but a∗b ∉ λF(ɛ)→(t0). This is a contradiction, and thereforex, y ∈ U. Consequently,  is an ɛ-cubic soft subalgebra over U.Conversely, suppose that ɛ ∈ A, if a cubic soft set  over U is an ɛ-cubic soft subalgebra over U, then λF(ɛ)(0) ≤ λF(ɛ)(x), for all x ∈ U.Given a parameter x ∈ U, we haveλF(ɛ)(0) = λF(ɛ)(x∗x) ≤ max{λF(ɛ)(x), λF(ɛ)(x)} = λF(ɛ)(x).For every F, A) be an ɛ-cubic soft subalgebra over U for a parameter ɛ ∈ A. If there is a sequence in U such that n→∞λF(ε)(xn) = 0, then λF(ɛ)(0) = 0.Let (λF(ɛ)(0) ≤ λF(ɛ)(x), for all x ∈ U, we haven. Note that λF(ɛ)(0) ≤ λF(ɛ)(xn) = 0. Hence λF(ɛ)(0) = 0.Since ɛ ∈ A, if a cubic soft set  over U is an ɛ-cubic soft subalgebra over U, then the sets UλF(ɛ)∶ = {x ∈ U | λF(ɛ)(x) = λF(ɛ)(0)} are subalgebras of U.Given a parameter x, y ∈ U. If λF(ɛ)(x∗y) ≤ max{λF(ɛ)(x), λF(ɛ)(y)} = max{λF(ɛ)(0), λF(ɛ)(0)} = λF(ɛ)(0). Combining this and λF(ɛ)(x∗y) = λF(ɛ)(0). This shows that x∗y ∈ UλF(ɛ). Therefore UλF(ɛ) are subalgebras of U.Let ɛ ∈ A, if a cubic soft set  over U is an ɛ-cubic soft subalgebra over U, then the set U.Given a parameter The proof is straightforward.P-order and the R-order. In [R-union of two internal cubic soft sets an internal cubic soft set? We have given an example to show that the answer to this question is negative, and then we have provided a condition for the R-union of two internal cubic soft sets to be an internal cubic soft set. We also have investigated several properties of cubic soft subalgebras of BCK/BCI-algebras based on any given parameter. Some important issues to be explored in the future includedeveloping strategies for obtaining more valuable results,applying these notions and results for studying related notions in other (soft) algebraic structures.In this paper, we first have considered operations of cubic soft sets, that is, “AND” operation and “OR” operation based on the rder. In , Muhiudd


In this paper, further properties and characterizations of int-soft left (right) ideals are studied, and the notion of int-soft  bi-ideals is introduced. Relations between int-soft generalized bi-ideals and int-soft semigroups are discussed, and characterizations of (int-soft) generalized bi-ideals and int-soft bi-ideals are considered. Given a soft set  As a new mathematical tool for dealing with uncertainties, the notion of soft sets is introduced by Molodtsov . The autα, S) over U, we establish int-soft  bi-ideals generated by .Our aim in this paper is to apply the soft sets to one of abstract algebraic structures, the so-called semigroup. So, we take a semigroup as the parameter set for combining soft sets with semigroups. This paper is a continuation of . We firsS be a semigroup. Let A and B be subsets of S. Then the multiplication of A and B is defined as follows:Let S is said to be regular if for every x ∈ S there exists a ∈ S such that xax = x.A semigroup S is said to be left  zero if xy = x  for all x, y ∈ S.A semigroup S is said to be left (right) simple if it contains no proper left (right) ideal.A semigroup S is said to be simple if it contains no proper two-sided ideal.A semigroup A of S is calledS if AA⊆A, that is, ab ∈ A for all a, b ∈ A,a subsemigroup of S if SA⊆A , that is, xa ∈ A  for all x ∈ S and a ∈ A,a left  ideal of S if it is both a left and a right ideal of S,a two-sided ideal of S if ASA⊆A,a generalized bi-ideal of S if it is both a semigroup and a generalized bi-ideal of S.a bi-ideal of A nonempty subset A soft set theory is introduced by Molodtsov , and ÇağU be an initial universe set and E be a set of parameters. Let P(U) denote the power set of U and A, B, C,…, ⊆E.In what follows, let α, A) over U is defined to be the set of ordered pairsα : E → P(U) such that α(x) = ∅ if x ∉ A.A soft set . The subscript A in the notation α indicates that α is the approximate function of .The function α, A) over U and a subset γ of U, the γ-inclusive set of , denoted by iA, is defined to be the setFor a soft set  and  over U, we defineα, S) and  is defined to be the soft set U in which α, S) and  is defined to be the soft set U in which α, S) and  is defined to be the soft set U in which S to P(U) given byFor any soft sets  over U is called an int-soft semigroup over U if it satisfiesA soft set  over U is called an int-soft left  ideal over U if it satisfiesA soft set  over U is both an int-soft left ideal and an int-soft right ideal over U, we say that  is an int-soft two-sided ideal over U.If a soft set  ideal over ral (see ).α, S) be an int-soft left ideal over U. If G is a left zero subsemigroup of S, then the restriction of  to G is constant; that is, α(x) = α(y) for all x, y ∈ G.Let  = α(y) for all x, y ∈ G.Let Similarly, we have the following proposition.α, S) be an int-soft right ideal over U. If G is a right zero subsemigroup of S, then the restriction of  to G is constant; that is, α(x) = α(y) for all x, y ∈ G.Let  be an int-soft left ideal over U. If the set of all idempotent elements of S forms a left zero subsemigroup of S, then α(x) = α(y) for all idempotent elements x and y of S.Let , we have uv = u and vu = v. Henceα(u) = α(v) for all idempotent elements u and v of S.Assume that the setSimilarly, we have the following proposition.α, S) be an int-soft right ideal over U. If the set of all idempotent elements of S forms a right zero subsemigroup of S, then α(x) = α(y) for all idempotent elements x and y of S.Let  with Φ⊋Ψ, define a map χA as follows:χA, S) is a soft set over U, which is called the -characteristic soft set. The soft set , S) is called the -identity soft set over U. The -characteristic soft set with Φ = U and Ψ = ∅ is called the characteristic soft set and is denoted by . The -identity soft set with Φ = U and Ψ = ∅ is called the identity soft set and is denoted by .For a nonempty subset χA, S) and , S) be -characteristic soft sets over U where A and B are nonempty subsets of S. Then the following properties hold: Let  Let x ∈ S, suppose x ∈ AB. Then there exist a ∈ A and b ∈ B such that x = ab. Thus we havex ∈ AB, we get χAB(x) = Φ. Suppose x ∉ AB. Then x ≠ ab for all a ∈ A and b ∈ B. If x = yz for some y, z ∈ S, then y ∉ A or z ∉ B. Hencex ≠ yz for all x, y ∈ S, then(2) For any χS, S), let  be a soft set over U such that α(x)⊆Φ for all x ∈ S. Then the following assertions are equivalent:α, S) is an int-soft left ideal over U,-identity soft set  is an int-soft left ideal over U. Let x ∈ S. If x = yz for some y, z ∈ S, thenSuppose that  is an int-soft left ideal over U.Conversely, assume that Similarly, we have the following theorem.χS, S), let  be a soft set over U such that α(x)⊆Φ for all x ∈ S. Then the following assertions are equivalent: α, S) is an int-soft right ideal over U,-identity soft set , S), let  be a soft set over U such that α(x)⊆Φ for all x ∈ S. Then the following assertions are equivalent:α, S) is an int-soft two-sided ideal over U,-identity soft set  ideal over U. In fact, the soft intersection of int-soft left  ideals containing a soft set  over U is the smallest int-soft left  ideal over U.Note that the soft intersection of int-soft left  ideals over α, S) over U, the smallest int-soft left  ideal over U containing  is called the int-soft left  ideal over U generated by  and is denoted by l  r, 2).For any soft set l = , wherea ∈ S.Let a ∈ S. Since a = ea, we havex, y ∈ S, we haveβ, S) is an int-soft left ideal over U. Now let  be an int-soft left ideal over U such that α(a)⊆γ(a) for all a ∈ S andα, S)l = .Let Similarly, we have the following theorem.S be a monoid with identity e. Then r = , wherea ∈ S.Let α, S) over U is called an int-soft generalized bi-ideal over U if it satisfiesA soft set  be a soft set over U = Z defined as follows:α, S) is an int-soft generalized bi-ideal over U = Z. But it is not an int-soft semigroup over U = Z since α(c)∩α(c) = 4Z⊈4N = α(b) = α(cc).Let α, S) over U is both an int-soft semigroup and an int-soft generalized bi-ideal over U, then we say that  is an int-soft bi-ideal over U.If a soft set  be an int-soft generalized bi-ideal over U and let x and y be any elements of S. Then there exists a ∈ S such that y = yay, and soα, S) is an int-soft semigroup over U.Let  bi-ideal.A of S, the following are equivalent: A is a generalized bi-ideal of S,χA, S) over U is an int-soft generalized bi-ideal over U for any Φ, Ψ ∈ P(U) with Φ⊋Ψ.the -characteristic soft set  with Φ⊋Ψ and x, y, z ∈ S. If x, z ∈ A, then χA(x) = Φ = χA(z) and xyz ∈ ASA⊆A. Hencex ∉ A or z ∉ A, then χA(x) = Ψ or χA(z) = Ψ. HenceχA, S) is an int-soft generalized bi-ideal over U for any Φ, Ψ ∈ P(U) with Φ⊋Ψ.Assume that χA, S) over U is an int-soft generalized bi-ideal over U for any Φ, Ψ ∈ P(U) with Φ⊋Ψ. Let a be any element of ASA. Then a = xyz for some x, z ∈ A and y ∈ S. ThenχA(a) = Φ. Thus a ∈ A, which shows that ASA⊆A. Therefore A is a generalized bi-ideal of S.Conversely, suppose that the -characteristic soft set  over U is an int-soft generalized bi-ideal over U if and only if the nonempty γ-inclusive set of  is a generalized bi-ideal of S for all γ⊆U.A soft set  is an int-soft semigroup over U. Let γ⊆U be such that iS ≠ ∅. Let a ∈ S and x, y ∈ iS. Then α(x)⊇γ and α(y)⊇γ. It follows from . Thus iS is a generalized bi-ideal of S.Assume that  is a generalized bi-ideal of S for all γ⊆U. Let x, y, z ∈ S be such that α(x) = γx and α(z) = γz. Taking γ = γx∩γz implies that x, z ∈ iS. Hence xyz ∈ iS, and soα, S) is an int-soft generalized bi-ideal over U.Conversely, suppose that the nonempty α, S) over U is an int-soft semigroup over U if and only if the nonempty γ-inclusive set of  is a subsemigroup of S for all γ⊆U.A soft set  over U is an int-soft bi-ideal over U if and only if the nonempty γ-inclusive set of  is a bi-ideal of S for all γ⊆U.A soft set  and a soft set  over U, the following are equivalent: α, S) is an int-soft generalized bi-ideal over U, is an int-soft generalized bi-ideal over U. Let a be any element of S. If x, y, u, v ∈ S such that a = xy and x = uv. Since  is an int-soft generalized bi-ideal over U, it follows from  is an int-soft generalized bi-ideal over U.Therefore  be an int-soft semigroup over U. Then  is an int-soft bi-ideal over U if and only if χS, S) is the identity soft set over U.Let  and  are int-soft generalized bi-ideals over U, then so is the soft intersection If  and  are int-soft bi-ideals over U, then so is the soft intersection If  be an int-soft generalized bi-ideal over U. For any x ∈ S, we haveα(x) = α(e). Therefore α is a constant function.Let α, S) over U, the smallest int-soft  bi-ideal over U containing  is called an int-soft  bi-ideal over U generated by  and is denoted by b.For any soft set  be a soft set over U such that α(x)⊆α(e) for all x ∈ S. Then b = , wherea ∈ S.Let a ∈ S. Since a = eea it follows from hypothesis thatx, y, z ∈ S, we getβ(x)∩β(z)⊆β(xyz). Let y = e. Then β(x)∩β(z)⊆β(xz) for all x, z ∈ S. Hence  is an int-soft  bi-ideal over U. Let  be an int-soft  bi-ideal over U such that a ∈ S, we haveα, S)b = .Let


Activated Gαq subunits have been shown to interact directly with Fhit, up-regulate Fhit expression and enhance its suppressive effect on cell growth and migration. Other signaling molecules may be involved in modulating Gαq/Fhit interaction.G proteins are known to modulate various growth signals and are implicated in the regulation of tumorigenesis. The tumor suppressor Fhit is a newly identified interaction partner of Gq and Fhit, co-immunoprecipication assay was performed on HEK293 cells co-transfected with different combinations of Flag-Fhit, Gα16, Gα16QL, pcDNA3 vector, and PLCβ isoforms. Possible associations of Fhit with other effectors of Gαq were also demonstrated by co-immunoprecipitation. The regions of Gαq for Fhit interaction and PLCβ stimulation were further evaluated by inositol phosphates accumulation assay using a series of Gα16/z chimeras with discrete regions of Gα16 replaced by those of Gαz.To test the relationship of PLCβ with the interaction between Gαq. Expression of PLCβ increased the affinities of Fhit for both wild-type and activated Gαq. Swapping of the Fhit-interacting α2-β4 region of Gαq with Gαi eliminated the association of Gαq with Fhit without affecting the ability of the mutant to stimulate PLCβ. Other effectors of Gαq including RGS2 and p63RhoGEF were unable to interact with Fhit.PLCβ1, 2 and 3 interacted with Fhit regardless of the expression of Gαq in a unique way. PLCβ interacts with Fhit and increases the interaction between Gαq and Fhit. The Gαq/PLCβ/Fhit complex formation points to a novel signaling pathway that may negatively regulate tumor cell growth.PLCβ may participate in the regulation of Fhit by GThe online version of this article (doi:10.1186/s12885-015-1802-z) contains supplementary material, which is available to authorized users. FHIT gene at the chromosomal fragile site FRA3B is often regarded as an early target of DNA damage in precancerous cells. Its gene product, the ubiquitously expressed Fhit (Fragile Histidine Triad) protein, is a member of the HIT (histidine triad) superfamily with three signature histidines in the conserved nucleoside binding motif. Fhit binds and hydrolyzes various dinucleoside polyphosphates  into two nucleotides where one is a nucleoside monophosphate inositol overnight. The labeled cells were then washed once with IP3 assay medium  and then incubated with 500 μl IP3 assay medium at 37 °C for 1 h. Reactions were stopped by replacing the assay medium with 750 μL ice-cold 20 mM formic acid and the lysates were kept in 4 °C for 30 min before the separation of [3H]inositol phosphates from other labeled species by sequential ion-exchange chromatography as described previously [HEK293 cells were seeded at a density of 2×10eviously .t test. Differences at values of P < 0.05 were considered significant (* P < 0.05).Data were expressed as the mean ± S.E. of at least three independent sets of experiments. The probability of an observed difference being a coincidence was evaluated by Dunnett q family  via their α2-β4 region without affecting Gαq-induced PLCβ activation [q [q in co-immunoprecipitation assays. HEK293 cells were co-transfected with Flag-tagged Fhit and wild-type or activated Gα16 (Gα16QL) with or without PLCβ1, PLCβ2 or PLCβ3. Because activated Gαq signaling always increase the expression levels of Fhit [q subunits, we have opted for using Gα16 as a representative Gαq member. In vector transfected cells, Fhit pulled down detectably more Gα16QL than wild-type Gα16 . The reduction of Gα16QL in the PLCβ3-immunoprecipitates was not due to variations in the expression levels or pull down efficiency, as these parameters were essentially similar in the different samples , we have previously identified the α2-β4 region of Gα16 as critical for Fhit interaction [q is seemingly involved in binding to PLCβ [q, PLCβ and Fhit, we constructed two new chimeras named zα2β4 (Gα16 backbone with α2β4 region from Gαz) and 16α2β4 (Gαz backbone with α2β4 region from Gα16), wherein the α2-β4 region of Gα16 or Gαz was swapped with each other . Because the Gα-specific antibodies are N-terminal targeting, the zα2β4 and 16α2β4 chimeras were recognized by anti-Gα16 and anti-Gαz antisera, respectively. When examined for their ability to stimulate PLCβ, activated zα2β4QL efficiently stimulated the formation of inositol phosphates to an extent similar to that of Gα16QL  tested as well as the endogenously expressed PLCβ1 could be detected in the immunoprecipitates of Fhit in the absence of Gαq overexpression  [q could be detected   but thisted Fig. . The incβs Figs.  may resuq is essential for the binding of Fhit [q-PLCβ3 complex, PLCβ3 interacts with the α2 and α3 region of Gαq by a helix-turn-helix domain [q/p63RhoGEF complex [q was present  for PLCβ3 interaction are the same with members of Gαi subfamily while the exceptional residue on Gαi subunits corresponding to R214 of Gαq is identical to that of Gα16 [16 (a Gαq member) with the corresponding region of Gαz (a Gαi subfamily member) would still allow the zα2β4QL chimera to interact productively with PLCβ  of cell growth inhibition by Gαq and PLCβ.PLCβ is a key molecule in transducing activated Gt cancer  and it pt cancer . On the t cancer . PLCβ3-dt cancer . The intq binds to Fhit through the α2-β4 region of Gαq (Fig. q including the α2 region (Fig. q is the α2 region (Fig. 16 with the corresponding region of Gαz did not affect the activated Gα16-induced PLCβ activation (Fig. q and change the binding interface between Gαq and PLCβ without affecting the PLCβ activity. The altered binding interfaces of activated Gαq and PLCβ may trigger Gαq and PLCβ to form a ‘clamp’ around Fhit. Beside protein interactions that were found between two protein pairs among Gαq, PLCβ and Fhit, PLCβs increased the interaction between Gαq and Fhit. But Fhit or Gαq did not enhance the interaction between PLCβ and Gαq or Fhit, respectively (Fig. q by PLCβ is that PLCβ may interact with and stabilize the complex of Fhit and Gαq. Another possibility is that when binding to Gαq, PLCβ may provide direct binding sites on itself for Fhit which also leads to the formation of a heterotrimeric protein complex. In both possibilities, the activated Gαq recruits PLCβ which acts as a positive regulator for the association of Fhit with activated Gαq. In the future, the involvement of PLCβ in the regulation of Fhit by Gαq and their possible roles on cancer therapy should be demonstrated.Activated Gαion Fig. . As subsion Fig. , Fhit maely Fig. . One posq and Fhit. This regulatory effect appears to be unique to PLCβ because other Gαq effectors such as RGS2 and p63RhoGEF could not interact with Fhit. Substitution of the α2-β4 region of Gαq with Gαi did not affect Gαq-induced PLCβ activation but eliminated the interaction between Gαq with Fhit. This new Gαq/PLCβ/Fhit signaling complex represents a novel pathway of Gαq regulation on tumor suppression.We showed that PLCβ could interact with Fhit, and the expression of PLCβ increased the interaction between Gα


Following up on this idea we obtain several results and conclusions of interest. We also discuss the notion of a couplet  on X, consisting of a two-step derivation d and its square D = d ∘ d, for example, whose defining property leads to further observations on the underlying ranked trigroupoids also.We define a ranked trigroupoid as a natural followup on the idea of a ranked bigroupoid. We consider the idea of a derivation on such a trigroupoid as representing a two-step process on a pair of ranked bigroupoids where the mapping  The notion of derivations arising in analytic theory is extremely helpful in exploring the structures and properties of algebraic systems. Several authors , 2 studid-algebras which is another useful generalization of BCK-algebras and then investigated several relations between d-algebras and BCK-algebras as well as several other relations between d-algebras and oriented digraphs +[xd(1) − d(x)] = D(x) + xd(1) − d(x). It follows that, for any x ∈ R,xd(y) − d(x)y = xyd(1) − xd(1)y = 0. Hence, by (D(xy) = D(x)y − xD(y) + xd(y) − d(x)y = D(x)y − xD(y); that is,Thus : = 1 in , then D = D(x)·(−1) − xD(−1). Hence D(x) = −xD(−1) for all x < 0.If we let  = −1 in , then wed is a two-step derivation on , then d(xy) = d(x)y + xd(y) and d(x + y) = (d(x) + y)−(x + d(y)) for any x, y ∈ R. It follows that d(2x) = d(x + x) = (d(x) + x)−(x + d(x)) = 0, which proves that d(x) = d(2 · (x/2)) = 0 for all x ∈ R.In X : = ) 1 and 02 are both zeros, then x ≠ 01, 02 yields 01 = x∗x = 02.The collection of frame algebras includes the collection of BCK-algebras and it is a variety. In a frame algebra the element 0 is unique. Indeed, if 0X, ∗i, 0) forms a subsemigroup of the semigroup (Bin(X), □).The collection of all frame algebras , , if we let : =   □  , then x  □  y = (x∗y)•(y∗x) for all x, y ∈ X. It follows that x  □  x = (x∗x)•(x∗x) = 0•0 = 0, 0  x = (0∗x)•(x∗0) = 0  •  x = 0, and x  □  0 = (x∗0)•(0∗x) = x•0 = x, proving that  is a frame algebra. This proves the proposition.Given frame algebras ,  ∈ Bin(X), we defineGiven groupoids  and  be frame algebras. If ∈S, then  is a frame algebra.Let ∈S, then x  ∇  y ∈ {x∗y, x•y} for all x, y ∈ X. It follows that x  ∇  x ∈ {x∗x, x•x} = {0} implies that x  ∇  x = 0. Moreover, 0  ∇  x ∈ {0∗x, 0•x} = {0} shows that 0  ∇  x = 0, and x  ∇  0 ∈ {x∗0, x•0} = {x} shows that x  ∇  0 = x, proving that  is a frame algebra.If  is called an fr(3)-algebra if(G)X, ∗, 01), ,  are frame algebras,((H)1 = 02 = 03.0A ranked trigroupoid  be an fr(3)-algebra. If d : X → X is a two-step derivation on X, then d(0) = 0,d(x) = d(x)•x = d(x)  ◊  x for all x ∈ X.Let , then for any x, y ∈ x, we haved(0) = d(0∗y) = (d(0)∗y)•(0∗d(y)) = (d(0)∗y)•0 = d(0)∗y; that is, d(0) = d(0)∗y, for all y ∈ X. If we let y : = d(0), then by applying (A) we obtain d(0) = d(0)∗d(0) = 0.(i) If x ∈ X, we have d(x) = d(x∗  0) = (d(x)∗0)  •  (x∗d(0)) = d(x)•x and d(x) = d(x•  0) = (d(x)•0)  ◊  (x•d(0)) = d(x)  ◊  x.(ii) Given d on a trigroupoid , we denote its kernel by Ker(d): = {x ∈ X∣d(x) = 0}.Given a two-step derivation X, ∗, •, ◊) be an fr(3)-algebra. If d : X → X is a two-step derivation on X, then x∗d(x), x•d(x) ∈ Ker(d),x ∈ Ker(d) implies that x∗y, x•y ∈ Ker(d),Ker(d)⊆Ker(d2),for all x, y ∈ X.Let (y : = d(x) in (d(x∗d(x)) = (d(x)∗d(x))  •  (x∗d(d(x))) = 0  •  (x∗d2(x)) = 0 an d(x•  d(x)) = (d(x)  •  d(x))  ◊  (x•  d(d(x))) = 0  ◊  (x•  d2(x)) = 0, proving that x∗d(x), x•d(x) ∈ Ker(d) for any x ∈ X.(i) If we let  d(x) in  and 25)y : = d(xx ∈ Ker(d), then d(x∗y) = (d(x)∗y)•(x∗d(y)) = (0∗y)•(x∗d(y)) = 0 and d(x•y) = (d(x)•y)  ◊  (x•  d(y)) = 0 for any y ∈ X, proving that x∗y,   x•y ∈ Ker(d).(ii) If x ∈ Ker(d), then d2(x) = d(d(x)) = d(0) = 0 by x ∈ Ker(d2).(iii) If X, ∗, •, ◊) be an fr(3)-algebra. If d : X → X is a two-step derivation on X, theny ∈ X.Let (x ∈ Ker(d2), then d(d(x)) = 0 and hencey ∈ X, which proves that x∗y ∈ Ker(d2).If d2(x•y) may not be computable unless the behavior of d(u  ◊  v) is specified, since d2(x•y) = d((d(x)•y)  ◊  (x•d(y))) = d(u  ◊  v) for some u, v ∈ X.Note that X, ∗, •, ◊) be an fr(3)-algebra. If  is a couplet of , then Ker(D)∗X⊆Ker(D).Let  is a couplet of  and if x ∈ Ker(D), then D(x)∗y = 0∗y = 0 for any y ∈ X. It follows from (D(x∗y) = 0  ◊  [(d(x)∗y)•[(d(x)∗d(y))•(x∗D(y))]] = 0, proving that x∗y ∈ Ker(D).If  be a poset with minimal element 0. Define a binary operation “∗” on X byX, ∗, 0) is a BCK-algebra, called a standard BCK-algebra inherited from the poset .Let  be a standard BCK-algebra. Let  be an fr(3)-algebra and let  be a couplet of . If D(x∗y) ≠ 0 for some x, y ∈ X, then D(x∗y) = D(x)∗y = D(x) and x∗y = x.Let (D(x∗y) ≠ 0. We claim that D(x)∗y ≠ 0. Suppose that D(x)∗y = 0. Since  is a fr(3)-algebra, by applying (D(x∗y) = 0 is a contradiction. Since  is a standard BCK-algebra, we obtain D(x)∗y = D(x). We claim that x∗y = x. If x∗y = 0, then D(x∗y) = D(0) = 0 is a contradiction. It follows that D(x∗y) = D(x) = D(x)∗y, proving the proposition.Let applying , we obtaX : = R be the real field and let  be a ranked trigroupoid, where , , and   are linear groupoids; that is, x∗y : = A + Bx + Cy, x•y : = a + bx + cy, and x  ◊  y : = α + βx + γy for any x, y ∈ X, where A, B, C, a, b, c, α, β, γ ∈ X (fixed).Let d : X → X is a two-step derivation on , then for any x, y ∈ X, we haveThus, if d(x) = 0 for all x ∈ X, then A = 0,  α + (β + γ)a = 0. Hence, d(x) = 0 X, then (a+(b+c)A+X, then (a+(b+c)A+ Hence,  be a ranked trigroupoid, where , ,  are linear groupoids; that is, x∗y : = A + Bx + Cy, x•y : = a + bx + cy, and x  ◊  y : = α + βx + γy for any x, y ∈ X, where A, B, C, a, b, c, α, β, γ ∈ X (fixed). Let d : X → X be a two-step derivation such that d(x) = 0 for all x ∈ X. bc ≠ 0 and b + c ≠ 0, then x∗y = −a/(b + c), x•y : = a + bx + cy, x  ◊  y = 0;If bc ≠ 0 and b + c = 0, then x∗y = A, x•y = b(x − y), x  ◊  y = 0.If Let bc ≠ 0, then it follows from  and x  ◊  y = 0.(i) If ows from  that we bc ≠ 0 and b + c = 0, then 0 = a + (b + c)A = a and A is arbitrary, and hence we obtain the result.(ii) If X : = R be the real field and let  be a ranked trigroupoid, where , ,  are linear groupoids; that is, x∗y : = A + Bx + Cy, x•y : = a + bx + cy, and x   ◊  y : = α + βx + γy for any x, y ∈ X, where A, B, C, a, b, c, α, β, γ ∈ X (fixed). Let d : X → X be a two-step derivation such that d(x) = 0 for all x ∈ X:b = c = 0 and b + c ≠ 0, then x∗y = A + Bx + Cy, x•y : = 0, x  ◊  y = βx + γy;if b ≠ 0, c = β = 0, then x∗y = −a/b + Bx, x•y = a + bx, x  ◊  y = 0;if b ≠ 0, c = 0, β ≠ 0, then x∗y = α/bβ + Bx, x•y = −α/β + bx, x  ◊  y = α + βy;if b = 0, c ≠ 0, γ = 0, then x∗y = −a/c + Cy, x•y = a + cy, x  ◊  y = 0;if b = 0, c ≠ 0, γ ≠ 0, then x∗y = α/cγ + Cy, x•y = −α/γ + cy, x  ◊  y = α + γy.if Let The proof is similar to d(x) = 0 for all x ∈ X, and most of them are classified by the properties of b, c in x•y = a + bx + cy.In Propositions The notion of two-step derivations is a generalization of derivations. This leads to the study of trigroupoids, and we explore some relations with several algebras, for example, BCK-algebras, frame algebras, and so forth. The classification of linear ranked trigroupoids then explains a number of concrete algebraic structures with derivations.


In several advanced fields like control engineering, computer science, fuzzy automata, finite state machine, and error correcting codes, the use of fuzzified algebraic structures especially ordered semigroups plays a central role. In this paper, we introduced a new and advanced generalization of fuzzy generalized bi-ideals of ordered semigroups. These new concepts are supported by suitable examples. These new notions are the generalizations of ordinary fuzzy generalized bi-ideals of ordered semigroups. Several fundamental theorems of ordered semigroups are investigated by the properties of these newly defined fuzzy generalized bi-ideals. Further, using level sets, ordinary fuzzy generalized bi-ideals are linked with these newly defined ideals which is the most significant part of this paper. The concept that belongs to relation (∈) and quasicoincident with relation (q) relation of a fuzzy point to fuzzy set was introduced by Pu and Liu  belongs to  a fuzzy set μ, if μ(x) ≥ t  + t > 1) and is denoted by  ∈ μ , where t ∈ , where α, β ∈ {∈, q, ∈∨q, ∈∧q} and α ≠ ∈∧q. The idea of generalized fuzzy subgroups has increased the importance of algebraic structures by attracting the attention of many researchers and opened ways for future researchers in this field. Furthermore, Jun  of real numbers.AAfter the introduction of fuzzy set theory , Rosenfeμ of S is called a fuzzy subsemigroup if for all x, y ∈ S,A fuzzy subset μ of S is called a fuzzy generalized bi-ideal of S if for all x, y, z ∈ S the following conditions hold:x ≤ y → μ(x) ≥ μ(y),μ(xyz) ≥ min{μ(x), μ(z)}.A fuzzy subset λ is called a fuzzy bi-ideal of S if the following conditions hold for all x, y, z ∈ S:x ≤ y → μ(x) ≥ μ(y),μ(xyz) ≥ min{μ(x), μ(z)}.A fuzzy subsemigroup S. But the converse is not true, as given in  ∈ λ  if λ(x) ≥ t  + t > 1). If  ∈ λ or qλ, then we write ∈∨qλ. The symbol q does not hold.is called aμ of S is called an -fuzzy generalized bi-ideal of S if it satisfies the following conditions:x, a, y ∈ S)   ,   .-fuzzy generalized bi-ideal of S if and only if it satisfies the following conditions:x, a, y ∈ S)  (μ(xay) ≥ min{μ(x), μ(y), 0.5}),   ≥ min{μ(y), 0.5}).  x ≤ y),  x, y ∈ S)  (max{μ(x), 0.5} ≥ μ(y) with x ≤ y),(∀(4)x, a, y ∈ S)  (max{μ(xay), 0.5} ≥ min{μ(x), μ(y)})., we have μ(y) ≥ t and r + μ(y) ≤ 1, which implies that t ≤ 0.5, a contradiction. Hence (3) is valid.(1)⇒(3). If there exists x, y ∈ S with x ≤ y and r ∈  < r. If μ(x) ≥ μ(y), then μ(y) ≤ μ(x) < r. It follows that μ(x) < μ(y), then by (3), we have 0.5 ≥ μ(y). Let μ(y) < r and μ(y) ≤ 0.5. It follows that (3)⇒(1). Let x, a, y ∈ S such thatt ≤ 1, x; t] ∈ μ and  ∈ μ. By (2), we have μ(x) ≥ t and t + μ(x) ≤ 1) or (μ(y) ≥ t and t + μ(y) ≤ 1), which implies that t ≤ 0.5, a contradiction.(2)⇒(4). If there exists x, a, y ∈ S and r, t ∈  < min{r, t}.(4)⇒(2). Let μ(xay) ≥ min{μ(x), μ(y)}, then min{μ(x), μ(y)} < min{r, t} and consequently μ(x) < r or μ(y) < t. It follows that (a) If μ(xay) < min{μ(x), μ(y)}, then by (4), we have 0.5 ≥ min{μ(x), μ(y)}. Let μ(x) < r and μ(x) ≤ 0.5 or μ(y) < t and μ(y) ≤ 0.5. It follows (b) If γ, ∈γ∨qδ)-fuzzy filters, -fuzzy ideals, and -fuzzy generalized bi-ideals of ordered semigroups.From the time that fuzzy subgroups gained general acceptance over the decades, it has provided a central trunk to investigate similar type of generalization of the existing fuzzy subsystems of other algebraic structures. A contributing factor for the growth of fuzzy subgroups is increased by the reports from Yin and Zhan  and Ma eγ, δ ∈  be such that γ < δ. For a fuzzy point  and a fuzzy subset μ of X, we say thatx; t]∈γμ if μ(x) ≥ t > γ.qδμ if μ(x) + t > 2δ.∈γ∨qδμ if ∈γμ or qδμ.∈γ∧qδμ if ∈γμ and qδμ.αμ does not hold for α ∈ {∈γ, qδ, ∈γ∨qδ, ∈γ∧qδ}.In what follows let α = ∈γ∧qδ is omitted. Because the set { | ∈γ∧qδ} is empty for μ(x) ≤ δ that is, if x ∈ S and r ∈  ≥ t > γ and μ(x) + t > 2δ. It follows that 2(x) = (x) + (x) ≥ (x) + t > 2δ so that μ(x) > δ which is contradiction to μ(x) ≤ δ. This means that { | ∈γ∧qδ} = ∅.Note that, the case μ of S is called an -fuzzy generalized bi-ideal of S if it satisfies the following conditions:(B1)x, y ∈ S)  .(∀(B2)x, a, y ∈ S)    .-fuzzy generalized bi-ideal of S.Then by μ of S is an -fuzzy generalized bi-ideal of S if and only if the following conditions hold for all x, a, y ∈ S:(B3)x ≤ y)→(max{μ(x), γ} ≥ min{μ(y), δ}),((B4)μ(xay), γ} ≥ min{μ(x), μ(y), δ}). ≥ t > γ, μ(x) < t, and μ(x) + t < 2t ≤ 2δ; that is, ∈γμ but (B1)⇒(B3). If there exists x, y ∈ S with x ≤ y and t ∈  ≥ t > γ, μ(x) < t and μ(x) + t < 2δ. It follows that μ(x) < δ and so max{μ(x), γ} < min{t, δ} ≤ min{μ(y), δ}, a contradiction. Hence (B1) is valid.(B3)⇒(B1). Assume that there exists x, a, y ∈ S such thatμ(xay) + r < 2r ≤ 2δ; that is, ∈γμ, ∈γμ but μ(xay), γ} ≥ min{μ(x), μ(y), δ} for all x, y ∈ S.(B2)⇒(B4). If there exists x, a, y ∈ S and r, t ∈  + min{r, t} ≤ 2δ. It follows that μ(xay) < δ and so(B4)⇒(B2). Assume that there exist S, whenever μ is an -fuzzy generalized bi-ideal of S (α ≠ ∈γ∧qδ) and 2δ = 1 + γ.The set μ is an -fuzzy generalized bi-ideal of S. Let a ∈ S, μ(x) > γ, μ(y) > γ.Assume that Case  1. Then αμ and αμ, where α ∈ {∈γ, ∈γ∨  qδ}. By (B2),μ(xay) ≥ min{μ(x), μ(y)} > γ or μ(xay) + {μ(x), μ(y)} > 2δ, and so μ(xay)≥  min{μ(x), μ(y)} > γ or μ(xay) > 2δ − min{μ(x), μ(y)} ≥ 2δ − 1 = γ. Hence It follows that Case  2. Then αμ and αμ, where α = qδ, since 2δ = 1 + γ. Analogous to the proof of Case 1, we have x, y ∈ S and x ≤ y if S.μ of S defined as follows:A is a nonempty subset of S. Then  A is a generalized bi-ideal of S if and only if μ is an -fuzzy generalized bi-ideal of S.Consider a fuzzy subset A is a generalized bi-ideal of S and x, y ∈ S with x ≤ y, t ∈  ≥ t > γ  follows that y ∈ A.Case  2. If qδμ, then μ(y) + t > 2δ and so μ(y) > 2δ − t ≥ 2δ − 1 = γ  follow that y ∈ A.y ∈ A and hence x ∈ A. By definition of μ we have μ(x) ≥ δ. If t ≤ δ, then μ(x) ≥ δ ≥ t > γ and hence ∈γμ but if t > δ, then μ(x) + t ≥ δ + t > 2δ; that is, qδμ. Thus for ∈γμ we have ∈γ∨qδμ.In the above two cases x, y, a ∈ S and r, t ∈  ≥ r > γ and μ(y) ≥ t > γ follow that x, y ∈ A.Case  2. If qδμ and qδμ, then μ(x) + r > 2δ and μ(y) + t > 2δ  and so μ(x) > 2δ − r ≥ 2δ − 1 = γ and μ(y) > 2δ − t ≥ 2δ − 1 = γ follow that x, y ∈ A.Case  3. Similarly, if ∈γμ and qδμ, then x, y ∈ A.Case  4. If qδμ and ∈γμ, then x, y ∈ A.x, y ∈ A. Hence xay ∈ A and by definition we have that μ(xay) ≥ δ. If min{r, t} ≤ δ, then μ(xay) ≥ δ ≥ min{r, t} > γ; that is,  ∈γμ. If min{r, t} > δ, then μ(xay) + min{r, t} > δ + δ = 2δ; that is, qδμ. Therefore ∈γ∨qδμ.Thus, in any case, μ is an -fuzzy generalized bi-ideal of S. It is easy to see that A is a generalized bi-ideal of S.Conversely, assume that γ∨qδ, ∈γ∨qδ)-fuzzy generalized bi-ideal of S is an -fuzzy generalized bi-ideal of S.Every -fuzzy generalized bi-ideal of S is an -fuzzy generalized bi-ideal of S.Every -fuzzy generalized bi-ideal of S;by μ is not an -fuzzy generalized bi-ideal of S, since ∈0.30μ and ∈0.30μ but μ is not an -fuzzy generalized bi-ideal of S, since ∈0.30∨q0.60μ and ∈0.30∨q0.60μ but Then,μ of S, we define the following sets for all t ∈ -fuzzy generalized bi-ideal and crisp generalized bi-ideal of S.The next theorem provides the relationship between -fuzzy generalized bi-ideal of S,μt(≠∅) is a generalized bi-ideal of S for all t ∈ -fuzzy generalized bi-ideal of S. Let x, y ∈ S with x ≤ y and t ∈ -fuzzy generalized bi-ideal of S, therefore ∈γ∨qδμ. If ∈γμ, then x ∈ μt and if qδμ, then μ(x) > 2δ − t > t > γ; that is,  x ∈ μt.(1)⇒(2). Let x, y, a ∈ S be such that x, y ∈ μt for some t ∈ -fuzzy generalized bi-ideal of S, therefore ∈γ∨qδμ. If ∈γμ, then xay ∈ μt and if qδμ, then μ(xay) > 2δ − t > t > γ; that is,  xay ∈ μt. Therefore μt is a generalized bi-ideal of S.Let μt(≠∅) is a generalized bi-ideal of S for all t ∈ , γ} < min{μ(y), δ}; then there exists t ∈ , γ} < t ≤ min{μ(y), δ}; this follows that ∈γμ; that is, y ∈ μt but μ(x), γ} ≥ min{μ(y), δ} for all x, y ∈ S with x ≤ y. Let x, y, a ∈ S and max{μ(xay), γ} < min{μ(x), μ(y), δ}; then max{μ(xay), γ} < t ≤ min{μ(x), μ(y), δ} for some t ∈ ⇒(1). Assume that μ is an -fuzzy generalized bi-ideal.Consequently, μ  of an ordered semigroup S, then μ is an -fuzzy generalized bi-ideal of S if and only if μrδ(≠∅) is a generalized bi-ideal of S for all t ∈ -fuzzy generalized bi-ideal of S. Let x, y ∈ S with x ≤ y and t ∈  > 2δ − t. Since t ∈ -fuzzy generalized bi-ideal of S; thereforeμ(x) ≥ 2δ − t. Hence x ∈ μtδ.Let x, y, a ∈ S be such that x, y ∈ μtδ for some t ∈  > 2δ − t, that is, μ(y) > 2δ − t. Since μ is an -fuzzy generalized bi-ideal of S, thereforeμ(xay) ≥ 2δ − t. Hence xay ∈ μtδ. Consequently, μtδ is generalized bi-ideal.Let μtδ(≠∅) be a generalized bi-ideal of S for all t ∈ , γ} < t = min{μ(y), δ}; this follows that y ∈ μtδ but μ(x), γ} ≥ min{μ(y), δ} for all x, y ∈ S with x ≤ y. Let x, y, a ∈ S and max{μ(xay), γ} < t = min{μ(x), μ(y), δ}; this implies that x ∈ μtδ and y ∈ μtδ but μ(xay), γ} ≥ min{μ(x), μ(y), δ}. Consequently, μ is an -fuzzy generalized bi-ideal.Conversely, let μ  of an ordered semigroup S is an -fuzzy generalized bi-ideal of S if and only if [μ]tδ(≠∅) is a generalized bi-ideal of S for all t ∈ -fuzzy subsemigroup if it satisfies the following condition:(B5)x, y ∈ S)    .-fuzzy bi-ideal of S if it is -fuzzy subsemigroup and -fuzzy generalized bi-ideal of S.A fuzzy subset μ of S is an -fuzzy bi-ideal of S if and only if the following conditions hold for all x, a, y ∈ S:(B6)x ≤ y)→(max{μ(x), γ} ≥ min{μ(y), δ}),((B7)μ(xy), γ} ≥ min{μ(x), μ(y), δ}),(max{(B8)μ(xay), γ} ≥ min{μ(x), μ(y), δ}). + r < 2r ≤ 2δ; that is, ∈γμ, ∈γμ but μ(xy), γ} ≥ min{μ(x), μ(y), δ} for all x, y ∈ S.(B5)⇒(B7). If there exists x, y ∈ S and r, t ∈  + min{r, t} ≤ 2δ. It follows that μ(xy) < δ and so(B7)⇒(B5). Assume that there exist The remaining proof follows from γ, ∈γ∨qδ)-fuzzy bi-ideals and -fuzzy generalized bi-ideals is provided.Next, the relationship between -fuzzy bi-ideal μ of S is -fuzzy generalized bi-ideal.Every -fuzzy generalized bi-ideal of S but not an -fuzzy bi-ideal, since ∈0.28μ but Then by A of an ordered semigroup S is a generalized bi-ideal of S if and only ifγ, ∈γ∨qδ)-fuzzy generalized bi-ideal of S.A nonempty subset The proof is straightforward and is omitted.In the last couple of decades, the importance of fuzzification of ordered semigroups and related structures is increased due to the pioneering role of aforementioned structures in advanced fields like computer science, error correcting codes, and fuzzy automata. In contribution to this fact, we define and investigate x ≤ y),(B10)A fuzzy subset μ of S such that μ(x) ≥ δ for any x ∈ S in the case μ(x) < r and μ(x) + r < 2δ. Thus μ(x) + μ(x) < μ(x) + r ≤ 2δ, which implies μ(x) < δ. This means that The case when S = {a, b, c, d, e} with the following multiplication table and order relation:Consider an ordered semigroup μ : S →  as follows:μ is an (S.Define a fuzzy subset μ of S is an (S if and only if the following conditions hold:(B11)x, y ∈ S)  (max{μ(x), δ} ≥ μ(y) with x ≤ y),(∀(B12)x, a, y ∈ S)  (max{μ(xay), δ} ≥ min{μ(x), μ(y)}). , δ} < μ(y). Then max{μ(x), δ} < t ≤ μ(y) for some t ∈ , δ} ≥ μ(y) for all x, y ∈ S with x ≤ y.(B9)⇒(B11). Assume that there exists x, y ∈ S with x ≤ y and t ∈  < t, μ(y) ≥ t, and μ(y) + t > 2δ and hence μ(y) > δ. This follows that(B11)⇒(B9). Assume that there exists x, a, y ∈ S such that max{μ(xay), δ} < min{μ(x), μ(y)}, then there exists t ∈ , δ} < t ≤ min{μ(x), μ(y)}. It follows that μ(xay), δ} ≥ min{μ(x), μ(y)} for all x, a, y ∈ S.(B10)⇒(B12). If x, a, y ∈ S and r, t ∈  < min{r, t}, μ(x) ≥ r, μ(y) ≥ t, μ(x) + r > 2δ, and μ(y) + r > 2δ. It follows that μ(x) > δ and μ(y) > δ, and so(B12)⇒(B10). Assume that there exist μ of S is called S if it satisfies the following conditions:(B13)A fuzzy subset μ of S is called S if it is S.A fuzzy subset μ of S is an (S if and only if the following conditions hold:(B14)x, y ∈ S)  (max{μ(x), δ} ≥ μ(y) with x ≤ y),(∀(B15)x, y ∈ S)  (max{μ(xy), δ} ≥ min{μ(x), μ(y)})(∀(B16)x, a, y ∈ S)  (max{μ(xay), δ} ≥ min{μ(x), μ(y)}). , δ} < t ≤ min{μ(x), μ(y)}. It follows that μ(xy), δ} ≥ min{μ(x), μ(y)} for all x, y ∈ S.(B13)⇒(B15). Suppose x, y ∈ S and r, t ∈  < min{r, t}, μ(x) ≥ r, μ(y) ≥ t, μ(x) + r > 2δ, and μ(y) + r > 2δ. It follows that μ(x) > δ and μ(y) > δ, and so(B15)⇒(B13). If there exist μ of S is  > δ. As μ is an S, thereforeμ(x) > δ and hence Assume that x, a, y ∈ S be such that μ(x) > δ, μ(y) > δ, and by (B8)Let S.that is, μ of S defined asConsider a fuzzy subset A is a generalized bi-ideal of S if and only if μ is an S.Then A is a generalized bi-ideal of S. Let x, a, y ∈ S and r, t ∈  < min{r, t} ≤ 1 and so μ(xay) = δ < min{r, t}; that is, xay ∉ A. It follows that x ∉ A or y ∉ A, and so μ(x) = δ < r or μ(y) = δ < t. Hence Case  2. μ(xay) + min{r, t} ≤ 2δ. If μ(xay) = δ, analogous to the proof of Case 1, we have μ(xay) = 1; then max{μ(x), μ(y)} + min{r, t}≤ 1 + min{r, t} =  μ(xay) + min{r, t} ≤ 2δ. It follows that μ(x) + r ≤ 2δ or μ(y) + t ≤ 2δ. Hence Case  3. x, y ∈ S with x ≤ y.In a similar way we can show that μ is a -fuzzy generalized bi-ideals and  is also constructed.Due to the significant role of ordered semigroups and their different characterizations in several applied fields such as control engineering, fuzzy automata, coding theory, and computer science, the latest research has been carried out in the last few decades by considering various characterizations of ordered semigroups in terms of different types of fuzzy ideals. In this paper, we determined a more generalized form of Davvaz and Khan  approach


Here, we aim to present some new fractional integral inequalities involving generalized Erdélyi-Kober fractional q-integral operator due to Gaulué, whose special cases are shown to yield corresponding inequalities associated with Kober type fractional q-integral operators. The cases of synchronous functions as well as of functions bounded by integrable functions are considered.In recent years, a remarkably large number of inequalities involving the fractional  If f and g are synchronous on , that is,x, y ∈ , then we have  = q(x) for any x, y ∈ , we get the Chebyshev inequality :1)T−f(3)T−f−f|≥m and |g′(x)|≥r for x ∈ , then we haveIf f and g are asynchronous on , then we havef and g are two differentiable functions on  with |f′(x)|≤M and |g′(x)|≤R for x ∈  and p is a positive integrable function on , then we haveIf nctional  has attrnctional  has alsoq-integral inequalities has been of great importance due to the fundamental role in the theory of differential equations. In recent years, a number of researchers have done deep study, that is, the properties, applications, and different extensions of various fractional q-integral operators .q-calculus analogs of some classical integral inequalities. In particular, we will find q-generalizations of the Chebyshev integral inequalities by using the generalized Erdélyi-Kober fractional q-integral operator introduced by Galué  is defined byDqf(0) = limt→0Dqf(t). It is noted thatf(t) is differentiable.We begin by noting that F. J. Jackson was the first to develop F(t) is a q-antiderivative of f(t) if DqF(t) = f(t). It is denoted byThe function  Jackson integral of f(t) is thus defined, formally, byThea < b. The definite q-integral is defined as follows:Suppose that 0 < A more general version of  is givenz) (n ∈ N0) to real numbers. The q-factorial function [n]q!  (n ∈ N0) of n! defined byn by a − 1 in (q-Gamma function Γq(a) byThe classical Gamma function Γ(follows:(1−q)−n∏kq-analogue of (t − a)n is defined by the polynomialγ ∈ R, thenThe R(β), R(μ) > 0 and η ∈ C. Then a generalized Erdélyi-Kober fractional integral Iqα,β,η for a real-valued continuous function f(t) is defined by (26)Iqηq-analogue of the Kober fractional integral operator is given by (27)Iqημ > 0 and k ∈ N0. If f : ,Iqη,μ,β{ny adding .q < 1, let f and g be two continuous and synchronous functions on [0, ∞), and let l, m, n : [0, ∞)→[0, ∞) be continuous functions. Then, the following inequality holds true:μ, ν, β, δ > 0 and η, ζ ∈ C.Let 0 < u = m and v = n in , after a little simplification, we getu, v by l, n and u, v by l, m in  and Iqη,μ,β{n}(t), respectively, we get the following two inequalities:Setting v = n in , we have = n in . The special case of 42) in Thequality  in Theorβ = 1 in , and let l, m, n : [0, ∞)→[0, ∞) be continuous functions. Then, the following inequality holds true:μ > 0 and η ∈ C.Let 0 < q < 1, let f and g be two continuous and synchronous functions on [0, ∞), and let l, m, n : [0, ∞)→[0, ∞) be continuous functions. Then, the following inequality holds true:μ, ν > 0 and η, ζ ∈ C.Let 0 < η = 0 and β = 1 in η = ζ = 0 and β = δ = 1 in If we take q-integral operator in the case where the functions are bounded by integrable functions and are not necessary increasing or decreasing as are the synchronous functions.In this section we obtain some new inequalities involving Erdélyi-Kober fractional q < 1, let f be an integrable function on [0, ∞), and let u, v : [0, ∞)→[0, ∞) be continuous functions. Assume the following.H1)  such thatThere exist two integrable functions Then, for t > 0, μ, β > 0, and η ∈ C, we haveLet 0 < H1), for all τ ≥ 0 and ρ ≥ 0, we haveβtβ(η+μ)−/Γq(μ))(tβ − τβq)μ−1)(τβ(η+1)−1u(τ), τ ∈ , and integrating both sides with respect to τ on , we obtainβtβ(η+μ)−/Γq(μ))(tβ − ρβq)μ−1)(ρβ(η+1)−1v(ρ), ρ ∈ , and integrating both sides with respect to ρ on , we get inequality f(ρ)e obtainIqη,μ,β{uequality  as requeAs special cases of Theorems q < 1, let f be an integrable function on [0, ∞) satisfying m ≤ f(t) ≤ M, for all t ∈ [0, ∞), let u, v : [0, ∞)→[0, ∞) be continuous functions, and let  m, M ∈ R. Then, for t > 0, μ, β > 0, and η ∈ C, we haveLet 0 < q < 1, let f be an integrable function on [1, ∞), and let u, v : [0, ∞)→[0, ∞) be continuous functions. Assume that there exists an integrable function φ(t) on [0, ∞) and a constant M > 0 such thatt > 0, μ, β > 0, and η ∈ C; we haveLet 0 < q < 1, let f be an integrable function on [0, ∞), let u, v : [0, ∞)→[0, ∞) be continuous functions, and let θ1, θ2 > 0 satisfying 1/θ1 + 1/θ2 = 1. Suppose that (H1) holds. Then, for t > 0, μ, β > 0, and η ∈ C, we haveLet 0 < x = φ2(τ) − f(τ) and y = f(ρ) − φ1(ρ), τ, ρ ≥ 0, we haveτ, ρ ∈ , and integrating with respect to τ and ρ from 0 to t, we deduce the desired result in 1θ1x we have1θ1(φ2(τ)esult in .q < 1, let f be an integrable function on [0, ∞) satisfying m ≤ f(t) ≤ M, for all t ∈ [0, ∞), let u, v : [0, ∞)→[0, ∞) be continuous functions, and let m, M ∈ R. Then, for t > 0, μ, β > 0, and η ∈ C, we haveLet 0 < q < 1, let f be an integrable function on [0, ∞), let u, v : [0, ∞)→[0, ∞) be continuous functions, and let θ1, θ2 > 0 satisfying θ1 + θ2 = 1. In addition, suppose that (H1) holds. Then, for t > 0, μ, β > 0, and η ∈ C, we haveLet 0 < x = φ2(τ) − f(τ) and y = f(ρ) − φ1(ρ), τ, ρ > 1, we haveτ, ρ ∈ , and integrating with respect to τ and ρ from 0 to t, we deduce inequality θ1x+ we haveθ1(φ2(τ)−equality .q < 1, let f be an integrable function on [0, ∞) satisfying m ≤ f(t) ≤ M, for all t ∈ [0, ∞), let u, v : [0, ∞)→[0, ∞) be continuous functions, and let m, M ∈ R. Then, for t > 0, μ, β > 0, and η ∈ C, we haveLet 0 < a ≥ 0, p ≥ q ≥ 0, and p ≠ 0. Then,Assume that q < 1, let f be an integrable function on [0, ∞), let u : [0, ∞)→[0, ∞) be a continuous function, and let constants p ≥ q ≥ 0, p ≠ 0. In addition, assume that (H1) holds. Then, for any k > 0, t > 0, μ, β > 0, and η ∈ C, the following two inequalities hold:Let 0 < H1) and p ≥ q ≥ 0, p ≠ 0, it follows thatk > 0. Multiplying both sides of (βtβ(η+μ)−/Γq(μ))(tβ − τβq)μ−1)(τβ(η+1)−1u(τ), τ ∈ , and integrating the resulting identity with respect to τ from 0 to t, one has inequality (i). Inequality (ii) is proved by setting a = f(τ) − φ1(τ) in By condition (ows that(φ2(τ)−f satisfying m ≤ f(t) ≤ M, for all t ∈ [0, ∞), let u, v : [0, ∞)→[0, ∞) be continuous functions, and let m, M ∈ R. Then, for t > 0, μ, β > 0, and η ∈ C, we haveLet 0 < q < 1, let f and g be two integrable functions on [0, ∞), and let u, v : [0, ∞)→[0, ∞) be continuous functions. Suppose that (H1) holds and moreover we assume the following. H2) There exist ψ1 and ψ2 integrable functions on [0, ∞) such that, from (H1) and (H2), we have for t ∈ [0, ∞) thatβtβ(η+μ)−/Γq(μ))(tβ − τβq)μ−1)(τβ(η+1)−1u(τ), τ ∈ , and integrating both sides with respect to τ on , we obtainβtβ(η+μ)−/Γq(μ))(tβ − ρβq)μ−1)(ρβ(η+1)−1v(ρ), ρ ∈ , and integrating both sides with respect to ρ on , we get the desired inequality (i).To prove g(ρ)e obtaing(ρ)Iqη,μii)–(iv), we use the following inequalities:To prove , let u, v : [0, ∞)→[0, ∞) be continuous functions, and let θ1, θ2 > 0 satisfying 1/θ1 + 1/θ2 = 1. Suppose that (H1) and (H2) hold. Then, for t > 0, μ, β > 0, and η ∈ C, the following inequalities hold:Let i)–(iv) can be proved by choosing the parameters in the Young inequality [The inequalities (i)f and g be two integrable functions on [0, ∞), let u, v : [0, ∞)→[0, ∞) be continuous functions, and let θ1, θ2 > 0 satisfying θ1 + θ2 = 1. Suppose that (H1) and (H2) hold. Then, for t > 0, μ, β > 0, and η ∈ C, the following inequalities hold:Let i)–(iv) can be proved by choosing the parameters in the Weighted AM-GM [The inequalities (ed AM-GM :(80)(i)f and g be two integrable functions on [0, ∞), let u, v : [0, ∞)→[0, ∞) be continuous functions, and let constants p ≥ q ≥ 0, p ≠ 0. Assume that (H1) and (H2) hold. Then, for any k > 0, t > 0, μ, β > 0, and η ∈ C, the following inequalities hold:Let i)–(iv) can be proved by choosing the parameters in The inequalities (q-integral inequalities and fractional calculus. Moreover, they are expected to find some applications for establishing uniqueness of solutions in fractional boundary value problems and in fractional partial differential equations. In last, use of the generalized Erdélyi-Kober fractional q-integral operator due to Gaulué is the advantage of our results because after setting suitable parameter values in our main results, we get known results established by number of authors.We conclude our present investigation with the remark that the results derived in this paper are general in character and give some contributions to the theory of


Scientific Reports5: Article number: 1076810.1038/srep10768; published online: 06112015; updated: 09212015LF = β·L’ should read ‘LF = (α + β) · L’. The correct Fig. 3a appears below as This Article contains a typographical error in Fig. 3a, where ‘


In the description of the test for cell viability, “4 μM ethidium bromide” should be replaced by “4 μM ethidium homodimer-1 (EthD-1)”.In the Results and discussion section, the sentence “The surface morphology of gelatin matrix is visualized using AFM and SEM, respectively ” should read “The surface morphology of gelatin matrix is visualized using SEM and AFM, respectively .”In the Results and discussion section, the sentence “AFM images also depicted a relatively smooth gelatin surface with an RMS Rq of 155.6 ± 18 nm ” should read “AFM images also depicted a relatively smooth gelatin surface with an RMS Rq of 155.6 ± 12.3 nm ”.In the original version of this article , there w


F*(K) be the set of all fuzzy complex numbers. In this paper some classical and measure-theoretical notions are extended to the case of complex fuzzy sets. They are fuzzy complex number-valued distance on F*(K), fuzzy complex number-valued measure on F*(K), and some related notions, such as null-additivity, pseudo-null-additivity, null-subtraction, pseudo-null-subtraction, autocontionuous from above, autocontionuous from below, and autocontinuity of the defined fuzzy complex number-valued measures. Properties of fuzzy complex number-valued measures are studied in detail.Let  It is well known that additivity of a classical measure primly depicted measure problems under error-free condition. But when measure error was unavoidable, additivity could not fully depict the measure problems under certain condition. To overcome such difficulties, fuzzy measure has been developed. Research on fuzzy measures was very deep in those aspects: research based on a certain number of subsets of a classic set and a real value nonaddable measure  –37 34–37R is the set of all real numbers set, K is the set of all complex numbers, X is an ordinary set, F*(R) is the set of all real fuzzy numbers on R, Δ(R) is the set of all interval numbers,  is a measurable space , and F*(K) is the set of all fuzzy complex numbers on K. Let F+*(K) = In this paper, K is denoted by F*(K).Let A, B of R, write ≜A + iB = {x + iy∣x ∈ A, y ∈ B}. The operation ∘∈{+, −, ·} is described as follows: c1′∘c2′ =  for any c1′, c2′ ∈ F*(K);c · c1′ =  for any c1′ ∈ F*(K) and c =  ∈ K (c′ ∈ F*(K) is said to be a fuzzy infinity [c1′, c2′ ∈ F*(K), one makes the following appointments:c1′ ≤ c2′  Rec1′ ≤ Rec2′ and Imc1′ ≤ Imc2′;if and only if  c1′ = c2′ if and only if c1′ ≤ c2′ and c2′ ≤ c1′; c1′ < c2′ if and only if c1′ ≤ c2′ and Rec1′ ≤ Rec2′ or Imc1′ < Imc2′; c′ ≥ 0 if and only if Rec′ ≥ 0, Imc′ ≥ 0.One uses A* to denote a family (which is obviously nonempty) of subsets of F*(K) that satisfies the following conditions:for each for each For any subsets infinity  (writtenF*(R):for any for any for any A mapping F*(R).The mapping F*(K):for any for any for any Analogously, a mapping F*(R). Then the mapping ρ′ : F*(K) × F*(K) → F*(R) defined byF*(K).Let cn′} ⊂ F*(K) and let c′ ∈ F*(K). {cn′} is said to converge to c′ according to a fuzzy metric ρ′ on F*(K) (written as limn→∞cn′ = c′) if, for each ε > 0, there exists a positive integer N such that ρ′ < ε for all n ≥ N.Let {The notion of complex fuzzy measure on family of classical sets was given in .σ-algebra A composed of subsets of X is a mapping μ(ϕ) = 0;A ⊂ B, then |μ(A)| ≤ |μ(B)|;if An}1∞↑, then μ(⋃n=1∞An) = limn→∞μ(An);if {An}1∞↓ and |μ(An0)| < +∞ for some n0, then μ(⋂n=1∞An) = limn→∞μ(An).if {Let In this paper we need an expansion of this notion. First we defined the concept of fuzzy complex value distance.F*(K):for any for any for any A mapping F*(K) if and only if F*(K).It can be easily seen that a mapping ε > 0, there exists a positive integer N such that n ≥ N hold, then Let Z be a nonempty complex number set, let F(Z) be the set of all complex fuzzy sets on Z, and let F(Z). A complex fuzzy set-value complex fuzzy measure is a mapping for any n = 1,2,…), then (lower semicontinuous) if n = 1,2,…) and n0, then (upper semicontinuous) if Let Apparently, a complex fuzzy set-value complex fuzzy measure is also a kind of special generalized fuzzy measures.0-add if null-additive  if autocontinuous from above  if autocontinuous from below  if autocontinuous if it is both autoc. ↓ and autoc. ↑.A mapping F(Z) is said to be pseudo-null-additive  if, for each ε = ε1 + iε2 > 0, there exists a δ = δ1 + iδ2 > 0 such that A complex fuzzy set-value complex fuzzy measure X be a set and F(X) the set of all fuzzy sets on X. Then a subfamily F ⊂ F(X) is addable if and only if it satisfies the following conditions :X ∈ F;x ∈ X).where We first have the following result.F ⊂ F(Z) is a complex fuzzy value fuzzy complex measure on F.Every fuzzy complex measure n0. By monotonicity of n ≥ n0. Since We only prove the upper continuity and lower continuity of Assume σ-algebra F ⊂ F(Z) is exhaustive.Every complex fuzzy set-value complex fuzzy measure z0 ∈ Z or z0 ∈ Z; that is, n ≥ 1 or n ≥ 1. Without loss of generality, we assume the first. Then there are two distinct indexes kn1 and kn2 such that Suppose σ-algebra F, On the other hand, since If k, Since n ≥ k, As Similarly, From properties of upper limits and lower limits we can see the following theorem holds.F(Z), then, for any Let Let Similar to F(Z), then, for any Let F(Z) which is pseudo-zero addable about Assume that Let Similarly, we have the following.F(Z) which is pseudo-zero subtractable about Suppose that σ-algebra F ⊂ F(Z) and autoc. ↓, then Let ε0 = ε0′ + iε0′′  such that, for any natural numbers n, m, there exist n0 ≥ 1. This conflicts with the hypothesis.Suppose that δn} of real numbers satisfying δn > 0  and δn↘0 and a subsequence k ≥ 1). Furthermore, Suppose that ε′, ε′′, δ1′, δ1′′ > 0, let ε = ε′ + iε′′, δ1 = δ1′ + iδ1′′, δ1′ ∈ , and δ1′′ ∈ . Since δ1 ∈  such that n1 such that δ2 = δ2′ + iδ2′′ such that δ2′ ∈ , δ2′′ ∈ , δ2 = δ2′ + iδ2′′ > 0, there exists an n2 > n1 such that For any real numbers δ2 = δ2′ + iδ2′′ > 0 and there exists δ3 = δ3′ + iδ3′′ > 0, δ3′ ∈ , and δ3′′ ∈  such that δ3 = δ3′ + iδ3′′ > 0, δ3′ ∈ , δ3′′ ∈  since n3 > n2, such that For nk+1 > nk > nk−1 > ⋯>n1, and δk < δk−1∧ε/2k−1, such that k = 1,2, 3,…, r + 1, r ≥ 1).Generally we can get Let Hence


As an application of our results, periodic points of weakly contractive mappings are obtained. We also derive certain new coincidence point and common fixed point theorems in partially ordered G-metric spaces. Moreover, some examples are provided here to illustrate the usability of the obtained results.The aim of this paper is to present some coincidence and common fixed point results for generalized ( G-metric space, was introduced by Mustafa and Sims . Let f : X → X be defined by fx = x1/3 and let α : X × X × X → [0, ∞) be defined by α = y + z − 2x + 1. Then, x ≤ x1/3 = fx for all x ∈ X. That is, 2x1/3 − 2x + 1 ≥ 1. Thus, f is an α-dominating map.Let X, G) be a G-metric space. We say that X is α-regular if and only if the following hypothesis holds.Let  ≥ 1 such that xn → z as n → ∞, it follows that α ≥ 1 or α ≥ 1 or α ≥ 1, for all n ∈ N.For any sequence {X be a set and let f, g : X → X be given mappings. We say that the pair  is partially weakly G-α-admissible if and only if α ≥ 1 for all x ∈ X.Let X be a nonempty set and f : X → X a given mapping. For every x ∈ X, let f−1(x) = {u ∈ X∣fu = x}.Let X be a set and let f, g, R : X → X be given mappings. We say that the pair  is partially weakly G-α-admissible with respect to R if and only if for all x ∈ X, α ≥ 1, where y ∈ R−1(fx).Let f = g, we say that f is partially weakly G-α-admissible with respect to R.If R = IX (the identity mapping on X), then the previous definition reduces to the partially weakly G-α-admissible pair.If f, g, h, R, S, and T for which ordered pairs , , and  are partially weakly G-α-admissible with respect to R, S, and T, respectively.Following is an example of mappings X = [0, ∞). We define functions f, g, h, R, S, T : X → X byLet Jungck in  introducX, G) be a G-metric space and let f, g : X → X. The pair  is said to be compatible if and only if limn→∞G = 0, whenever {xn} is a sequence in X such that limn→∞fxn = limn→∞gxn = t for some t ∈ X.Let -contractive mappings , , and  which are partially weakly α-admissible with respect to R, S, and T, respectively, in a G-metric space.The aim of this paper is to prove some coincidence and common fixed point theorems for nonlinear weakly  be a metric space and let f, g, h, R, S, T : X → X be six self-mappings. In the rest of this paper, unless otherwise stated, for all x, y, z ∈ X, letα : X3 → [0, ∞) be a function having the following property:α : [0, ∞)3 → [0, ∞) by α = ex−y−z2.Let  be a G-complete G-metric space. Let f, g, h, R, S, T : X → X be six mappings such that f(X)⊆R(X), g(X)⊆S(X), and h(X)⊆T(X). Suppose that, for every three elements x, y, and z with α ≥ 1, one hasψ, φ : [0, ∞)→[0, ∞) are altering distance functions. Let f, g, h, R, S, and T be continuous, the pairs , , and  compatible, and the pairs , , and  partially weakly α-admissible with respect to R, S, and T, respectively. Then, the pairs , , and  have a coincidence point z in X. Moreover, if α ≥ 1, then z is a coincidence point of f, g, h, R, S, and T.Let (x0 ∈ X be an arbitrary point. Since f(X)⊆R(X), we can choose x1 ∈ X such that fx0 = Rx1. Since g(X)⊆S(X), we can choose x2 ∈ X such that gx1 = Sx2. Also, as h(X)⊆T(X), we can choose x3 ∈ X such that hx2 = Tx3.Let zn} defined byn ≥ 0.Continuing this process, we can construct a sequence {x1 ∈ R−1(fx0), x2 ∈ S−1(gx1), and x3 ∈ T−1(hx2) and , , and  are partially weakly α-admissible with respect to R, S, and T, respectively, we obtain thatNow, since n ∈ N.Continuing this process, from , we getGk = G. Suppose Gk0 = 0, for some k0. Then, zk0 = zk0+1 = zk0+2. In the case that k0 = 3n, then zn3 = zn+13 = zn+23 gives zn+13 = zn+23 = zn+33. Indeed,Define zn+13 = zn+23 = zn+33.Thus,M, we can get this result.Analogously, for other values of k0 = 3n + 1, then zn+13 = zn+23 = zn+33 gives zn+23 = zn+33 = zn+43. Also, if k0 = 3n + 2, then zn+23 = zn+33 = zn+43 implies that zn+33 = zn+43 = zn+53. Consequently, zk0 is a coincidence point of the pairs , , and . Indeed, let k0 = 3n. Then, we know that zn3 = zn+13 = zn+23 = zn+33.Similarly, if So,T(xn3) = f(xn3), R(xn+13) = g(xn+13), and S(xn+23) = h(xn+23).This means that f, T), , and  are compatible. So, they are weakly compatible. Hence, fT(xn3) = Tf(xn3), gR(xn+13) = Rg(xn+13), and hS(xn+23) = Sh(xn+23) or, equivalently, fzn3 = Tzn+13, gzn+13 = Rzn+23, and hzn+23 = Szn+33.On the other hand, the pairs , similarly, one can show that zn+13 (zn+23) is a coincidence point of the pairs , , and .In the other cases, when k; that is, zk ≠ zk+1 for each k.So, suppose thatWe complete the proof in three steps as follows.Step  1. We will prove that limk→∞G = 0.α ≥ 1, using  = G, then (If 2), then  becomesIfand then  will beIfand then  becomesIfand then  becomesFinally, ifand then  becomesSimilarly it can be shown thatG} is a nondecreasing sequence of nonnegative real numbers. Thus, there is an r ≥ 0 such thatHence, we conclude that {Reviewing the above argument, from , we haveIn general, we can show thatk → ∞ in  ≤ ψ(r) − φ(r), and hence φ(r) = 0. This gives us thatφ. Also, from Letting nd using , 35), a, an → ∞ Step  2. We will show that {zn} is a G-Cauchy sequence in X. So, we will show that, for every ɛ > 0, there exists k ∈ N such that, for all m, n ≥ k, G < ɛ.ɛ > 0 for which we can find subsequences {zm(k)3} and {zn(k)3} of {zn3} such that n(k) > m(k) ≥ k satisfying thatn(k) is the smallest number such that ,quality,G(z3m(k),∞, using  and 39)ɛ > 0 fork → ∞, using ,∞, using  and 38)(42)G, ∞, from , we havek → ∞, using ,∞, using  and 38)(46)G+∞, using  and 45)(48)G3, Rxn(k)+13, Sxn(k)+23) ≥ 1, putting x = xm(k)3, y = xn(k)+13, and z = xn(k)+23 in ,∞, using  and 38)(53)G(z3m(k)+2 in , for allM(xm(k)3, xn(k)+13, xn(k)+23) = G(zm(k)3, zn(k)+13, zn(k)+23), from (k → ∞ in (If 2), from  and 41)M(x3m(k)k → ∞ in , we havection to .k → ∞ in , M, a, a(60)M(62)M(64)M65) yieldStep  3. We will show that f, g, h, R, S, and T have a coincidence point.zn} is a G-Cauchy sequence in the complete G-metric space X, there exists z ∈ X such thatSince {f, T) is compatible, soAs  = 0, limn→∞G = 0, and the continuity of T and f, we obtainMoreover, from limBy the rectangle inequality, we haven → ∞ in n → ∞ in gz = Rz and hz = Sz.Similarly, we can obtain that α ≥ 1. By  > 0; that is, fz ≠ gz = hz or fz = gz ≠ hz.Let M = G = G, from  = 0, a contradiction.If z), from , we haveM =  + G + G)/6 and fz ≠ gz = hz, thenfz = gz = hz, a contradiction to G(fz = gz = hz) > 0.If so, from , we havefz = gz = hz = Tz = Rz = Sz.In the other cases, by a similar manner, we can show that In the following theorem, we will omit the compatibility and continuity assumptions.X, G) be an α-regular G-metric space and f, g, h, R, S, T : X → X six mappings such that f(X)⊆R(X), g(X)⊆S(X), and h(X)⊆T(X) and RX, SX, and TX are G-complete subsets of X. Suppose that, for elements x, y, and z with α ≥ 1, we haveψ, φ : [0, ∞)→[0, ∞) are altering distance functions. Then, the pairs , , and  have a coincidence point z in X provided that the pairs , , and  are weakly compatible and the pairs , , and  are partially weakly α-admissible with respect to R, S, and T, respectively. Moreover, if α ≥ 1, then z ∈ X is a coincidence point of f, g, h, R, S, and T.Let (z ∈ X such thatR(X) is G-complete and {zn+13}⊆R(X), therefore z ∈ R(X), so there exists u ∈ X such that z = Ru andv, w ∈ X such that z = Sv = Tw andw is a coincidence point of f and T.Following the proof of Txn3 → z = Tw = Ru = Sv as n → ∞, α-regularity of X implies that α ≥ 1. Therefore, from  = G.Let n → ∞ in , it can be shown that z is a coincidence point of the pairs  and .Similarly, in other cases for The rest of the proof is similar to the proof of Assume thatR = S = T = IX (the identity mapping on X) in the previous theorems, we obtain the following common fixed point result.Taking X, G) be a G-complete G-metric space. Let f, g, h : X → X be three mappings. Suppose that, for every three elements x, y, and z with α ≥ 1, we haveψ, φ : [0, ∞)→[0, ∞) are altering distance functions. Let the pairs , , and  be partially weakly α-admissible. Then, the triple  has a common fixed point z in X provided that (a) f, g, and h are continuous or (b) X is α-regular.Let  be a G-metric space. Let f, R : X → X be two mappings such that f(X)⊆R(X). Suppose that, for every three elements x, y, and z with α ≥ 1, we haveψ, φ : [0, ∞)→[0, ∞) are altering distance functions. Let the pair  be compatible and f is partially weakly α-admissible w.r.t. R. Then,  has a coincidence point z in X provided that (a) f and R are continuous and  is a G-complete G-metric space or (b) X is α-regular and R(X) is G-complete.Let  ≥ 1 or α ≥ 1, where u and v are common fixed points of f and R.Under the hypotheses of z ∈ X such that fz = Rz. Since f and R are weakly compatible  is compatible), we have fRz = Rfz. Let w = Rz = fz. Therefore, we haveR is an α-dominating map,Since z = w, then z is a common fixed point of f and R. If z ≠ w, then, from  ≥ 1, from  =  + G + G)/6. Then, from ψ + G + G)/6) = 0. So, fz = fw. Now, since w = Rz = fz and fw = Rw, we have w = Rw = fw.Therefore, α ≥ 1 or α ≥ 1, where u and v are common fixed points of f and R. We claim that common fixed point of f and R is unique. Assume on the contrary that fu = Ru = u and fv = Rv = v and u ≠ v. Without any loss of generality, we may assume that α = α ≥ 1. Using  =  + G + G)/6. Then we haveLet u = v, a contradiction.Therefore, In the other cases the proof will be done in a similar way.X = [0, ∞), G on X given by G = |x − y| + |y − z| + |z − x|, for all x, y, z ∈ X, and α : X3 → [0, ∞) given by α = ex−y−z2. Define self-maps f, g, h, R, S, and T on X byf, g) is partially weakly α-admissible with respect to R, let x ∈ X and y ∈ R−1fx; that is, Ry = fx. By the definition of f and R, we have ln(1 + x) = ey3 − 1. So, y = ln(ln(1 + x) + 1)/3 and henceα ≥ 1.Let g, h) is partially weakly α-admissible with respect to S, let x ∈ X and y ∈ S−1gx; that is, Sy = gx. By the definition of g and S, we have ln(1 + (x/2)) = ey2 − 1. So,α ≥ 1.To prove that  is partially weakly α-admissible with respect to T, let x ∈ X and y ∈ T−1hx; that is, Ty = hx. By the definition of h and T, we have ln(1 + (x/3)) = ey6 − 1. So,α ≥ 1.To prove that .Furthermore, ψ, φ : [0, ∞)→[0, ∞) as ψ(t) = bt and φ(t) = (b − 1)t, for all t ∈ [0, ∞), where 1 < b ≤ 36.Define x, y, and z with α ≥ 1 we haveM = G. Therefore, all the conditions of Using the mean value theorem for all F(f) = {x ∈ X : fx = x} be the fixed point set of f.Let f is also a fixed point of fn, for every n ∈ N; that is, F(f) ⊂ F(fn). However, the converse is false. For example, the mapping f : R → R, defined by fx = (1/2) − x, has the unique fixed point 1/4, but every x ∈ R is a fixed point of f2. If F(f) = F(fn), for every n ∈ N, then f is said to have property P. For more details, we refer the reader to [Clearly, a fixed point of eader to , 42–45 aAssume thatR = IX (the identity mapping on X) in Taking X, G) be a G-complete G-metric space. Let f : X → X be a mapping such that f is partially weakly α-admissible and, for every x, y, z ∈ X such that α ≥ 1,φ : [0, ∞)→[0, ∞) is an altering distance function. Then, f has a fixed point z in X provided that (a) f is continuous or (b) X is α-regular.Let (X and f be as in f has property P if f is an α-dominating map.Let F(f) ≠ ∅. Let u ∈ F(fn) for some n > 1. We will show that u = fu. We have α ≥ 1, as f is α-dominating. Using  = G, then, from  and repeating the above process, we getφ implies thati ≤ n − 1. Now, taking i = n − 1, we have u = fu.Starting from Now, letSo, we haveRepeating the above process, we getFrom the above inequalities, we haveφ implies thati ≤ n − 1. Now, taking i = n − 1, we have u = fu.Therefore,In other three cases, the proof will be done in a similar way.Fixed point theorems for monotone operators in ordered metric spaces are widely investigated and have found various applications in differential and integral equations  be a partially ordered G-metric space. We say that X is regular if and only if the following hypothesis holds.Let  be a partially ordered set and f, g, h : X → X given mappings such that fX⊆hX and gX⊆hX. We say that f and g are weakly increasing with respect to h if and only if for all x ∈ X, fx⪯gy, for all y ∈ h−1(fx), and gx⪯fy, for all y ∈ h−1(gx).Let  be a partially ordered set and f and g two self-maps on X. An ordered pair  is said to be partially weakly increasing with respect to h if fx⪯gy, for all y ∈ h−1(fx).Let (h = I (the identity mapping on X), then the previous definition reduces to the weakly increasing mapping [If  mapping  .f, g) is weakly increasing with respect to h if and only if ordered pairs  and  are partially weakly increasing with respect to it.Note that a pair  be a partially ordered set and letLet  be a partially ordered G-complete G-metric space. Let f, g, h, R, S, T : X → X be six mappings such that f(X)⊆R(X), g(X)⊆S(X), and h(X)⊆T(X). Suppose that, for every three elements Tx⪯Ry⪯Sz, one hasψ, φ : [0, ∞)→[0, ∞) are altering distance functions. Let f, g, h, R, S, and T be continuous, the pairs , , and  compatible, and the pairs , , and  partially weakly increasing with respect to R, S, and T, respectively. Then, the pairs , , and  have a coincidence point z in X. Moreover, if Tz⪯Rz⪯Sz, then z is a coincidence point of f, g, h, R, S, and T.Let  be a regular partially ordered G-metric space, f, g, h, R, S, T : X → X six mappings such that f(X)⊆R(X), g(X)⊆S(X), and h(X)⊆T(X), and RX, SX, and TXG-complete subsets of X. Suppose that, for elements Tx⪯Ry⪯Sz, one hasψ, φ : [0, ∞)→[0, ∞) are altering distance functions. Then, the pairs , , and  have a coincidence point z in X provided that the pairs , , and  are weakly compatible and the pairs , , and  are partially weakly increasing with respect to R, S, and T, respectively. Moreover, if Tz⪯Rz⪯Sz, then z ∈ X is a coincidence point of f, g, h, R, S, and T.Let (


BL-algebra. The notions of fuzzy prime ideals, fuzzy irreducible ideals, and fuzzy Gödel ideals of a BL-algebra are introduced and their several properties are investigated. We give a procedure to generate a fuzzy ideal by a fuzzy set. We prove that every fuzzy irreducible ideal is a fuzzy prime ideal but a fuzzy prime ideal may not be a fuzzy irreducible ideal and prove that a fuzzy prime ideal ω is a fuzzy irreducible ideal if and only if ω(0) = 1 and |Im(ω)| = 2. We give the Krull-Stone representation theorem of fuzzy ideals in BL-algebras. Furthermore, we prove that the lattice of all fuzzy ideals of a BL-algebra is a complete distributive lattice. Finally, it is proved that every fuzzy Boolean ideal is a fuzzy Gödel ideal, but the converse implication is not true.In this paper we investigate further properties of fuzzy ideals of a  It is well-known that an important task of the artificial intelligence is to make computer simulate human being in dealing with certainty and uncertainty in information. Logic gives a technique for laying the foundations of this task. Information processing dealing with certain information is based on the classical logic. Nonclassical logic includes many valued logic and fuzzy logic which takes the advantage of the classical logic to handle information with various facets of uncertainty , such asBL, in short). A well known example of a BL-algebra is the interval  endowed with the structure induced by a continuous t-norm. MV-algebras ∧[(y → x) → x],x ≤ y implies y− ≤ x−,x = x,x → x = 1,x → 1 = 1,1 → x ≤ y → x, or equivalently, x → (y → x) = 1,x → y) → y) → y = x → y,− = x−∧y−, (x∧y)− = x−∨y−,−− = y → x− for any x, y ∈ A (see . Let μ be a fuzzy set in A and t ∈ , the set μt = {x ∈ A | μ(x) ≥ t} is called a level subset of μ.A fuzzy set in A and 0A represent two special fuzzy sets in A satisfying 1A(x) = 1 for any x ∈ A and 0A(x) = 0 for any x ∈ A, respectively.The notations 1μ, ν, μλ (λ ∈ Λ) in A where Λ is an index set, we define μ∨ν, μ∧ν, ∨{μλ : λ ∈ Λ} and ∧{μλ : λ ∈ Λ} as follows: for all x ∈ A,For any fuzzy sets μ ≤ ν we mean that μ(x) ≤ ν(x) for all x ∈ A.By A be a BL-algebra. A fuzzy set μ in A is called a fuzzy ideal of A if, for all x, y ∈ A,(FI1)μ(0) ≥ μ(x),(FI2)μ(y) ≥ μ(x)∧μ((x− → y−)−).Let μ be a fuzzy set in A. Then μ is a fuzzy ideal if and only if, for each t ∈ , μt is either empty or an ideal of A.Let BL-algebras. The set of all fuzzy sets in BL-algebra A is denoted by F(A) and the set of all fuzzy ideals in A is denoted by FI(A).In this section, we give a procedure to construct the fuzzy ideal generated by a fuzzy set. First of all we give some further properties of fuzzy ideals in μ ∈FI(A) and μ((x− → y−)−) = μ(0), then μ(x) ≤ μ(y) for any x, y ∈ A. In particular, μ(x) ≤ μ(y) where x− ≤ y− for any x, y ∈ A.If μ(y) ≥ μ((x− → y−)−)∧μ(x) = μ(0)∧μ(x) = μ(x), we have μ(x) ≤ μ(y).Since As an immediate consequence of the proposition we have the following.μ ∈FI(A) and y ≤ x then μ(x) ≤ μ(y) for any x, y ∈ A.If μ ∈FI(A). Then for any x ∈ A, μ(x) = μ(0) if and only if μ(x−−) = μ(0).Let μ(x) = μ(0). By μ ∈ FI(A), we haveμ(x−−) = μ(0).(⇒) Suppose that μ(x−−) = μ(0). Since for any x ∈ A, x ≤ x−−, by μ(0) = μ(x−−) ≤ μ(x), thus μ(x) = μ(0).(⇐) Assume that μ in A is a fuzzy ideal if and only if for all x, y, z ∈ A, z− → (y− → x−) = 1 implies μ(x) ≥ μ(y)∧μ(z).A fuzzy set μ is a fuzzy ideal of A and z− → (y− → x−) = 1, then z− ≤ y− → x− = (y− → x−)−−, and by μ(z) ≤ μ((y− → x−)−), so μ(x) ≥ μ((y− → x−)−)∧μ(y) ≥ μ(y)∧μ(z).Assuming that z− → (y− → x−) = 1 implies μ(x) ≥ μ(y)∧μ(z) for all x, y, z ∈ A. Since x− → (x− → 0−) = x− → (x− → 1) = 1, so μ(o) ≥ μ(x)∧μ(x) = μ(x); (FI1) holds. By 1 = (x− → y−) → (x− → y−) = (x−→y−)−− → (x− → y−), then μ(y) ≥ μ((x−→y−)−)∧μ(x), (FI2) holds.Conversely, suppose that By induction and μ be a fuzzy set in A. μ is a fuzzy ideal if and only if, for any x, y1,…, yn ∈ A, yn− → (yn−1− → (⋯→(y1− → x−)⋯)) = 1 implies μ(x) ≥ μ(y1)∧⋯∧μ(yn).Let A be a BL-algebra and x, a, b, a1,…, an, b1,…, bm ∈ A, if b∗a → x = 1, a1∗ ⋯ ∗an → a = 1, b1∗ ⋯ ∗bm → b = 1, thenLetting a1∗ ⋯ ∗an → a = 1, b1∗ ⋯ ∗bm → b = 1 it follows that a1∗ ⋯ ∗an ≤ a, b1∗ ⋯ ∗bm ≤ b. Since the operation “∗” is isotone, so we have a1∗ ⋯ ∗an∗b1∗ ⋯ ∗bm ≤ a∗b. While b∗a → x = 1 implies b∗a ≤ x, henceFrom f be a fuzzy set in A. A fuzzy ideal μ is called to be generated by f if f ≤ μ and f ≤ ν implies μ ≤ ν for any fuzzy ideal ν in A. The fuzzy ideal generated by f will be denoted by  we have f ≤ 1A and the intersection of any family of fuzzy ideals in A is a fuzzy ideal in A.It is worth noticing that this definition is well-defined because 1A = {0, a, b, 1}. Define ∗,→, ∨, and ∧ as follows:A; ∧, ∨, ∗, →, 0,1) is a BL-algebra. Define a fuzzy set μ in A by μ(a) = μ(b) = 0.5, μ(1) = 0, μ(0) = 0.8. It is easy to check that (μ](0) = 0.8, (μ](a) = (μ](b) = (μ](1) = 0.5.Let μ and ν be fuzzy sets in A.μ is a fuzzy ideal in A then  = 1  for any x, y, z ∈ A. Given any arbitrary small ɛ > 0, there exists a1,…, an; b1,…, bm ∈ A, such thatμ(z) ≥ μ(x)∧μ(y), and μ is a fuzzy ideal in A by First, we prove x− → x− = 1 for any x ∈ A, it follows that f(x) ≤ μ(x), so f ≤ μ.Next, since ν is any fuzzy ideal in A with f ≤ ν, then f(x) ≤ ν(x) for any x ∈ A,Finally, supposing that μ ≤ ν.by μ = p → x = 1.Let μ be a fuzzy ideal in A. If s, t ∈  satisfies s ≥ μ(a), t ≥ μ(b), s∧t ≤ μ(a∧b) where a, b ∈ A, thenLet μ ≤  − ɛ < (μ∨as)(a1)∧⋯∧(μ∨as)(an).(μ∨as)(ai) = μ(ai)  , we obtainSince  − ɛ < (μ∨bt)(b1)∧⋯∧(μ∨bt)(bm).l∗a1−∗ ⋯ ∗an− → x− = 1,(μ∨as](x) − ɛ < (μ∨as)(a1)∧⋯∧(μ∨as)(an)∧(μ∨as)(a) = μ(a1∧⋯∧μ(an)∧s.b−)k∗b1−∗ ⋯ ∗bm− → x− = 1,((iv) μ∨bt](x) − ɛ < (μ∨bt)(b1)∧⋯∧(μ∨bt)(bm)∧(μ∨bt)(b) = μ(b1)∧⋯∧μ(bm)∧t.a−)l → (a1−∗ ⋯ ∗an−∗b1−∗ ⋯ ∗bm− → x−) = 1.(Because (iii′)b−)k → (a1−∗ ⋯ ∗an−∗b1−∗ ⋯ ∗bm− → x−) = 1.(By the similar argument and (iii) we can getp ∈ N such thatBy (i′), (iii′), and μ∨as]∧ ≤ μ(x)∨μ(y).A nonconstant fuzzy ideal μ in A is a fuzzy prime ideal in A if and only if μt = ∅ where t > μ(0); μt is a prime ideal of A where inf{μ(x) | x ∈ A} < t ≤ μ(0); μt = A where 0 ≤ t ≤ inf{μ(x) | x ∈ A}.A nonconstant fuzzy set It is easy and omitted.a, b ∈  and a < b. If I is a prime ideal of A and J is an proper ideal of A with I ⊂ J, then the function fIJab : A →  is a fuzzy prime ideal in A whereLet I is a prime ideal of A, then the characteristic function χI of I is a fuzzy prime ideal in A whereIf μ be a fuzzy ideal in A, then μ is a fuzzy prime ideal in A if and only if μμ(0) is a prime ideal of A.Letting μμ(0) is a prime ideal of A. μt = ∅ where t > μ(0); μμ(0)⊆μt ≠ A where inf{μ(x) | x ∈ A} < t ≤ μ(0), so μt is a prime ideal of A; μt = A where 0 ≤ t ≤ inf{μ(x) | x ∈ A}. Thus μ is a fuzzy prime ideal in A by The “only if” part is easy. We now prove the part “if” as follows. Suppose that μ in A is a fuzzy prime ideal in A if and only if μ(x∧y) = μ(0) implies μ(x) = μ(0) or μ(y) = μ(0).A nonconstant fuzzy ideal μ is a fuzzy prime ideal in A; then μμ(0) is a prime ideal of A by μ(x∧y) = μ(0), then x∧y ∈ μμ(0), and so x ∈ μμ(0) or y ∈ μμ(0). Hence μ(x) = μ(0) or μ(y) = μ(0).Suppose that x, y ∈ A, μ(x∧y) = μ(0) implies μ(x) = μ(0) or μ(y) = μ(0). That is, x∧y ∈ μμ(0) implies x ∈ μμ(0) or y ∈ μμ(0), and thus μμ(0) is a prime ideal of A. Therefore μ is a fuzzy prime ideal in A by Conversely, suppose, for any Note. The above theorem shows that the definition on fuzzy prime ideals in this paper and one in  + ω(a∧b)} and μ = ∨ω(b) < s < ω(a∧b), by μ∧ν = ω, but μ ≠ ω, ν ≠ ω, a contradiction.Suppose But the converse of the above theorem is not true.A = {0, a, b, 1}. Define ∗, →, ∨, and ∧ as follows:A; ∧, ∨, ∗, →, 0,1) is a BL-algebra. It is easy to check that I1 = {0, a}, I2 = {0, b} are prime ideals of A.Let ω in A by ω(0) = ω(a) = 1/2, ω(1) = ω(b) = 1/4, and then ω is a fuzzy prime ideal in A. Indeed, ωt = ∅ where t > 1/2; ωt = {0, a} where 1/4 < t ≤ 1/2; ωt = A where 0 ≤ t ≤ 1/4. Let μ be a fuzzy ideal in A defined by μ(0) = 3/4, μ(a) = 3/4, μ(b) = μ(1) = 1/4. Let ν be a fuzzy ideal in A defined by ν(0) = ν(a) = 1/2, ν(b) = ν(1) = 3/8. It is easy to verify that μ∧ν = ω but μ ≠ ω, ν ≠ ω. Therefore ω is not a fuzzy irreducible ideal in A.Define a fuzzy set This example shows the converse of ξ be a fuzzy set in A defined by ξ(0) = ξ(a) = 1, ξ(b) = ξ(1) = t(0 < t < 1), it is easy to check that ξ is a fuzzy irreducible ideal in A.Letting BL-algebras, an ideal is prime if and only if it is irreducible . we get the fuzzy set f∨g:Let f, g are fuzzy ideals in A, but f∨g is not a fuzzy ideal in A since (f∨g)s = {0, a, b} is not an ideal of A.It is easy to see that μ, ν ∈ FI(A), denote μ⊔ν = :(λ ∈ Λ ≠ ∅),Now we give the following definition: for any A be a BL-algebra, then (FI(A); ⊔, ∧, 0A, 1A) is a complete distributive lattice where 0A and 1A are the least lower bound and the largest upper bound ofFI(A), respectively, and satisfies the following infinitely distributive law:(DL)μ∧⊔{να | α ∈ Λ} = ⊔{μ∧να | α ∈ Λ}, for all μ, να ∈FI(A)(α ∈ Λ ≠ ∅).Letting μ, ν ∈ FI(A),  μ⊔ν and μ∧ν are the supremum and the infimum in FI(A) of μ and ν, and 0A ≤ μ ≤ 1A, so (FI(A); ⊔, ∧, 0A, 1A) is a bounded lattice. This lattice is obviously complete.It is easy to verify that for any νλ ∈ FI(A)(λ ∈ Λ ≠ ∅),x ∈ A and arbitrary small ɛ > 0, there exist a1, a2,…, an ∈ A such thatνα | α ∈ Λ}(ai)  , there are α1,…, αn ∈ Λ such thatIn order to check (DL) it suffices to prove, for any Thus by μ(x) ≤ μ(b1), μ(b2),…, μ(bn).(i) a1− ≤ b1−. By a similar way we may prove that a2− ≤ b2−,…, an− ≤ bn−. ThereforeSinceBy να1(a1) ≤ να1(b1),…, ναn(an) ≤ ναn(bn).(ii) By (i) and (ii) we haveand soμ∧να | α ∈ Λ} is a fuzzy ideal of A, and by ɛ is arbitrary small, we haveIt is obviousThe proof is completed.BL-algebra and investigate some of their properties.In this section, we introduce the notion of fuzzy Gödel ideals of μ be a fuzzy ideal of A. μ is called a fuzzy Gödel ideal if μ((x− → (x−)2)−) = μ(0) for all x ∈ A.Let It is obvious that each fuzzy ideal in Gödel algebra is a fuzzy Gödel ideal. In μ be a fuzzy subset of A. μ is a fuzzy Gödel ideal if and only if, for each t ∈ , μt is a Gödel ideal of A where μt ≠ ∅.Let Next theorem is called the extension theorem of fuzzy Gödel ideals.μ and ν be fuzzy ideal of A with μ ≤ ν and μ(0) = ν(0). If μ is a fuzzy Gödel ideal, then so is ν.Let μ be a fuzzy ideal of A. Define a fuzzy set χμ in A byLet χμ is also a fuzzy ideal in A.Obviously, μ of A is a fuzzy Gödel ideal if and only if χμ is a fuzzy Gödel ideal of A.A fuzzy subset μ is a fuzzy Gödel ideal of A, then by t ∈ , μt is a Gödel ideal of A. In particular, μμ(0) = {x ∈ A | μ(x) = μ(0)} is a Gödel ideal of A. We notice for any t ∈ t ∈ , (χμ)t is a Gödel ideal of A where (χμ)t ≠ ∅. By χμ is a fuzzy Gödel ideal of A.If χμ is a fuzzy Gödel ideal of A. It is clear that χμ ≤ μ and χμ(0) = μ(0). By μ is a fuzzy Gödel ideal of A.Conversely, suppose μ be a fuzzy ideal of A. The following conditions are equivalent:μ is a fuzzy Gödel ideal,μ(((x−)2 → y−)−) = μ(0) implies μ((x− → y−)−) = μ(0),μ(((x−∗y−) → z−)−) = μ(0) implies μ(((x− → y−)→(x− → z−))−) = μ(0).Let μ is a fuzzy Gödel ideal and μ(((x−)2 → y−)−) = μ(0). Then we haveμ((x− → y−)−) = μ(0).(i)⇒(ii) Suppose that μ(((x−∗y−) → z−)−) = μ(0). Since y− → z− ≤ (x− → y−)→(x− → z−), then we havex−)2 → ((x− → y−) → z−))− ≤ ((x−∗y−) → z−)−. Henceμ(((x−)2 → ((x− → y−) → z−))−) = μ(0). From (ii) we get thatμ(((x−)2 → (x−)2)−) = μ(0), by (iii) we haveμ is a fuzzy Gödel ideal.(ii)⇒(iii) Suppose that μ be a fuzzy ideal. μ is a fuzzy Gödel ideal if and only if μ((x− → (y− → z−))−) = μ(0) and μ((x− → y−)−) = μ(0) imply μ((x− → z−)−) = μ(0).Let μ is a fuzzy Gödel ideal. Letμ((x− → z−)−) = μ(0).Suppose that μ((x− → (x− → (x−)2))−) = μ(0) and μ((x− → x−)−) = μ(0), then by the assumption we have μ((x− → (x−)2)−) = μ(0), so μ is a fuzzy Gödel ideal.Conversely, since μ be a fuzzy ideal. Then μ is a fuzzy Gödel ideal if and only if the following condition holds:(∗)μ(x− → ((y−)2 → z−)−) = μ(0) and μ(x) = μ(0) imply μ((y− → z−)−) = μ(0) for any x, y, z ∈ A.Let μ is a fuzzy Gödel ideal of A. Letting μ((x− → ((y−)2 → z−))−) = μ(0) and μ(x) = μ(0), thenμ(((y−)2 → z−)−) = μ(0). Since μ is a fuzzy Gödel ideal, by μ((y− → z−)−) = μ(0). (∗) holds.Suppose that μ((x− → (y− → z−))−) = μ(0) and μ((x− → y−)−) = μ(0). Sinceμ((x− → (y− → z−))−) = μ(0), we get thatμ((x− → y−)−) = μ(0) we get μ((x− → z−)−) = μ(0). Thus μ is a fuzzy Gödel ideal by Conversely, suppose that (∗) is true. Let BL-algebra and proved that fuzzy Boolean ideals are equivalent to fuzzy implicative ideals. In the following, we investigate the relation between fuzzy Boolean ideals and fuzzy Gödel ideals.Zhang et al. . introduμ in A is called a fuzzy Boolean ideal if μ(x∧x−) = μ(0) for all x ∈ A.A fuzzy ideal A is a fuzzy Gödel ideal in A.Each fuzzy Boolean ideal in μ is a fuzzy Boolean ideal in A and μ(((x−)2 → y−)−) = μ(0). Thenμ((x− → y−)−) = μ(0). By μ is a fuzzy Gödel ideal in A.Suppose that But the converse of the above theorem is not true.A = {0, a, b, c, d, 1}. Define ∗, →, ∨, and ∧ as follows:A; ∧, ∨, ∗, →, 0,1) is a BL-algebra. Define a fuzzy set μ in A by μ(a) = μ(b) = μ(c) = μ(1) = 0.5, μ(0) = μ(d) = 0.8; one can easily check that μ is a fuzzy Gödel ideal in A, but it is not a fuzzy Boolean ideal, since μ(a∧a−) = μ(a∧c) = μ(c) = 0.5 ≠ 0.8 = μ(0).Let BL-algebras is technically more difficult, so far little research literature. Zhang et al. [BL-algebras. The notions of fuzzy prime ideals, fuzzy irreducible ideals, and fuzzy Gödel ideals are introduced and studied. We give a procedure to generate a fuzzy ideal by a fuzzy set. Using this result we prove that any fuzzy irreducible ideal is a fuzzy prime ideal and meanwhile we give an example to show that a fuzzy prime ideal may not be a fuzzy irreducible ideal; we also give the Krull-Stone representation theorem of fuzzy ideals in BL-algebras. Furthermore we prove that the set of all fuzzy ideals forms a complete distributive lattice. In addition, we prove that any fuzzy Boolean ideal in BL-algebras is a fuzzy Gödel ideal, but the converse is not true.Study of fuzzy ideal theory in g et al.  initiateBL-algebras should be related to (1) several special types of fuzzy ideals; (2) decomposition properties of fuzzy ideals. Our obtained results can be applied in information science, engineering, computer science, and medical diagnosis.In our opinion, the future study of fuzzy ideals in