Based on linguistic term sets and hesitant fuzzy sets, the concept of hesitant fuzzy linguistic sets was introduced. The focus of this paper is the multicriteria decision-making (MCDM) problems in which the criteria are in different priority levels and the criteria values take the form of hesitant fuzzy linguistic numbers (HFLNs). A new approach to solving these problems is proposed, which is based on the generalized prioritized aggregation operator of HFLNs. Firstly, the new operations and comparison method for HFLNs are provided and some linguistic scale functions are applied. Subsequently, two prioritized aggregation operators and a generalized prioritized aggregation operator of HFLNs are developed and applied to MCDM problems. Finally, an illustrative example is given to illustrate the effectiveness and feasibility of the proposed method, which are then compared to the existing approach. Since the fuzzy set was proposed by Zadeh in 1965 –5. Due tHesitant fuzzy sets (HFSs), an extension of traditional fuzzy sets, can address this problem. HFSs were first introduced by Torra and Narukawa , 7, and When faced with problems that are too complex or ill-defined to be solved by quantitative expressions, linguistic variables can be an effective tool because the use of linguistic information enhances the reliability and flexibility of classical decision models . Linguiss2, 〉 is an HFLN and 0.3, 0.4, and 0.5 are the possible membership degrees to the linguistic term s2. HFLSs have enabled great progress in describing linguistic information and to some extent may be considered an innovative construct. The main advantage of HFLSs is that they can describe two fuzzy attributes of an object: a linguistic term and a hesitant fuzzy element (HFE). The former provides an evaluation value, such as “excellent” or “good.” The latter describes the hesitancy for the given evaluation value and denotes the membership degrees associated with the specific linguistic term. However, the operations proposed by Lin et al.  in the case of it being applied to X.Let hE(x) is a set of values in , denoting the possible membership degrees of the element x ∈ X to the set E. hE(x) is called a hesitant fuzzy element (HFE)  and denotes the possible membership degrees that x belongs to sθ(x).Let X has only one element, the HFLS A is reduced to 〈sθ(x), hA(x)〉. For computational convenience, we call α = 〈sθ(α), hα〉 as an HFLN.When hA(x) = {r} has only one element, it indicates that the degree that x belongs to sθ(x) is r. For example, 〈s2, 0.3〉 is called a fuzzy linguistic number, which is a special case of HFLN.When X = {x1, x2} be a universal set. If an HFLS A = {〈x, sθ(x), hA(x)〉∣x ∈ X} = {〈x1, s5, {0.4,0.6,0.7}〉, 〈x2, s6, {0.1,0.5,0.7}〉} is divided into two subsets that contain only one object, respectively, then 〈s5, {0.4,0.6,0.7}〉 and 〈s6, {0.1,0.5,0.7}〉 are HFLNs. 0.4, 0.6, and 0.7 are the possible membership degrees that x1 belongs to s5; 0.1, 0.5, and 0.7 are the possible membership degrees that x2 belongs to s6.Let An HFLN is an extension of a linguistic term and an HFE. Compared to linguistic terms, HFLNs embody the possible membership degrees that an evaluation object attaches to the linguistic term, and they can depict the fuzziness more accurately than an uncertain linguistic variable does. When compared to HFEs, HFLNs add linguistic terms and assign the membership function to a specific linguistic evaluation value, which make the membership degrees no longer relative to a fuzzy concept, but to linguistic terms, such as “poor” or “good.”s5)” is an acceptable evaluation result for the car and is given by three decision makers. Then, each decision maker uses a value to express his/her opinion about the car under the evaluation of “good (s5).” Decision maker A may give the value 0.4 for “good,” whilst decision maker B may give 0.6 and decision maker C may give 0.7. In this case, HFLNs may be a better choice, and the evaluation result can be denoted by 〈s5, {0.4,0.6,0.7}〉.In fuzzy set theory, the hesitant values in HFLNs are called possible membership degrees, which are caused by the hesitancy and uncertainty of decision makers. In the example of the performance evaluation of a car, suppose that “good   are diffTo use data more efficiently and to express the semantics more flexibly, linguistic scale functions assign different semantic values to linguistic terms under different situations . They arθi ∈  is a numeric value, then the linguistic scale function f that conducts the mapping from si to θi   is defined as follows:θ0 < θ1 < ⋯<θt2.If θi   reflects the preference of the decision makers when they are using the linguistic term si ∈ S  . Therefore, the function/value in fact denotes the semantics of the linguistic terms. (1) ConsiderClearly, the symbol  (2) ConsiderThe evaluation scale of the linguistic information given above is divided on average. (3) ConsiderWith the extension from the middle of the given linguistic term set to both ends, the absolute deviation between adjacent linguistic subscripts also increases.With the extension from the middle of the given linguistic term set to both ends, the absolute deviation between adjacent linguistic subscripts will decrease.f*(si) = θi, and is a strictly monotonically increasing and continuous function. Therefore, the mapping from R+ is one-to-one because of its monotonicity, and the inverse function of f* exists and is denoted by f∗−1.To preserve all the given information and facilitate the calculation, the above function can be expanded to α = 〈sθ(α), hα〉 and β = 〈sθ(β), hβ〉 be two HFLNs. Some operations of α and β are defined as follows:α ⊕ β = 〈sθ(α)+θ(β), ∪r1∈hα,r2∈hβ{r1 + r2 − r1r2}〉;λα = 〈sλθ(α), ∪r∈hα{1 − (1 − r)λ}〉;α ⊗ β = 〈sθ(α)×θ(β), ∪r1∈hα,r2∈hβ{r1r2}〉;αλ = 〈sθ(α)λ, ∪r∈hα{rλ}〉.Let α = 〈s2, {0.3,0.4}〉; for example, 0.3 and 0.4 are the possible membership degrees that the object belongs to s2; that is, {0.3,0.4} is the explanatory part of s2 and should be closely related to s2 in the additive operation.The operations proposed in  have somIn order to overcome the existing limitations given above, new operations of HFLNs based on linguistic scale functions are defined as follows.α = 〈sθ(α), hα〉 and β = 〈sθ(β), hβ〉 be two HFLNs. Some operations of α and β are defined as follows:(1)(2)(3)(4)(5)Let f* is a mapping from the linguistic term si to the numeric value θi and f∗−1 is a mapping from θi to si. So, the first part of (1)–(5) is a linguistic term. In addition, it is obvious that the second part of (1)–(5) is an HFE. In summary, according to According to f* is utilized. Thus, decision makers can flexibly select the linguistic scale function f* depending on their preferences and the actual semantic situations. In addition, the new addition operation of HFLNs is more reasonable and reliable, because the final hesitant fuzzy membership has closely combined each element of the original HFLNs.The operations defined above are based on linguistic scale functions, which can get different results when a different linguistic scale function α ⊕ β, λα, α ⊗ β, and αλ necessarily appear in defining basic operations, but their results have no practical meaning. In the aggregation process, for example, using weighted operators, α ⊕ β is combined with λα and α ⊗ β is combined with αλ; therefore, the calculation results are interpretable in practice.S = {s0, s1, s2, s3, s4, s5, s6} = {very poor, poor, slightly poor, fair, slightly good, good, very good}, α = 〈s2, {0.1,0.3}〉, β = 〈s3, {0.2}〉, and λ = 2.Let f1*(si) = i/6  (0 ≤ i ≤ 6), thenα) = 〈s4, {0.7,0.9}〉;neg = 〈s4, {0.7,0.9}〉;neg = 〈s4, {0.7,0.9}〉;neg, hαi〉   be any three HFLNs; thus the following properties are true.α1 ⊕ α2 = α2 ⊕ α1;α1 ⊗ α2 = α2 ⊗ α1;α1 ⊕ α2) ⊕ α3 = α1 ⊕ (α2 ⊕ α3);(α1 ⊗ α2) ⊗ α3 = α1 ⊗ (α2 ⊗ α3);(λ(α1 ⊕ α2) = λα1 ⊕ λα2, (λ ≥ 0);α1 ⊗ α2)λ = α1λ ⊗ α2λ, (λ ≥ 0).(Let  (3) ConsiderAccording to α1 ⊕ α2) ⊕ α3 = α1 ⊕ (α2 ⊕ α3).So,  and (6) can be easily proven.α = 〈sθ(α), hα〉 be an HFLN. The score function E(α) of α can be represented as follows:s(hα) is the score function of hα.Let α = 〈s3, {0.2,0.4,0.5,0.7}〉. If t = 3 and f1*(si) = i/2t, by applying , hα〉 = 〈sθ(α), ∪r∈hα{r}〉 be an HFLN. A variance function hα can be denoted by D(α) of α can be represented as follows:l(hα) is the number of the values in hα.Let α = 〈s3, {0.3,0.6}〉. If t = 3 and f1*(si) = i/2t, by applying , hα1〉 and α2 = 〈sθ(α2), hα2〉 be any two HFLNs.E(α1) > E(α2), then α1 > α2.If E(α1) = E(α2), thenIf  D(α1) > D(α2), then α1 > α2;if  D(α1) = D(α2), then α1 = α2.if Let α = 〈s3, {0.3,0.6}〉 and β = 〈s6, {0.15,0.3}〉. If t = 3 and f1*(si) = i/2t, then E(α) = E(β) = 0.225, D(α) = 0.489, D(β) = 0.994, and thus α < β.Let In this section, two prioritized aggregation operators for HFLNs are proposed based on the PA operator, and some desirable properties are also analyzed. Subsequently, these operators are extended to a generalized form. Finally, a method for solving MCDM problems with HFLNs, where the criteria are in different priority levels, is developed.The PA operator was originally introduced by Yager  and is sG = {G1, G2,…, Gn} be a collection of criteria and ensure that there is a prioritization between the criteria expressed by the linear ordering G1≻G2≻G3≻⋯≻Gn, which indicates that the criteria Gj has a higher priority than Gk, if j < k. Gj(x) is an evaluation value denoting the performance of the alternative x under the criteria Gj and satisfies Gj(x)∈. Ifwj = Tj/∑i=1nTi, T1 = 1 and Tj = ∏k=1j−1Gk(x)  , then PA is called the PA operator.Let α1, α2,…, αn} and {β1, β2,…, βn} are two sets of criteria values under criteria {G1, G2,…, Gn}, where G1≻G2≻G3≻⋯≻Gn. Now the PA operators under a hesitant fuzzy linguistic environment will be analyzed in the following subsections.PA operators have usually been used in situations where input arguments are exact values. Therefore, PA operators could be extended to accommodate situations where the input arguments are hesitant fuzzy linguistic information. Based on In this subsection, the prioritized weighted average operator under a hesitant fuzzy linguistic environment is investigated. The definition of the HFLPWA operator and its relevant theorems are given as follows.αj   be a collection of HFLNs, and then the HFLPWA operator can be defined as follows:Tj = ∏k=1j−1E(αk)  , T1 = 1, and E(αk) is the score function of αk.Let Based on the operations of HFLNs described in αj = Gj(x) = 〈sθ(αj), hαj〉   be a collection of HFLNs. Then the aggregated value, obtained by using the HFLPWA operator, is also an HFLN, andTj = ∏k=1j−1E(αk)  , T1 = 1, and E(αk) is the score function of αk.Let n.(1)n = 2, sinceFor (2)n = k, thenIf  holds foWhen n = k + 1, by the operations described in n = k + 1. Thus,  = 〈sθ(αj), hαj〉   be a collection of HFLNs. If α− = 〈minj{sθ(αj)}, r−〉, α+ = 〈maxj{sθ(αj)}, r+〉, where r− = minrj∈∪hα1,hα2,…,hαn{rj} and r+ = maxrj∈∪hα1,hα2,…,hαn{rj}, thenLet α1, α2,…, αn) = α = 〈sθ(α), hα〉, and then E(α) = f*(sθ(α)) · s(hα).Let HFLPWA(j{sθ(αj)} ≤ sθ(αj) ≤ maxj{sθ(αj)} for all j, we haveSince minαj = Gj(x) = 〈sθ(αj), hαj〉   be a collection of HFLNs and α1, α2,…, αn). ThenLet αj is decided by the priority and value of αj and will not be influenced by its position in the permutation. So, The weight of α = 〈s2, {0.2,0.4}〉, for example, and if α1 = α2 = α, then HFLPWA = 〈s2, {0.2,0.218,0.382,0.4}〉 ≠ α. In addition, we do not consider the monotonicity of HFLPWA because the weights will be recalculated and vary if the values used in the HFLPWA operator change. It is difficult to consider the monotonic property when the parameters are irregularly variable.It should be noted that the HFLPWA operator cannot satisfy idempotency. Take In this subsection, the prioritized weighted geometric operator under a hesitant fuzzy linguistic environment is investigated. The definition of the HFLPWG operator and its relevant theorems are given as follows.αj   be a collection of HFLNs, and then the HFLPWG operator can be defined as follows:Tj = ∏k=1j−1E(αk)  , T1 = 1, and E(αk) is the score function of αk.Let Similar to the HFLPWA operator, the HFLPWG operator satisfies the following properties.αj = 〈sθ(αj), hαj〉   be a collection of HFLNs. Then the aggregated value, obtained by using the HFLPWG operator, is also an HFLN, andTj = ∏k=1j−1E(αk)  , T1 = 1, and E(αk) is the score function of αk.Let n.(1)n = 2, sinceFor we have(2)n = k, thenIf  holds foWhen n = k + 1, by the operations described in n = k + 1. Thus,  = 〈sθ(αj), hαj〉   be a collection of HFLNs. If α− = 〈minj{sθ(αj)}, r−〉, α+ = 〈maxj{sθ(αj)}, r+〉, where r− = minrj∈∪hα1,hα2,…,hαn{rj} and r+ = maxrj∈∪hα1,hα2,…,hαn{rj}, thenLet α1, α2,…, αn) = α = 〈sθ(α), hα〉, and then E(α) = f*(sθ(α)) · s(hα).Let HFLPWG(j{sθ(αj)} ≤ sθ(αj) ≤ maxj{sθ(αj)} for all j, we haveSince minαj = Gj(x) = 〈sθ(αj), hαj〉   be a collection of HFLNs and α1, α2,…, αn). ThenLet Similarly to α = 〈s2, {0.2,0.4}〉, for example; if α1 = α2 = α, then HFLPWG = 〈s2, {0.2,0.213,0.376,0.4}〉 ≠ α. Similarly, we do not consider the monotonicity of HFLPWG because of the variability of weights.Besides, the HFLPWG operator cannot satisfy idempotency. Take In general, the HFLPWA operator emphasizes the impact of the overall evaluation data and the compensation between different evaluation results, while the HFLPWG operator emphasizes the balance in the system and the coordination between different evaluation results. In this section, the generalized form of the HFLPWA and HFLPWG operators will be proposed, that is, the hesitant fuzzy linguistic generalized prioritized weighted aggregation (HFLGPWA) operator.αj   be a collection of HFLNs, and then the HFLGPWA operator can be defined as follows:λ > 0, Tj = ∏k=1j−1E(αk)  , T1 = 1, and E(αk) is the score function of αk.Let (1)λ → 0, then the HFLGPWA operator degenerates into the HFLPWG operator:If (2)λ = 1, then the HFLGPWA operator degenerates into the HFLPWA operator:If Obviously, the HFLPWA and HFLPWG operators are the special cases of the HFLGPWA operator.In this subsection, the HFLGPWA operator will be applied to MCDM problems with hesitant fuzzy linguistic information.C1, C2,…, Cn}, and the prioritization relationships that exist among them are C1≻C2≻⋯≻Cn. Every criterion in Ci has a higher priority than every criterion in Cj if i < j. Under these criteria, there is a set of alternatives {x1, x2,…, xm} and the criteria values of the alternatives are expressed as HFLNs Cj(xi)  . Suppose that R = (Cj(xi))m×n is the decision matrix. Subsequently, a ranking of alternatives is required.For MCDM problems with hesitant fuzzy linguistic information, assume that there is a set of criteria {In the following paragraphs, the HFLGPWA operator is applied to MCDM problems with hesitant fuzzy linguistic information.This method involves the following steps.The common types of criteria in MCDM problems are maximizing criteria and minimizing criteria. For the minimizing criteria the negation operator in xi   with respect to Cj   are also denoted by Cj(xi) = 〈sθ(Cij), hCij〉.For convenience, the normalized criteria values of Xi   of xi by applying the formula as follows:Obtain the comprehensive evaluation values E(Xi)   of the comprehensive values Xi of the alternatives xi  , in order to rank all alternatives xi   and then select the best one(s). If the score values are E(Xi) = E(Xj)  (i ≠ j), it is necessary to calculate the accuracy values D(Xi) and D(Xj) and then rank the alternatives xi and xj in accordance with these accuracy values.Use E(Xi) and D(Xi)  .Use The following case is adapted from .ABC Nonferrous Metals Co. Ltd. is a large state-owned company whose main business is producing and selling nonferrous metals. It is also the largest manufacturer of multispecies nonferrous metals in China, with the exception of aluminum. To expand its main business, the company is always engaged in overseas investment, and a department which consists of executive managers and several experts in the field has been established specifically to make decisions on global mineral investment.x1, x2, x3, x4, x5}. There are many factors that affect the investment environment and four factors are considered based on the experience of the department personnel, including C1: resources ; C2: politics and policy ; C3: economy ; C4: infrastructure (such as railway and highway facilities).Recently, the overseas investment department decided to select a pool of alternatives from several foreign countries based on preliminary surveys. After thorough investigation, five countries  are taken into consideration, that is, {S = {s0, s1, s2, s3, s4, s5, s6} = {very poor, poor, slightly poor, fair, slightly good, good, very good} is used. The evaluation information is given in the form of HFLNs, where Cj(xi) is the evaluation value of the alternative xi on the criterion Cj. In Cj(xi) there is a consensus on the chosen linguistic term and each decision maker can use a value to express his/her opinion; in other words, the value denotes to what degree xi matches this given linguistic term under Cj. Each decision maker gave his/her own evaluations (membership degrees) based on the surveys of the five countries as well as his/her knowledge and experience. Then all the possible membership degrees under each given linguistic term are gathered together. The same membership degrees for a given linguistic term will appear only once in an HFLN. Consequently, following a heated discussion, they came to a consensus on the final evaluations which are expressed by HFLNs as shown in The decision makers, including the experts and executive managers, have gathered to determine the decision information. The linguistic term set C1≻C2≻C3≻C4.Assume that the prioritization relationship for the criteria is f1*(si) = i/2t and adopt the following steps.To get the optimal alternative(s), let xi   do not need to be normalized.Considering that all the criteria are of maximizing type, the performance values of the alternatives Use E(Xi)   of the comprehensive evaluation values Xi of the alternatives xi   can be calculated and are shown in The score values λ, the score values E(Xi) would increase slightly, where i = 1,2, 3,4, 5.It has been ascertained that, with the increase of E(Xi)  , the rankings of the alternatives are shown in In accordance with the score values x5 is the best choice. This result can reveal the stability of the proposed method.From f* and parameter λ on the decision-making process in this example, different f* and λ are used in Steps In order to illustrate the influence of the linguistic scale function f* or parameter λ is utilized. If λ ≤ 5, the best alternative is x5; if λ ≥ 8, the best one is x3. Moreover, the worst alternative is always x4 except for one situation.We can conclude that the rankings of the alternatives may be a little different when a different linguistic scale function To further illustrate the advantages of the proposed MCDM approach under a hesitant fuzzy linguistic environment, the method in  is used Lin et al.  utilizedS(x5) > S(x1) > S(x3) > S(x2) > S(x4), the ranking is x5≻x1≻x3≻x2≻x4, and the most desirable car is x5.Since x1 and x3, and this may be caused by the different operations and comparison method for HFLNs. The operations and comparison method for HFLNs in [f* used in this paper are applicable and effective under different semantic environment. In addition, the HFLWA operator in [Obviously, the rankings obtained by the proposed method in this paper may be a little different from that obtained by the method in . The onlHFLNs in  considerrator in  emphasizAccording to the above comparison analyses, the proposed method for MCDM problems with HFLNs has the following advantages.First, HFLNs used in this paper can express the evaluation information more flexibly. They can depict fuzzy linguistic information more accurately and retain the completeness of the original data or the inherent thoughts of decision makers, which is the prerequisite of guaranteeing accuracy of final outcomes.f* is used. Thus, decision makers can flexibly select the linguistic scale function f* depending on their preferences and the actual semantic situations.Second, the operations of HFLNs in this paper are defined based on linguistic scale functions, which can achieve different results when a different linguistic scale function Third, the proposed hesitant fuzzy linguistic prioritized aggregation operators can deal with MCDM problems under the hesitant fuzzy linguistic environment in which the criteria are in different priority levels. What is more, the criteria weights, which are calculated by the prioritized aggregation operator according to the criteria priority levels, are more objective and reasonable than a set of known criteria weights.To address situations where decision-making problems use qualitative variables rather than numerical ones and to reflect the uncertainty, hesitancy, and inconsistency of decision makers, HFLSs have been introduced and used in this paper. Considering the limitations in the existing literature, new operations of HFLNs were introduced. Then, on the basis of the PA operator, two prioritized aggregation operators for HFLNs were proposed and extended to a generalized form. Furthermore, an MCDM method based on the generalized prioritized aggregation operator under a hesitant fuzzy linguistic environment was developed. Finally, an illustrative example demonstrated the application of the proposed method and comparison analysis was made with the representative method. The results indicated that the method proposed in this paper is feasible and effective in solving MCDM problems with HFLSs.λ may also influence the results. Decision makers can select the most appropriate linguistic scale function f* according to their interests and actual semantic situations. In a word, the main advantages of the proposed method are not only that the proposed operators accommodate a hesitant fuzzy linguistic environment, but also its consideration of the priority among criteria, which is more feasible and practical. In the future research, the linguistic scale function can be applied in other linguistic sets, such as ILSs, LHFSs, and HFLSs. Moreover, we believe that the study of information measures and outranking relations for HFLSs does make great sense.It is well known that the PA operator has many advantages over other operators as it does not need to provide weight vectors and, when using this approach, it is only necessary to know the priority among criteria. The foremost characteristic of these proposed operators is that they take into account the priority among criteria. Although traditional prioritized aggregation operators are generally suitable for aggregating information which is in the form of numerical values or simple fuzzy values, they are unable to deal with hesitant fuzzy linguistic information. The proposed HFLGPWA operator can accommodate situations where the input arguments consist of hesitant fuzzy linguistic information. In addition, the results may change using different linguistic scale functions, and the parameter


L is strongly semicontinuous if and only if L is semicontinuous and meet semicontinuous. It is proved that semi-FS domains are strongly semicontinuous. Some interpolation properties of semiway-below relations in (strongly) semicontinuous bc-domains are given. In terms of these properties, it is proved that strongly semicontinuous bc-domains, in particular strongly semicontinuous lattices, are all semi-FS domains.We are mainly concerned with some special kinds of semicontinuous domains and relationships between them. New concepts of strongly semicontinuous domains, meet semicontinuous domains and semi-FS domains are introduced. It is shown that a dcpo  A logic-oriented approach to domain theory to formalize the properties of computation is provided in  to denote all order-preserving functions from L to M, use [L↪M] to denote all the functions preserving suprema of prime ideals from L to M and use [L → M] to denote all the Scott-continuous functions from L to M. All of them are under the pointwise order. It is easy to see that [L → M]⊆[L↪M]⊆.Let L, M be dcpos. Then [L↪M] is a dcpo.Let G be a directed subset of [L↪M] and f(x) = ∨g∈Gg(x) for all x ∈ L. Then it is easy to see that f is order-preserving. For any prime ideal P ∈ PI(L), we havef ∈ [L↪M], showing that [L↪M] is a dcpo.Let L be a dcpo. If D⊆ is directed and supδ∈Dδ = IdL, then we say that D is an approximate identity for L.Let L be a dcpo. If L has an approximate identity D⊆[L↪L] such that δ(x) ⇐ x for all δ ∈ D and for all x ∈ L, then L is a strongly semicontinuous domain.Let x ∈ L, let A = {δ(x) : δ ∈ D}. Then A⊆⇓x ∩↓x. Note that D is directed and x ∈ ↓(∨A) = A↑↓. By the L is a strongly semicontinuous domain.For any L be a dcpo. A function δ : L → L on L is finitely separating if there is a finite set Fδ such that, for each x ∈ L, there exists y ∈ Fδ such that δ(x) ≤ y ≤ x.Let L is called a semi-FS domain if there is an approximate identity D⊆[L↪L] consisting of finitely separating functions.A dcpo L, there is an approximate identity D⊆[L → L]⊆[L↪L] for L consisting of finitely separating functions. So, an FS-domain is a semi-FS domain.For an FS-domain L be a dcpo. If δ ∈ [L↪L] is finitely separating, then, for all x ∈ L, δ(x) ⇐ x. Thus a semi-FS domain is a strongly semicontinuous domain.Let x ∈ L and P ∈ PI(L) with x ≤ ∨P. Since δ is finitely separating, there exists a finite set Fδ such that for each d ∈ P there exists yd ∈ Fδ with δ(d) ≤ yd ≤ d. Let Fδ′ = {yd ∈ Fδ : d ∈ P}, a nonempty finite subset of Fδ. Then for each y ∈ Fδ′, we can get dy ∈ P such that δ(dy) ≤ y ≤ dy. As P is a prime ideal, there exists d0 ∈ P such that y ≤ dy ≤ d0 for all y ∈ Fδ′. Hence for all d ∈ P, δ(d) ≤ yd ≤ d0 and δ(x) ≤ δ(∨P) = ∨d∈Pδ(d) ≤ d0. Therefore, δ(x) ∈ P. It follows from δ(x) ⇐ x.Suppose that L is a strongly semicontinuous domain.By The next example gives a strongly semicontinuous domain which is not a semi-FS domain, showing that the reverse of L = {0, b1, b2}∪{aj : j = 1,2,…} be the domain showing in bi < aj for i = 1,2 and j = 1,2,…. It is clear that L is an L-domain but not compact in the Lawson topology. By ⊆[L↪L]⊆ = [L → L]. So, [L → L] = [L↪L]. Since L is not an FS-domain, L is not a semi-FS domain either.Let Every strongly semicontinuous bc-domain is a semi-FS domain.L be a strongly semicontinuous bc-domain. For each x ∈ L, S ∈ Pfin(L), define δS : L → L by δS(x) = ∨{y ∈ S∩↓x : y ⇐ x}. If {y ∈ S∩↓x : y ⇐ x} = ∅, then δS(x) = ⊥, the least element of L. So, δS(x) is well-defined. It is easy to see that δS is order-preserving with δS(x) ≤ x for all x ∈ L. Next we show that δS preserves suprema of prime ideals. For each P ∈ PI(L), it suffices to show that δS(∨P) = δS(k) ≤ ∨p∈PδS(p), where k = ∨P. If {y ∈ S∩↓k : y ⇐ k} = ∅, then, by the definition of δS, we see that δS(k) = ⊥ ≤∨p∈PδS(p). Let {y1,…, yl} = {y ∈ S∩↓k : y ⇐ k} ≠ ∅, then m = ∨i=1lyi exists in L. By the definition of δS, δS(k) = m ≤ k. Since yi ⇐ k for i = 1,2,…, l, we have that m ⇐ k. By (SI≤) in m* ≥ m such that m ⇐ m* ⇐ k. It follows from m* ⇐ k = ∨P that m, m* ∈ P. Noticing that m ≤ m* and yi ≤ m ⇐ m* ∈ P, we have {y1,…, yl}⊆{y ∈ S∩↓m* : y ⇐ m*} which yields that ∨p∈PδS(p) ≥ δS(m*) ≥ δS(k). So, δS preserves suprema of prime ideals, and δ ∈ [L↪L].Let S, T ∈ Pfin(L) with S⊆T. It is easy to see that δS ≤ δT and D = {δS}S∈Pfin(L) is directed. Let FδS = {δS(x) : x ∈ L}. Then FδS is finite by the finiteness of S. So, δS is a finitely separating function. Since L is strongly semicontinuous, for each x ∈ L and δS ∈ D, δS(x) = ∨{y ∈ S∩↓x : y ⇐ x} ≤ x andSuppose that D = {δS}S∈Pfin(L)⊆[L↪L] is an approximate identity for L consisting of finitely separating functions, and L is a semi-FS domain.Therefore, Every strongly semicontinuous lattice is a semi-FS domain.


The aim of this paper is to investigate the general approximation structure, weak approximation operators, and Pawlak algebra in the framework of fuzzy lattice, lattice topology, and auxiliary ordering. First, we prove that the weak approximation operator space forms a complete distributive lattice. Then we study the properties of transitive closure of approximation operators and apply them to rough set theory. We also investigate molecule Pawlak algebra and obtain some related properties. The theory of rough sets was originally proposed by Pawlak  in 1982 The relationships between lattice theory and rough sets are another topic receiving much attention in recent years. Cattaneo and Ciucci  focus onThe remaining part of the paper is organized as follows. In this section, we introduce some basic definitions see , 19, 20), 2019, 2L, ≤) is said to be a lattice if inf{x, y} and sup{x, y}, denoted by ∧ and ∨, respectively, exist, for all x, y ∈ L. A lattice L is said to be complete if, for every A⊆L, ⋀α∈Aα and ⋁α∈Aα exist.A partially ordered set  be a complete lattice with the maximum element 1 and minimum element 0 and “≺” a binary relation on L. If the following conditions hold: for all α, β, μ, η, γ ∈ L,α≺β⇒α ≤ β,μ ≤ α≺β ≤ η⇒μ≺η,α≺γ, β≺γ⇒α∨β≺γ,α ∈ L, 0≺α≺1,∀then the relation “≺” is called an auxiliary ordering on L. If the relation “≺” satisfies conditions (i), (ii), and (iv), it is called a weak auxiliary ordering on L. The weak auxiliary ordering “≺” is called completely approximate if(iii)′αt≺β(t ∈ T)⇒⋁t∈Tαt≺β,then “≺” is called a strong auxiliary ordering on L.Let  be a completely distributive lattice. If the mapping c : L → L satisfies the following conditions:α, β ∈ L, if α ≤ β, then βc ≤ αc,reverse law: ∀α ∈ L, (αc)c = α,recovery law: ∀then  is called a fuzzy lattice.Let  be a fuzzy lattice and π⊆L. If the subset π satisfies the following conditions:π,0 ∉ α, β ∈ π, α∧β ≠ 0⇒α ≤ β  or  β ≤ α,α, β, γ ∈ π, α ≤ γ, β ≤ γ⇒α∧β ≠ 0,η = ∨{α∣α ∈ π, α ≤ η} for all η ∈ L,π0⊆π and π0 is linear order subset, then ∨π0 ∈ π,if then π is called a molecular set, and the element of π is called a molecule. The fuzzy lattice with molecule (L(π), ∨, ∧,c, 0,1) is called a molecular lattice.Let  be a fuzzy lattice and δ⊆L. If the subset δ satisfies the following conditions:δ,0, 1 ∈ αt ∈ δ(t ∈ T)⇒⋁t∈Tαt ∈ δ,α, β ∈ δ⇒α∧β ∈ δ,then  is called a lattice topology space.Let  be a fuzzy lattice. If the dual mappings  (P1)  (P2)  (P3)  (P4) then L. If α is called a definable element. If α is called a rough element.Let ( (P3)∗then L, and the system If (P3) is replaced by the following condition (P3)∗:L, ∨, ∧,c, 0,1) be a fuzzy lattice. Denote by APR(L) ) the set of all dual approximation operators  on L, respectively, and by σ0apr(L) the set of all definable elements in the Pawlak algebra Let  be a fuzzy lattice and  rougher than Let (APR(L)⊆APRw(L).Let α, β ∈ L with α ≤ β, we have α = α∧β, and soLet I, O ∈ APR(L) and Define two dual approximation operators APRW(L) according to the following result.It is possible to characterize L, ∨, ∧,c, 0,1) be a fuzzy lattice. Then (APRW(L), ≺) is a complete distributive lattice with maximum element I and minimum element O.Let  Consider  (P1)  (P2)  α ≤ β, then t ∈ T, implying that (P3) if  t ∈ T.(P4)  (2) Consider  ≺t ∈ T. In fact, let t ∈ T. Then t ∈ T and α ∈ L by definition. It follows that(3) Consider On the other hand, it is obvious that t ∈ T. Hence, Suppose that APRW(L), ≺) is a complete lattice. It is obvious that I is the maximum element and O is the minimum element and that (APRW(L), ≺) is distributive. This completes the proof.Summing up the above analysis,  be a fuzzy lattice and  a lattice topology space. Define the operators α ∈ L. Then the system Let ) is a zero-dimensional lattice topology space.Let T is an index set. Now, for any s ∈ T, we have αs ≤ ⋁t∈Tαt, and so t∈Tαt ≥ T is an index set. For any s ∈ T, we have αs ≥ ⋀t∈Tαt, and so ∩αc∈∩It is evident that L, σ0apr(L)) is a zero-dimensional lattice topology space.Summing up the above analysis,  be a fuzzy lattice. Define the operator “∘” on APR(L) as follows:k, define  transitive closure of Let  be a fuzzy lattice. ThenLet ((1) It is straightforward and omitted.α, β ∈ L, we have (P1)  (P2)  (P3) (P4)α ∈ L, ∀(2) It suffices to prove that (3) From the proof of L.If L, ∨, ∧,c, 0,1) be a fuzzy lattice and α ∈ L, one hasLet  be a fuzzy lattice and “≺” a strong auxiliary ordering on L. Define two operators L → L as follows:Let ((1)The strong auxiliary ordering ≺ implies that 1≺1 is true, and so we have(2)α, β, γ ∈ L be such that γ≺α∧β. Then γ ≤ γ≺α∧β ≤ α and γ ≤ γ≺α∧β ≤ β. By condition (ii) in γ≺α, γ≺β, and soLet ThereforeIt is obvious that condition (P1) holds. In what follows, we prove that conditions (P2)–(P4) are satisfied.(3)By the duality of approximation operators, it suffices to prove that On the other hand, by condition (iii) in Summing up the above statements, L.Let α, β ∈ Lbe such that α≺β. Then α ≤ β since (1) Let α, β, η, λ ∈ L be such that η ≤ α≺β ≤ λ. Then η≺λ.(2) Let α, β, λ ∈ L be such that α≺γ and β≺γ. Then α∨β≺γ.(4)α for all α ∈ L.It follows from (3) Let L.Summing up the above statements, “≺” is an auxiliary ordering on In this section, we focus on the approximate structure on fuzzy lattices. This structure can be regarded as abstract system of rough set.D, ≥) a directed set, and {S(d)}d∈D a molecular net and α ∈ π. If there exists d0 ∈ D such that d ≥ d0, then α is called a rough limit of S, denoted by S is denoted by limS.Let In the sequel, we provide two examples of molecular Pawlak algebra in topology space.U, ρ) be a metric space, L = Φ(U), and π = {{x}∣x ∈ U}, where Φ(U) denotes the set of all subsets of U. Definex0 be the apr-limit of the molecular sequence {xn}n∈N  ≤ 1 for n ≥ no. Obviously, Let  fuzzy power set on U, π = {{xλ}∣x ∈ U, λ ∈ , define two fuzzy sets FA and FA on U/R asA, B ∈ F(U),FA∪B = FA ∪ FB,FA∩B = FA∩FB,FA∩B⊆FA∩FB,FA∪B⊇FA ∪ FB,FA⊆FA.Moreover, for any A ∈ F(U), define x ∈ U. Then Let xλ is represented asd0 ∈ D such that xRyd and λd ≤ λ for d ≥ d0, where S(d) = yλdd and Xi is poly-point set.In the system U/R = {{x}∣x ∈ U}, namely, R = 1 if and only if x = y, then we havexλ and it is know that d0 ∈ D such that S(d) = xλ for d ≥ d0.If R-fuzzy-rough-convergent if and only if it is apr-convergent in the molecular Pawlak algebra A molecular net is called formulas  and 32)R-fuzzy-rThe following is now straightforward.R-fuzzy-rough-convergent classes satisfy the Moore-Smith conditions can induce a topology, called R-rough fuzzy topology, which is a nullity topology with square members.The L(π), ∨, ∧,c, 0,1) be a molecular lattice. Then any mapping h : π → L satisfying α ≤ h(α) can at least induce a molecular Pawlak algebra.Let (R2)(R3)(R4)(R5)(R6)Let U) denotes the set of all subsets of U.R⊆U × U be a binary relation on U. ThenR is reflexive if and only if R is symmetric if and only if R is transitive if and only if  the reflexive relation Let Let It is straightforward and omitted.(1)Rp defined asU;the binary relation (2)U is a finite universe or the cover U is finite, then Let  It follows from X⊆U, we have(2) For any U is finite and let X = {y1, y2,…, ym}⊆U. Then we haveSuppose that universe U is finite; that is, there exist xk ∈ U, k = 1,2,…, m such that U. Analogous to the above proof, to prove that X⊆U, U that Now suppose that the cover R the set of all similar relations on U. Then it is evident that R is infinite-intersection-closed. And the following result holds.In the sequel, denote by R be the set of all similar relations on U. Then we havex]t∈TRt⋂ = ⋂t∈T[x]Rt for {Rt}t∈T⊆R, where T is an index set;t∈TRt⋂ = {y∣ ∈ ⋂t∈TRt} = {y∣∀ t ∈ T,  ∈ Rt} = ⋂t∈T{y∣ ∈ Rt} = ⋂t∈T[x]Rt.(1) Consider R1⊆[x]R2. Now, let X⊆U. If x]R2⊆X, implying that [x]R1 = [x]R1∩R2 = [x]R1∩[x]R2⊆[x]R2⊆X; that is, (2) Let x, y) ∈ R1. It follows that X⊆U by the assumption, we know that x, y) ∈ R2. Therefore, R1⊆R2.Conversely, suppose that U be a finite set and R a similar relation on U. Then there exists an equivalent relation R.Let U as follows:U and R, which also implies that U.It follows from the proof of R* is an equivalent relation such that R⊆R*. By Suppose that In this paper, we have investigated the general approximation structure, weak approximation operators, and Pawlak algebra in the framework of fuzzy lattice, lattice topology, and auxiliary ordering. The relationships between the Pawlak approximation structures and these mathematic structures are established, and some related properties are presented. These works would provide a new direction for the study of rough set theory and information systems. As for future research, it will be interesting to continue the study of molecular Pawlak algebra and general partial approximation spaces.


We put forward a new general iterative process. We prove a convergence result as well as a stability result regarding this new iterative process for weak contraction operators. N, we denote the set of all positive integers. In this paper, we obtain results on the stability and strong convergence for a new iteration process  be complete metric space and T : E → E a self-map on E; and the set of fixed points of T in E is defined by FT = {p ∈ E : Tp = p}. Let {xn}n∈N ⊂ E be the sequence generated by an iteration involving T which is defined byx0 ∈ E is the initial point and f is a proper function. Suppose that sequence {xn}n∈N converges to a fixed point p of T. Let {yn}n∈N ⊂ E and setLet  and ∑n=0∞μn = ∞, then limn→∞sn = 0.If {T : C → E is said to be contraction if there is a fixed real number a ∈ ,d ≤ c[d + d].Let  such thatx, y ∈ E.A mapping E, ||·||) be Banach space. Assume that C⊆E is a nonempty closed convex subset and T : C → C is a mapping satisfying (F(T) ≠ ∅.Let  ≠ ∅.From Theorems From , we haveIn addition,Substituting  in 13),,13), we p − Tp|| = 0, we haven ∈ N.Since ||δ < 1,  ℘n ∈ , and ∑n=0∞℘n = ∞, we haveSince 0 < n→∞||xn − p|| = 0 yields xn → p ∈ F(T). This completes the proof of theorem.So limE, ||·||) be Banach space and T : E → E a self-mapping with fixed point p with respect to weak contraction condition in the sense of Berinde yn + ζnTyn. Suppose that limn→∞ϵn = 0. Then, we shall prove that limn→∞yn = p. Using contraction condition ,,19), we p − Tp|| = 0, we haveSince ||℘n(1 − δ) < 1 and using n→∞yn = p.Since 0 < 1 − n→∞yn = p, we show that limn→∞εn = 0 as follows:Conversely, letting limn→∞||yn − p|| = 0, it follows that limn→∞εn = 0. Therefore the iteration scheme is T stable.Since limT :  → , Tx = x/2, ℘n, ξn, ζn, ϑn = 0, n = 1,2,…, 15, and n ≥ 16.  It is easy to show that T is a weak contraction operator satisfying  − y(t)| denotes the space of all continuous functions. It is well known that C is a real Banach space with respect to ||·||∞ norm; more details can be found in  such that Kg > 0,there exists the following inequality:5)(CKg(b − t0) < 1, and according to a solution of problem ∩C1. The problem can be reconstituted as follows:2 problem -29) we  we Kg(b6) → C is defined by the following form:Assume that the following conditions are satisfied:Using weak-contraction mapping, we obtain the following.C1)–(C5) are performed. Then the problem ∩C1.We suppose that conditions ( problem -29) has hasC1)–T. The fixed point of T is shown via p such that Tp = p.We consider iterative process  for the t ∈ , it is clear that limn→∞xn = p. Therefore, letting t ∈ , we obtainFor the first part, that is, for Hence, we obtainBy continuing this way, we haveHence, we obtainSubstituting  into 3434, we obKg(b − t0)) < 1, we haveSince (1 − 2ζn(1 − 2Kg(b − t0)) = μn < 1 and ||xn−p||∞ = sn, and then the conditions of n→∞||xn−p||∞ = 0.We take


Biliary atresia (BA) is a human infant disease with inflammatory fibrous obstructions in the bile ducts and is the most common cause for pediatric liver transplantation. In contrast, the sea lamprey undergoes developmental BA with transient cholestasis and fibrosis during metamorphosis, but emerges as a fecund adult. Therefore, sea lamprey liver metamorphosis may serve as an etiological model for human BA and provide pivotal information for hepatobiliary transformation and possible therapeutics.hmgcr), and bile acid biosynthesis, cyp7a1. Injection of hsp90 siRNA for 4 days altered gene expressions of met, hmgcr, cyp27a1, and slc10a1. Bile acid concentrations were increased while bile duct and gall bladder degeneration was facilitated and synchronized after hsp90 siRNA injection.We hypothesized that liver metamorphosis in sea lamprey is due to transcriptional reprogramming that dictates cellular remodeling during metamorphosis. We determined global gene expressions in liver at several metamorphic landmark stages by integrating mRNA-Seq and gene ontology analyses, and validated the results with real-time quantitative PCR, histological and immunohistochemical staining. These analyses revealed that gene expressions of protein folding chaperones, membrane transporters and extracellular matrices were altered and shifted during liver metamorphosis. HSP90, important in protein folding and invertebrate metamorphosis, was identified as a candidate key factor during liver metamorphosis in sea lamprey. Blocking HSP90 with geldanamycin facilitated liver metamorphosis and decreased the gene expressions of the rate limiting enzyme for cholesterol biosynthesis, HMGCoA reductase  contains supplementary material, which is available to authorized users. Manduca sexta and the single burst of triiodothyronine (T3) for metamorphosis of Xenopus larvae to the froglet stage 3keto-petromyzonol sulfate; Bridge Organic Inc., Vicksburg, MI, USA) was added to the whole liver (24.9 ± 1.7 mg). Liver tissues were homogenized and incubated in a shaker with 70 rpm at room temperature overnight. The homogenized tissues were then centrifuged at 13,000 × g for 10 min. The supernatant was transferred to a new tube, freeze-dried overnight, and stored at −20 °C until analyses. Samples were reconstituted in 1 mL of methanol:water (1:1) and placed in an autosampler for LC-MS/MS analysis.Bile acid analyses followed the method developed by Li et al.  with minRTQ-PCR was performed using the TaqMan MGB or SYBR Green system (Life Technologies) as described previously . Gene se40S ribosomal protein: 5′acctacgcaggaacagctatgaccATCTCGAGCAGCTGAAgctccaatgtggtggaattcgtcg3′. 60S ribosomal protein: 5′cgcatccgcgcaatgaAGACCATCCAGAGCAAtcagatcgtggacatacccgac3′. Bsep: 5′gtgtctcaggagccggtgttgTTCGACTGCAGCATTGccgacaacattcgctacggtgcc3′. Slc10a1: 5′ctgtcccggagggaacctctccaacgtgttcgcgctggcgcTCGACGGAGACATGAAcctcagcatcctcatgaccacgtg3′. Col2a1: 5′ttcacttactctgtgctggaggatgggTGCACTACGCACACCGgcgtgtggggcaagacggtgatcgagtacagg3′For each gene, the following information is shown: gene name and synthetic oligo used as the standard for RTQ-PCR. 5′ and 3′ primer sequences are underlined. (Note: the 3′ primer is complementary to the sequence shown). TaqMan MGB probes are shown in uppercase. Hgf : 5′cggcattgcttggaaggaaaaggggaaaattaccgcggccttgtgaacaaaacagccaccgac3′.Hsp90:cgtgctgcacctgaaggaggaCCAATCTGAGTACCTGGAGgagaagcgcatcaaagacatcg3′. Met: 5′ctgcagacgcagaggttcaccACCAAGTCGGATGTGTGgtcgtttggcgttctgctg3′.5′


Meanwhile, on the basis of functional analysis skill, the existence of the smooth solution and the uniform validity of the asymptotic expansion are proved.We study a kind of vector singular perturbed delay-differential equations. By using the methods of boundary function and fractional steps, we construct the formula of asymptotic expansion and confirm the interior layer at  Singular perturbed differential equations are often used as mathematical models describing processes in biological sciences and physics, such as genetic engineering and the El Nino phenomenon of atmospheric physics . In ordeIn addition, in the study of population models and propagation of epidemic virus, we sometimes require the construction of models. The models are often expressed by singular perturbed delay-differential equations. We can get the equilibrium points of singular perturbed delay-differential equations and confirm the laws of processes. Therefore, using the research methods and theoretical results of singular perturbed delay-differential problems to solve natural and social processes is essential.In recent years, more and more attention was paid to the study of singular perturbed delay-differential problems , especiaIn this paper, we will discuss the interior layer for a class of nonlinear singularly perturbed differential-difference equations and construct its asymptotic expansion formula. Then, the existence of the smooth interior layer solution and the uniform validity of the asymptotic expansion are proved. The results of this paper are new and complement the previous ones.Fi   are sufficiently smooth on the domain D = {(μ y′(t), y(t), t) | ||μ y′(t)|| ≤ l1, ||y(t)|| ≤ l2, 0 ≤ t ≤ 2σ}, and l1, l2 are given positive real numbers. The restriction on 2σ will not influence the essence of the problems.We consider the following nonlinear singularly perturbed differential-difference equationsμ = 0; then we can obtain the degenerate equations   μ = 0; tquations 3)F = T and using the method of boundary function [σ]:t = 0, andt = σ, and limτ0→+∞Πkx(τ0) = 0, limτ→−∞Qk(−)x(τ) = 0 hold. The system  are undetermined n-dimensional vector functions.In , where Reλi(t) < 0, i = 1,…, n, Reλi(t) > 0, i = n + 1,…, 2n, and I is the n × n identity matrix.Suppose that the characteristic equation of the systems  given byBy (H1) and , y¯k(t),0x(τ0), we haveFor Π we have14)LetLet0x, 0) on the phase plane G1 of vector function Π0y.(H3)n-dimensional stable manifold is Π0z = ΦL(Π0y), and α(0) − φ(0) ∈ G1.Suppose that the By (H2), there exists an 0y, Π0z, which are both satisfied with exponential decay.By (H2) and (H3), systems  and 14)14) have kx, we have0z, φ(0) + Π0y, α(−σ), 0) and Gk(τ0) is a known vector function about Π1y(τ0),…, Πk−1y(τ0).For ΠIn fact, the homogeneous system of 18)dΠkyation of . Thus, idΠkyationtituting , we haveky = Φ1(τ0)C be the general solution of C be the general solution of kx(τ0) is completely determined. Obviously, Πkx(τ0) decays exponentially as τ → ∞.Let Πution of , under tution of , we obtaution of  as21) irtue of  and 17)ky = Φ1(lx(τ0) satisfy the following inequality:C1, k1 are all positive constants.Under conditions (H1)–(H3), the boundary functions ΠQ0(−)xφ(σ), 0) is a hyperbolical singular point on the plane n-dimensional stable manifold passing through (φ(σ), 0).(H4)n-dimensional stable manifold is G2 is a domain of Suppose that this We first consider the system of 0(−)xdQ0(−)ydτQ0(−)x can be completely determined, but it contains the unknown vector p0.By (H4) and , Q0(−)Qk(−)x is determined by the following system:Hk(−)(τ) is a known vector; elements of matrix Q0(−)z, φ(σ) + Q0(−)y, α(0), σ). Because Qk(−)x(τ) can be completely determined. We can easily obtain the exponential decay of Qk(−)x(τ). system:dQk(−)ydτQlx(τ) satisfy the following inequality:C2, k2 are positive constants.Under the condition (H4), the boundary functions σ, 2σ], let μ y′ = z; the system (x(+) = T and using the method of boundary function :t = σ, andt = 2σ, and limτ→+∞Qk(+)x(τ) = 0, limτ∗→−∞Rkx(τ∗) = 0 hold.In , where Reλi(t) < 0, i = 1,…, 2n, Reλi(t) > 0, i = 2n + 1,…, 4n.Suppose that the characteristic equations of the systems  have 4n For the zeroth approximation of the left boundary layer  we have(40)wheΠ0y and  and 14)Q0(+)xn-dimensional stable manifold passing the equilibrium point of (ψ(σ), 0, φ(0), 0).(H7)n-dimensional stable manifold isSuppose that this 2where (+) = (Φ1(+), Φ2(+))T, and (α(0) − φ(0), p0 − ψ(σ)) ∈ G3.By condition (H6), there exists a 2Q0(+)x(τ) can be determined.By (H7) and , 40), Q, Q40), QQk(+)x(τ) satisfies the following boundary value problem:2(τ) and Ψ2(τ) is similar to that of Φ and Ψ, respectively, but Qk+x(τ) contains the unknown vector functions p1,…, pk.Ql(+)x(τ) satisfy the following inequality:C3, k3 are all positive constants.Under conditions (H5)–(H7), the boundary functions R0x(τ∗), we haveFor n-dimensional stable manifold, which is in some region G4 of vector function R0y.(H8)n-dimensional stable manifold is R0z = ΦR(R0y), and Suppose that the Consider the first approximate system of (48), the system  has a soRkx, we haveGk(τ0) is a known vector function.For Rkx(τ∗) is treated in the same way as Πkx(τ0) and is omitted here. Obviously, Rkx(τ∗) satisfy the following Lemma.The determination of Rlx(τ∗) satisfy the following inequality:C4, k4 are all positive constants.Under the condition (H8), the boundary functions Qk(−)x(τ), Qk(+)x(τ) contain the unknown vector functions p0,…, pk. Next, we will use the continuous conditions to determine them.Obviously, t = σ, the solution x of equivalent system needs to satisfyp0. (H4), (H7), and ((H9)Suppose that  has a soAs the solution of the original problem is continuous at  satisfy(52)Firpk can be completely determined.Next, bringing  and 39)39) into Nk(pk)=y¯Letμ0 > 0, c > 0, such that, for 0 < μ ≤ μ0, the solution x of the systems –(H9) and –(H9) andy = T. For 0 ≤ t ≤ 1, the degenerate solution of (0y(τ0), R0y(τ∗), Q0(−)y(τ), and Q0(+)y(τ) are given by the following systems, respectively,0y = ((1/2)eτ0−, (1/2)eτ0−)T, R0y = ((1/2)eτ∗, (1/2)eτ∗)T, Q0(−)y(τ) = T, Q0(+)y(τ) = (p01eτ− + (1/2)eτ− + (1/4)τeτ−, p02eτ− + (1/2)eτ− + (1/4)τeτ−)T. By the smooth connection (dQ0(−)y/dτ)|τ=0 = (dQ0(+)y/dτ)|τ=0, we have H(p0) = (2p01 + (1/2) − (1/4), 2p02 + (1/2) − (1/4))T = 0. Obviously, p0 = T and |H(p0)/d p0|p0=−1/8 = 4 ≠ 0. Thus, we obtain that the zero order asymptotic solution of (Let us consider the systemsμ2y′′=y(tμ2y′′=y(tWe can see that the zero order asymptotic solution is close to the reduced solution.n-dimensional singularly perturbed differential equations with time delay. Under some assumptions, we obtain the asymptotic solution of the system (Using the boundary layer function method, we consider a class of e system . In compe system  in this e system  can be u


Multiscale information system is a new knowledge representation system for expressing the knowledge with different levels of granulations. In this paper, by considering the unknown values, which can be seen everywhere in real world applications, the incomplete multiscale information system is firstly investigated. The descriptor technique is employed to construct rough sets at different scales for analyzing the hierarchically structured data. The problem of unravelling decision rules at different scales is also addressed. Finally, the reduct descriptors are formulated to simplify decision rules, which can be derived from different scales. Some numerical examples are employed to substantiate the conceptual arguments. As one of the important mathematical tools for granular computing , 2, the Pawlak's rough set was proposed on the basis of an indiscernibility relation, which can generate a granulation space on the universe of discourse. Such granulation space is actually a partition since the indiscernibility relation is an equivalence relation. With respect to different requirements, a variety of the expanded rough sets models have been proposed. For example, the tolerance relation –7, similObviously, the above rough sets are constructed on the basis of one and only one set of the information granules, which can be generated from a binary relation or a covering. From this point of view, we may call these rough sets the single-granulation rough sets. In single-granulation rough sets, a partition is a granulation space, a binary neighborhood system induced by a binary relation is a granulation space, and a covering is also a granulation space. Nevertheless, it should be noticed that, in , the autPresently, the development of multigranulation rough sets approaches is progressing rapidly. For instance, Qian et al. classified their multigranulation rough sets into two categories: one is the optimistic case and the other is the pessimistic case. Yang et al.  generaliIt must be noticed that the multiscale information system is a very important knowledge representation approach; it can help us to analyze data from the viewpoint of different levels of granulations. For example, maps can be hierarchically organized into different scales, from large to small and vice versa. The smaller the scale, the finer the partition that can be obtained; conversely, the bigger the scale, the coarser the partition that can be obtained. However, what Wu and Leung investigated is complete multiscale information systems. Gore, in his influential book Earth in the Balance , notes tIn the next section, we first introduce some basic notions related to Pawlak's rough set and multiscale information system. The incomplete multiscale information system and rule induction problem are explored in S = , in whichU is a nonempty finite set of objects; it is called the universe;a ∈ AT, Va is the domain of attribute a.AT is a nonempty finite set of attributes, such that, ∀Formally, an information system  can be cx ∈ U, let us denote by a(x) the value that x holds on a (a ∈ AT). For an information system S, one then can describe the relationship between objects through their attributes values. With respect to a subset of attributes such that A⊆AT, an indiscernibility relation A = {y ∈ U :  ∈ IND(A)} is the A-equivalence class containing x. The pair X with respect to the set of attributes A.The relation IND, whereU = {x1,…, xn} is a nonempty, finite set of objects called the universe of discourse;a1,…, am} is a nonempty, finite set of attributes, and each aj ∈ A is a multiscale attribute; that is, for the same object in U, attribute aj can take on different values at different scales.AT = {A multiscale information system  is a tupI of levels of granulations. Therefore, a multiscale information system can be rewritten as a system such that , where ajk : U → Vajk is a surjective function and Vajk is the domain of the kth scale attribute ajk. For the set of the kth scale attributes ATk = {a1k, a2k,…, amk}, one then denotes an equivalence relation such thatIn multiscale information system, Wu and Leung  assumed RkAT is denoted by U/RkAT such thatx]RkAT is the equivalence class that includes x at scale k.The partition induced by I-scales, such hierarchical structure can be expressed by the inclusion relation among m equivalence relations; that is,It should be noticed that, in multiscale information system, since different scales represent different levels of granulations and then there is a hierarchical structure among S =  =  is referred to as a multiscale decision system, where  =  is a multiscale information system and d ∉ {ajk : k = 1,2,…, I, j = 1,2,…, m}, d : U → Vd is a special attribute called decision. Such multiscale decision system can be decomposed into I decision systems Sk =  = , k = 1,2,…, I, with the same decision d. Each decomposed decision system Sk represents information on a special level of granulation, that is, scale.A system S1 =  = , is consistent; otherwise it is referred to as inconsistent. Following such work, we will propose a more generalized definition of the concept of consistency in multiscale decision system.In , Wu and S =  =  be a multiscale decision system; then S is referred to as k-scale consistent if and only if k-scale decision system, that is, Sk =  = , is consistent; otherwise S is referred to as k-scale inconsistent.Let k = 1, 1-scale consistency is same to what have been proposed by Wu and Leung. In such case, we have R1AT⊆Rd}{. Moreover, since R1AT⊆R2AT⊆⋯⊆RIAT, then we do not always have R2AT⊆Rd}{,…, RIAT⊆Rd}{; it follows that, in I-scale levels of granulations, we need only at least one of the levels of granulations to satisfy the condition of consistent; then this type of the consistent, that is, 1-scale consistent, can be referred to as the optimistic consistent in multiscale decision system.By I-scale consistent. In such case, we have RIAT⊆Rd}{. Moreover, since R1AT⊆R2AT⊆⋯⊆RIAT, then we also have R2AT⊆Rd}{,…, RIAT⊆Rd}{; it follows that, in I-scale levels of granulations, we need all the levels of granulations to satisfy the condition of consistent; then this type of the consistent, that is, I-scale consistent, can be referred to as the pessimistic consistent in multiscale decision system.On the other hand, let us consider the k-scale consistent in multiscale decision system.From discussions above, it is not difficult to observe that the optimistic and pessimistic consistent are all special cases of S =  =  be a multiscale decision system; if S is k-scale consistent, then S is also k′-scale consistent, where k′ ∈ {1,2,…, I} and k′ ≤ k.Let The above proposition tells us that if a multiscale decision system is consistent in a given scale, then such multiscale decision system is also consistent in the scale, which is smaller than the given scale.It should be noticed that the inverse of S =  =  be a multiscale decision system; then ∀X⊆U, the k-scale lower and upper approximations of X are denoted by Let k-scale rough set of X in multiscale decision system. Since the k-scale rough set shown in k-scale rough set. We omit these properties in this paper.The pair x]R1AT⊆X. Since R1AT⊆R2AT⊆⋯⊆RmAT, then we do not have that [x]R2AT⊆X,…, [x]RIAT⊆X; it follows that, in I levels of granulations, we need only at least one of the levels of granulations to satisfy the inclusion condition between equivalence class and target concept; such explanation is compatible with that in Qian et al.'s optimistic multigranulation lower approximation [x]R1AT∩X ≠ ∅. Since R1AT⊆R2AT⊆⋯⊆RmAT, then we always have [x]R2AT∩X ≠ ∅ ⋯ [x]RIAT∩X ≠ ∅; it follows that, in I levels of granulations, we need all the levels of granulations to satisfy the intersection condition between equivalence class and target concept; such explanation is also compatible with that in Qian et al.'s optimistic multigranulation upper approximation. From this point of view, 1-scale rough set is also referred to as the optimistic multiscale rough set in multiscale decision system.By ximation , 33. MorI-scale lower approximation; we have [x]RIAT⊆X. Since R1AT⊆R2AT⊆⋯⊆RIAT, then we also have [x]R1AT⊆X,…, [x]RI−1AT⊆X; it follows that, in I levels of granulations, we need all the levels of granulations to satisfy the inclusion condition between equivalence class and target concept; such explanation is compatible with that in Qian et al.'s pessimistic multigranulation lower approximation RIAT∩X ≠ ∅. Since R1AT⊆R2AT⊆⋯⊆RmAT, then we do not always have [x]R1AT∩X ≠ ∅ ⋯ [x]RI−1AT∩X ≠ ∅; it follows that, in I levels of granulations, we need at least one of the levels of granulations to satisfy the intersection condition between equivalence class and target concept; such explanation is also compatible with that in Qian et al.'s pessimistic multigranulation upper approximation. From this point of view, I-scale rough set is also referred to as the pessimistic multiscale rough set in multiscale decision system.On the other hand, let us consider ximation . MoreoveS =  =  be a multiscale decision system; then, ∀X⊆U, one hasLet k-scale lower approximations become smaller while the k-scale upper approximations become bigger. In other words, we can obtain a string of rough sets through different levels of granulations in multiscale decision system.The above proposition tells us that, with the monotonous increasing of levels of granulations, the k-scale rough approximation is defined byX| denotes the cardinal number of set X. Obviously, 0 ≤ αk(X) ≤ 1 holds.The accuracy of S =  =  be a multiscale decision system; then, ∀X⊆U, one hasLet S =  in this paper. Given an incomplete multiscale information system S, if ajk(x) = ∗, then we say that the value of object x is unknown on the attribute aj in terms of the k-scale. Moreover, we assume that the unknown value ∗ can be compared with any other values in the domain of the corresponding attributes [An incomplete multiscale information system is still denoted by tributes , 5. TherS, if Ak⊆ATk, then any attribute-value pair  is called an Ak-atomic property where ajk ∈ Ak and vjk ∈ Vajk. Any Ak-atomic property or conjunction of different Ak-atomic properties is called the Ak-descriptor. If  is the atomic property occurring in Ak-descriptor tk, we simply say that  ∈ tk. Obviously, tk is constructed at scale k; it can also be called a k-scale descriptor.In the discussion to follow, the symbols ∧ and ∨ denote the logical connectives “and” (conjunction) and “or” (disjunction), respectively . Given atk be an Ak-descriptor; if, for all  ∈ tk, we have  ∈ sk, that is, tk is constructed from a subset of atomic properties occurring in sk, then we say tk is coarser than sk or sk is finer than tk and is denoted by tk⪰sk or sk⪯tk. If tk is constructed from a proper subset of atomic properties occurring in sk, then we say tk is properly coarser than sk and is denoted by tk≻sk or sk≺tk.Let tk be an Ak-descriptor; the attributes set occurring in tk is denoted by Ak(tk). Moreover, if tk is an Ak-descriptor and Ak(tk) = Ak, then tk is called full Ak-descriptor. Here, suppose that ∧ajk∈Ak is a full Ak-descriptor; we denoteajk∈Ak|| is referred to as the support of ∧ajk∈Ak.Let Here, let us denoteU could be partitioned into several subsets that may overlap at scale k, and the result is denoted by U/Ak such thatBy the descriptor technique, the universe k, k′ ∈ {1,2,…, I} and k′ ≤ k, U/ATk′⊑U/ATk means that the following two conditions hold: tk|| ∈ U/ATk, there must be ||tk′|| ∈ U/ATk′ such that ||tk′||⊆||tk||;∀||tk′|| ∈ U/ATk′, there must be ||tk|| ∈ U/ATk such that ||tk′||⊆||tk||.∀||In complete multiscale decision system, the hierarchical structure is represented by a partial relation among different equivalence relations or among different partitions. In incomplete multiscale decision system, since, for each level of granulation, we can obtain a family of the supports of the descriptors, which form coverings on the universe of discourse, and then we can use those supports of the descriptors to represent the hierarchical structure such thatU/ATk′ and U/ATk are all partitions, then condition (1) implies condition (2) or condition (2) implies condition (1); it follows that only one of the above conditions is needed. However, since, in incomplete multiscale information system, U/ATk′ and U/ATk may be the coverings instead of the partitions, then the above two conditions are needed simultaneously.It should be noticed that if S =  = , ∀k, k′ ∈ {1,2,…, I} and k′ ≤ k, a surjective function can be defined asThe above two conditions for expressing the hierarchical structure in incomplete multiscale information system are consistent with the basic thinking of surjective function. In other words, in an incomplete multiscale information system S =  = , where U = {x1, x2,…, x20}, AT = {a1, a2, a3, a4} is the set of the condition attributes, and d is the decision attribute. The system has three levels of granulations, where “G,” “F,” “B,” “L,” “M,” “H,” “Y,” and “N” stand for, respectively, “good,” “fair,” “bad,” “low,” “medium,” “high,” “yes,” and “no.”1, AT2, and AT3 descriptors are shown in Tables By the descriptor technique we mentioned above, it is not difficult to obtain the descriptors and their supports in each level of granulation. The results of full ATS =  =  be an incomplete multiscale decision system; then S is referred to as k-scale consistent if and only if k-scale decision system Sk =  =  is consistent; that is, U/ATk⊑U/IND({d}); otherwise S is referred to as k-scale inconsistent.Let I-scale consistent incomplete multiscale decision system is referred to as the pessimistic consistent incomplete multiscale decision system.Similar to the complete case, the 1-scale consistent incomplete multiscale decision system is referred to as the optimistic consistent incomplete multiscale decision system while the S =  =  be an incomplete multiscale decision system; if S is k-scale consistent, then S is also k′-scale consistent, where k′ ∈ {1,2,…, I} and k′ ≤ k.Let It should be noticed that the inverse of S =  =  be an incomplete multiscale decision system; then, ∀X⊆U, the k-scale lower and upper approximations of X are denoted by k-scale boundary region of X is thenLet X1 = {x1, x2, x3, x13, x14, x15}, X2 = {x4, x5, x6, x7, x8, x9, x10, x11, x12}, X3 = {x16, x17, x18, x19, x20}, then by  1-scale lower and upper approximations:(1)t31||, ||t51||, ||t71||,||t81||, ||t91||, ||t101||, ||t141||, ||t151||, ||t191||, ||t231||, ||t261||},t41||, ||t61||, ||t111||, ||t121||,||t131||, ||t161||, ||t171||, ||t181||, ||t201||, ||t211||, ||t221||, ||t251||},t41||, ||t61||, ||t91||, ||t111||, ||t121||, ||t131||, ||t141||, ||t161||, ||t171||, ||t181||, ||t191||, ||t201||, ||t211||, ||t221||, ||t251||}, 2-scale lower and upper approximations:(2) 3-scale lower and upper approximations:(3)Take, for instance, By the above computations, we can see that S =  =  be an incomplete multiscale decision system; then, ∀X⊆U, one hasLet The results in S =  =  be an incomplete multiscale decision system; then a k-scale decision rule is represented bytk ∈ FDES(ATk), s = , w ∈ Vd, and tk and s are, respectively, called the condition and decision parts of the rule rk.The end result of rough set is a representation of the information contained in the data system considered in terms of “if… then…” decision rules , 44. Sink-scale decision rule rk : tk → s, we associate a quantitative measure, called the certainty, of rk and it is defined byFor each k-scale decision rule rk : tk → s is referred to as certain if and only if Cer(rk) = 1; a k-scale decision rule rk : tk → s is referred to as possible if and only if 0 < Cer(rk) < 1. Similar to the traditional rough set approach, the certain decision rules are supported by the descriptors in lower approximation while the possible rules are supported by the descriptors in boundary region. In other words, if tk → s is a certain decision rule; if ||tk|| ∈ BNDESkAT||), then tk → s is a possible decision rule.A S =  = ,U/AT1⊑U/AT2⊑⋯⊑U/ATI, then, ∀||tk|| ∈ U/ATk, there must be ||tk′|| ∈ U/ATk′ such that ||tk′||⊆||tk||; it tells us that if we have a certain k-scale decision rule such that rk : tk → s, then we can also obtain a certain k′-scale decision rule; that is, rk′ : tk′ → s;since U/AT1⊑U/AT2⊑⋯⊑U/ATI, then, ∀||tk′|| ∈ U/ATk′, there must be ||tk|| ∈ U/ATk such that ||tk′||⊆||tk||; it tells us that if we have a possible k′-scale decision rule such that rk′ : tk′ → s, then we can also obtain a possible k-scale decision rule; that is, rk : tk → s.since In an incomplete multiscale decision system  1-scale decision rules:(1)(a) 1-scale certain decision rules:r11 : a11(x) = 1∧a21(x) = 3∧a31(x) = 5∧a41(x) = 2 → d(x) = 1 // supported by t31,r21 : a11(x) = 3∧a21(x) = 3∧a31(x) = 2∧a41(x) = 2 → d(x) = 1 // supported by t81,r31 : a11(x) = 4∧a21(x) = 3∧a31(x) = 5∧a41(x) = 2 → d(x) = 1 // supported by t101,r41 : a11(x) = 5∧a21(x) = 3∧a31(x) = 5∧a41(x) = 2 → d(x) = 1 // supported by t151,r51 : a11(x) = 7∧a21(x) = 3∧a31(x) = 5∧a41(x) = 2 → d(x) = 1 // supported by t231,r61 : a11(x) = 8∧a21(x) = 3∧a31(x) = 5∧a41(x) = 2 → d(x) = 1 // supported by t261,r71 : a11(x) = 1∧a21(x) = 1∧a31(x) = 3∧a41(x) = 1 → d(x) = 2 // supported by t21,r81 : a11(x) = 2∧a21(x) = 1∧a31(x) = 3∧a41(x) = 1 → d(x) = 2 // supported by t41,r91 : a11(x) = 3∧a21(x) = 1∧a31(x) = 3∧a41(x) = 1 → d(x) = 2 // supported by t61,r101 : a11(x) = 4∧a21(x) = 4∧a31(x) = 1∧a41(x) = 1 → d(x) = 2 // supported by t111,r111 : a11(x) = 4∧a21(x) = 4∧a31(x) = 3∧a41(x) = 3 → d(x) = 2 // supported by t121r121 : a11(x) = 5∧a21(x) = 1∧a31(x) = 3∧a41(x) = 1 → d(x) = 2 // supported by t131,r131 : a11(x) = 5∧a21(x) = 5∧a31(x) = 4∧a41(x) = 1 → d(x) = 2 // supported by t161,r141 : a11(x) = 5∧a21(x) = 5∧a31(x) = 4∧a41(x) = 4 → d(x) = 2 // supported by t171,r151 : a11(x) = 6∧a21(x) = 1∧a31(x) = 3∧a41(x) = 1 → d(x) = 2 // supported by t181,r161 : a11(x) = 6∧a21(x) = 6∧a31(x) = 5∧a41(x) = 4 → d(x) = 2 // supported by t201,r171 : a11(x) = 6∧a21(x) = 6∧a31(x) = 5∧a41(x) = 2 → d(x) = 2 // supported by t211,r181 : a11(x) = 7∧a21(x) = 1∧a31(x) = 3∧a41(x) = 1 → d(x) = 2 // supported by t221,r191 : a11(x) = 8∧a21(x) = 1∧a31(x) = 3∧a41(x) = 1 → d(x) = 2 // supported by t251,r201 : a11(x) = 7∧a21(x) = 7∧a31(x) = 6∧a41(x) = 5 → d(x) = 3 // supported by t241,r211 : a11(x) = 8∧a21(x) = 7∧a31(x) = 6∧a41(x) = 6 → d(x) = 3 // supported by t271;(b) 1-scale possible decision rules:r1′1 : a11(x) = 1∧a21(x) = 1∧a31(x) = 1∧a41(x) = 1 → d(x) = 1, Cer (r1′1) = 0.5 // supported by t11,r2′1 : a11(x) = 2∧a21(x) = 2∧a31(x) = 1∧a41(x) = 1 → d(x) = 1, Cer (r2′1) = 0.5 // supported by t51,r3′1 : a11(x) = 3∧a21(x) = 3∧a31(x) = 2∧a41(x) = 2 → d(x) = 1, Cer (r3′1) = 0.5 // supported by t71,r4′1 : a11(x) = 4∧a21(x) = 1∧a31(x) = 3∧a41(x) = 1 → d(x) = 1, Cer (r4′1) = 0.5 // supported by t91,r5′1 : a11(x) = 5∧a21(x) = 2∧a31(x) = 4∧a41(x) = 1 → d(x) = 1, Cer (r5′1) = 0.5 // supported by t141,r6′1 : a11(x) = 6∧a21(x) = 3∧a31(x) = 5∧a41(x) = 2 → d(x) = 1, Cer (r6′1) = 0.5 // supported by t191,r7′1 : a11(x) = 4∧a21(x) = 1∧a31(x) = 3∧a41(x) = 1 → d(x) = 2, Cer (r7′1) = 0.5 // supported by t91,r8′1 : a11(x) = 5∧a21(x) = 2∧a31(x) = 4∧a41(x) = 1 → d(x) = 2, Cer (r8′1) = 0.5 // supported by t141,r9′1 : a11(x) = 6∧a21(x) = 3∧a31(x) = 5∧a41(x) = 2 → d(x) = 2, Cer (r9′1) = 0.5 // supported by t191,r10′1 : a11(x) = 1∧a21(x) = 1∧a31(x) = 1∧a41(x) = 1 → d(x) = 3, Cer (r10′1) = 0.5 // supported by t11,r11′1 : a11(x) = 2∧a21(x) = 2∧a31(x) = 1∧a41(x) = 1 → d(x) = 3, Cer (r11′1) = 0.5 // supported by t51,r12′1 : a11(x) = 3∧a21(x) = 3∧a31(x) = 2∧a41(x) = 2 → d(x) = 1, Cer (r12′1) = 0.5 // supported by t71; 2-scale decision rules:(2)(a) 2-scale certain decision rules:r12 : a12(x) = F∧a22(x) = G∧a32(x) = M∧a42(x) = L → d(x) = 1 // supported by t72,r22 : a12(x) = F∧a22(x) = G∧a32(x) = L∧a42(x) = L → d(x) = 1 // supported by t82,r32 : a12(x) = F∧a22(x) = G∧a32(x) = H∧a42(x) = L → d(x) = 1 // supported by t92,r42 : a12(x) = F∧a22(x) = G∧a32(x) = M∧a42(x) = M → d(x) = 2 // supported by t22,r52 : a12(x) = F∧a22(x) = F∧a32(x) = M∧a42(x) = M → d(x) = 2 // supported by t42,r62 : a12(x) = F∧a22(x) = F∧a32(x) = M∧a42(x) = L → d(x) = 2 // supported by t52,r72 : a12(x) = F∧a22(x) = B∧a32(x) = M∧a42(x) = M → d(x) = 2 // supported by t62,r82 : a12(x) = B∧a22(x) = B∧a32(x) = H∧a42(x) = H → d(x) = 3 // supported by t102,r92 : a12(x) = B∧a22(x) = F∧a32(x) = H∧a42(x) = H → d(x) = 3 // supported by t112,r102 : a12(x) = B∧a22(x) = G∧a32(x) = H∧a42(x) = H → d(x) = 3 // supported by t122;(b) 2-scale possible decision rules:r1′2 : a12(x) = G∧a22(x) = G∧a32(x) = L∧a42(x) = L → d(x) = 1, Cer (r1′2) = 0.5 // supported by t12,r2′2 : a12(x) = F∧a22(x) = G∧a32(x) = M∧a42(x) = L → d(x) = 1, Cer (r2′2) = 0.5 // supported by t32,r3′2 : a12(x) = F∧a22(x) = G∧a32(x) = M∧a42(x) = L → d(x) = 2, Cer (r3′2) = 0.5 // supported by t32,r4′2 : a12(x) = G∧a22(x) = G∧a32(x) = L∧a42(x) = L → d(x) = 3, Cer (r4′2) = 0.5 // supported by t12; 3-scale decision rules:(3)(a) 3-scale certain decision rules:r13 : a13(x) = N∧a23(x) = N∧a33(x) = N∧a43(x) = Y → d(x) = 2 // supported by t43;(b) 3-scale possible decision rules:r1′3 : a13(x) = Y∧a23(x) = Y∧a33(x) = Y∧a43(x) = Y → d(x) = 1, Cer (r1′3) = 0.5 // supported by t13,r2′3 : a13(x) = N∧a23(x) = Y∧a33(x) = N∧a43(x) = Y → d(x) = 1, Cer (r2′3) = 0.5 // supported by t23,r3′3 : a13(x) = N∧a23(x) = Y∧a33(x) = N∧a43(x) = Y → d(x) = 2, Cer (r3′3) = 0.5 // supported by t23,r4′3 : a13(x) = N∧a23(x) = N∧a33(x) = N∧a43(x) = N → d(x) = 2, Cer (r4′3) = 0.667 // supported by t33,r5′3 : a13(x) = Y∧a23(x) = Y∧a33(x) = Y∧a43(x) = Y → d(x) = 3, Cer (r5′3) = 0.5 // supported by t13,r6′3 : a13(x) = N∧a23(x) = N∧a33(x) = N∧a43(x) = N → d(x) = 3, Cer (r6′3) = 0.333 // supported by t33.Following the results of approximations we obtained in S =  =  be an incomplete multiscale decision system, tk ∈ FDES(ATk) is a full ATk-descriptor, tk′ ∈ DES(ATk) is an ATk-descriptor, and tk′⪰tk; then tk′ is referred to as the reduct descriptor of tk if and only if the following two conditions hold: tk′|| = ||tk||;||tk′′|| ≠ ||tk|| for each tk′′≻tk′.||Let tk is a conjunction of the atomic properties in tk, which preserves the support of tk. The reduct descriptor allows us to classify objects with the smallest number of required atomic properties.By tk, let us definetk discernibility matrix.To compute the reduct descriptor of S =  =  be an incomplete multiscale decision system in which tk ∈ FDES(ATk), tk′ ∈ DES(ATk), and tk′⪰tk; then tk|| = ||tk′||⇔AT(tk′)∩DIS ≠ ∅ for each DIS ∈ Mtk.||Let tk, y) ∈ Mtk such that AT(tk′)∩DIS = ∅; then by the definition of DIS, we have y ∈ ||tk′|| because tk′⪰tk. Moreover, since ||tk′|| = ||tk||, then y ∈ ||tk||; such result is contradictive to the condition DIS ∈ Mtk⇒y ∉ ||tk||.⇒: Suppose ∃DIS ∈ Mtk, since AT(tk′)∩DIS ≠ ∅; then there must be ajk ∈ ATk(tk′) such that y ∉ ||||; it follows that y ∉ ||tk′||, from which we can conclude that y ∉ ||tk||⇒y ∉ ||tk′||; that is, ||tk′||⊆||tk||. That completes the proof.⇐: Since S =  =  be an incomplete multiscale decision system in which tk ∈ FDES(ATk); then the tk discernibility function is defined asLet S =  =  be a multiscale decision system in which tk ∈ FDES(ATk), tk′ ∈ DES(ATk), and tk′⪰tk; then tk′ is a reduct descriptor of tk if and only if ATk(tk′) is the prime implicant of the tk discernibility function Δtk.Let tk′)∩DIS ≠ ∅ for each DIS ∈ Mtk. We claim that, for each ajk ∈ ATk(tk′), there must be DIS ∈ Mtk such that AT(tk′)∩DIS = {ajk}. If fact, if ∀DIS ∈ Mtk, there exist ajk ∈ DIS such that |AT(tk′)∩DIS| > 2, where ajk ∈ ATk(tk′)∩DIS; let tk′′≻tk′, where ATk(tk′′) = ATk(tk′)−{ajk}; then by tk′′|| = ||tk||, which contradicts the fact that tk′ is a reduct descriptor of tk; it follows that ATk(tk′) is the prime implicant of the Δtk.⇒: By k(tk′) is the prime implicant of the Δtk, then by k(t′)∩DIS ≠ ∅ for each DIS ∈ Mtk. Moreover, for each ajk ∈ ATk(tk′), there exists DIS ∈ Mtk such that ATk(tk′)∩DIS = {ajk}. Consequently, ∀tk′′≻tk′ and ATk(tk′′) = ATk(tk′)−{ajk}, ||tk′′|| ≠ ||tk||; we then conclude that tk′ is a reduct descriptor of tk. That completes the proof.⇐: If ATIn S =  =  be an incomplete multiscale decision system in which tk ∈ FDES(ATk), tk′ ∈ DES(ATk), and tk′⪰tk; definetk′ is referred to as the lower approximation reduct descriptor of tk if and only if the following two conditions hold: tLk = tLk′;tLk′′ ≠ tLk for each tk′′≻tk′;Let tk′ is referred to as the boundary region reduct descriptor of tk if and only if the following two conditions hold: tBk = tBk′;tBk′′ ≠ tBk for each tk′′≻tk′.tk is a minimal conjunction of the atomic properties in tk, which preserves the inclusion relation between support of tk and the decision classes; a boundary region reduct descriptor of tk is a minimal conjunction of the atomic properties in tk, which preserves the intersection relation between support of tk and the decision classes.By tLk = ∅, then ||tk||⊈Xi for each Xi ∈ U/IND({d}); it follows that no certain decision rules are supported by the k-scale descriptor tk. Therefore, it is meaningless to compute the lower approximation reduct descriptor of tk if tLk = ∅.It should be noticed that if S =  = , if tLk = {Xi}, then ||tk||⊆Xi; we can obtain a certain decision rule such that rk : tk → Xi. Moreover, suppose that tk′ is a lower approximation reduct descriptor of tk; then tk′ is a minimal conjunction of the atomic properties in tk, which preserves the inclusion property; therefore, the decision rule rk : tk → Xi can be simplified to be rk′ : tk′ → Xi. The condition part of rk : tk → Xi is shortened because tk′⪰tk. Similar to what have been analyzed, by the boundary region reduct descriptor, the possible rules supported by tk can also be simplified.Given an incomplete multiscale decision system tk, let us definetk lower approximation and tk boundary region discernibility matrixes, respectively.To compute the lower approximation and boundary region reduct descriptor of S =  =  be an incomplete multiscale decision system in which tk ∈ FDES(ATk), tk′ ∈ DES(ATk), and tk′⪰tk: tLk = tLk′⇔ATk(tk′)∩DISL ≠ ∅ for each DISL ∈ MLtk;tBk = tBk′⇔ATk(tk′)∩DISB ≠ ∅ for each DISB ∈ MBtk.Let We only prove (1); the proof of (2) is similar to the proof of (1).L ∈ MLtk such that ATk(tk′)∩DISL = ∅; then we have y ∈ ||tk′|| because tk′⪰tk. Moreover, by condition we have tLk = tLk′; that is, ||tk||⊆Xi⇔||tk′||⊆Xi; we know y ∈ Xi; such result is contradictive to the assumption because DISL ∈ MLtk⇒y ∉ Xi.⇒: Suppose ∃DIStk′⪰tk, then ||tk′||⊇||tk||, from which we obtain tLk⊇tLk′. Thus, it must be proved that tLk⊆tLk′. ∀Xi ∈ tLk, we have ||tk||⊆Xi. Then ∀y ∉ Xi, we have DISL ∈ MLtk; by condition we know that ATk(tk′)∩DISL ≠ ∅; it follows that y ∉ ||t′||; then there must be ajk ∈ ATk(tk′) such that y ∉ ||ajk, vjk||; it follows that y ∉ ||tk′||, from which we can conclude that ||tk′||⊆Xi; that is, Xi ∈ tLk′, tLk⊇tLk′. That completes the proof.⇐: Since S =  =  be an incomplete multiscale decision system in which tk ∈ FDES(ATk); thentk lower approximation discernibility function;tk boundary region discernibility function.Let S =  =  be an incomplete multiscale decision system in which tk ∈ FDES(ATk), tk′ ∈ DES(ATk), and tk′⪰tk; then tk′ is a lower approximation reduct descriptor of tk if and only if ATk(tk′) is the prime implicant of the tk lower approximation discernibility function ΔLtk;tk′ is a boundary region reduct descriptor of tk if and only if ATk(tk′) is the prime implicant of the tk boundary region discernibility function ΔBtk.Let The proof of In By the discernibility matrixes we mentioned above, it is not difficult to obtain the lower approximation and boundary region reduct descriptors as Tables  1-scale certain decision rules:a11(x) = 1,4, 5,7, 8∧a21(x) = 3 → d(x) = 1,a11(x) = 1,3, 4,5, 7,8∧a31(x) = 5 → d(x) = 1,a11(x) = 1,4, 5,7, 8∧a41(x) = 2 → d(x) = 1,a11(x) = 1,2, 3,5, 6,7, 8∧a31(x) = 3 → d(x) = 2,a11(x) = 2,3, 5,6, 7,8∧a21(x) = 1 → d(x) = 2,a11(x) = 3,6, 7,8∧a41(x) = 1 → d(x) = 2,a21(x) = 4,5, 6 → d(x) = 2,a11 = 4∧a31 = 1 → d(x) = 2,a41(x) = 3,4 → d(x) = 2,a21(x) = 7∨a31(x) = 6∨a41(x) = 5,6 → d(x) = 3; 2-scale certain decision rules:a12(x) = F∧a32(x) = L, H → d(x) = 1,a32(x) = H∧a42(x) = L → d(x) = 1,a42(x) = M → d(x) = 2,a12(x) = F∧a22(x) = F, B → d(x) = 2,a22(x) = F, B∧a32(x) = M → d(x) = 2,a22(x) = F∧a42(x) = L → d(x) = 2,a12(x) = B∨a42(x) = H → d(x) = 3,a22(x) = B, F∧a32(x) = H → d(x) = 3; 3-scale certain decision rules:a23(x) = N∧a43(x) = Y → d(x) = 2.By the lower approximation reduct descriptors shown in  1-scale possible decision rules:a11(x) = 1∧a31(x) = 1 → d(x) = 1∨3 with certainty 0.5,a21(x) = 1∧a31(x) = 1 → d(x) = 1∨3 with certainty 0.5,a11(x) = 2∧a21(x) = 2 → d(x) = 1∨3 with certainty 0.5,a11(x) = 2∧a31(x) = 1 → d(x) = 1∨3 with certainty 0.5,a11(x) = 3∧a21(x) = 3 → d(x) = 1 with certainty 0.67,a11(x) = 3∧a21(x) = 3 → d(x) = 3 with certainty 0.33,a31(x) = 2 → d(x) = 1∨3 with certainty 0.5,a11(x) = 3∧a41(x) = 2 → d(x) = 1 with certainty 0.67,a11(x) = 3∧a41(x) = 2 → d(x) = 3 with certainty 0.33,a11(x) = 4∧a31(x) = 3 → d(x) = 2 with certainty 0.67,a11(x) = 4∧a31(x) = 3 → d(x) = 1 with certainty 0.33,a11(x) = 5∧a31(x) = 4 → d(x) = 2 with certainty 0.75,a11(x) = 5∧a31(x) = 4 → d(x) = 1 with certainty 0.25,a11(x) = 6∧a31(x) = 5 → d(x) = 2 with certainty 0.75,a11(x) = 6∧a31(x) = 5 → d(x) = 1 with certainty 0.25; 2-scale possible decision rules:a12 = G∧a32 = L → d(x) = 1∨3 with certainty 0.5,a12 = F∧a32 = M → d(x) = 2 with certainty 0.75,a12 = F∧a32 = M → d(x) = 1 with certainty 0.25; 3-scale possible decision rules:a13 = Y∨a33 = Y → d(x) = 1∨3 with certainty 0.5,a13 = N∨a23 = Y → d(x) = 1∨2 with certainty 0.5,a13 = N∨a43 = Y → d(x) = 2 with certainty 0.67,a13 = N∨a43 = Y → d(x) = 1 with certainty 0.33,a23 = Y∨a33 = N → d(x) = 1∨2 with certainty 0.5,a33 = N∨a43 = Y → d(x) = 2 with certainty 0.67,a33 = N∨a43 = Y → d(x) = 1 with certainty 0.33,a23 = N → d(x) = 2 with certainty 0.75,a23 = N → d(x) = 3 with certainty 0.25,a43 = N → d(x) = 2 with certainty 0.67,a43 = N → d(x) = 3 with certainty 0.33.By the boundary region reduct descriptors shown in In this paper, the incomplete multiscale information system is explored through the well-known rough set approach. The incomplete multiscale information system is different from the traditional incomplete information system since it reflects data represented at different levels of granulations, which are transformed from smaller to bigger scales. At each level of granulation, that is, scale, we can construct the corresponding rough set and then derive decision rules. Obviously, the decision rules derived from a coarser scale are more general than that from a finer scale. The incomplete multiscale information system is closely related to multigranulation rough set approach. The rough set at the smallest scale is optimistic multigranulation rough set while the rough set at the biggest scale is pessimistic multigranulation rough set.Obviously, the study of multiscale information system will give new life to the development of rough set theory. Furthermore, by considering the preference-ordered –47 domai


M be a 2‐torsion free prime Γ‐ring satisfying the condition aαbβc=aβbαc,∀a,b,c∈M and α,β∈Γ, U be an admissible Lie ideal of M and F=(fi)i∈N be a generalized higher ‐derivation of M with an associated higher ‐derivation D=(di)i∈N of M. Then for all n∈N we prove that Let Mathematics Subject Classification (2010): 13N15; 16W10; 17C50 M and Γ be additive abelian groups. If there is a mapping M×Γ×M→M  into xαy) such that (i) (x+y)αz=xαz+yαz,x(α+β)y=xαy+xβy,xα(y+z)=xαy+xαz, (ii) (xαy)βz=xα(yβz), for all x,y,z∈M and α,β∈Γ, then M is called a Γ‐ring. This concept is more general than a ring and was introduced by Barnes , where Z(M) denotes the center of M, then U is called an admissible Lie ideal of M. In ‐derivations in rings have been introduced by Faraj et al. ‐derivation extends the concept given in ≠2, U a square closed Lie ideal of R and d a ‐ derivation of R, then d(ur)=d(u)r+ud(r),∀,u∈U,r∈R. This result is a generalization of a result in ‐derivation, generalized ‐derivation and generalized higher ‐derivation, where U is a Lie ideal of a Γ‐ring M. Examples of a Lie ideal of a Γ‐ring, ‐derivation, generalized ‐derivation, higher ‐derivation and generalized higher ‐derivation are given here. A result in ‐derivation. Throughout the article, we use the condition aαbβc=aβbαc,∀a,b,c∈M and α,β∈Γ and this is represented by (*). We make the basic commutator identities β= βαy+xzy+xαβ, β= βαz+yxz+yαβ, ∀x,y,z∈M,∀α,β∈Γ. According to the condition (*), the above two identities reduces to β= βαy+xαβ,β= βαz+yαβ,∀x,y,z∈M,∀α,β∈Γ.The notion of a Γ‐ring has been developed by Nobusawa , as a gey Barnes . A Γ‐rinHerstein , HersteiHerstein , Awtar ej et al. , as a gein Awter . In thisin Awter , they prin Awter , Theoremand Paul , TheoremU,R)‐derivation of an ordinary ring developed by Faraj et al. ‐derivation of Γ‐rings as defined below.In view of the concept of ‐derivation of M if d(uαm + sαu) = d(u)αm + uαd(m) + d(s)αu + sαd(u),∀u∈U,m,s∈M and α∈Γ.‐ derivation of M if there exists a ‐derivation d of M such that f(uαm+sαu)=f(u)αm+uαd(m)+f(s)αu+sαd(u),∀u∈U,m,s∈M and α∈Γ.‐derivation and a generalized ‐derivation are confirmed by the following examples.The existence of a Lie ideal of a Γ‐ring,  and M is a Γ‐ring.Let N={:x∈R}⊆M and U1={:u∈U} then N is a sub Γ‐ring of M and U1 is a Lie ideal of N. Let f:R→R be a generalized ‐derivation. Then there exists a ‐derivation d:R→R such that f(uαx+sαu)=f(u)αx+uαd(x)+f(s)αu+sαd(u).If D :N→N by D)=(d(x),d(x)), then we have If we define a mapping D(u1αx1+y1αu1)=D(u1)αx1+u1αD(x1)+D(y1)αu1+y1αD(u1), where After calculation we have D is a − derivation on N.Hence F :N→N be the additive mapping defined by F)=(f(x),f(x)), then considering Let F is a generalized −derivation on N.Hence ), Lemma 2.4) Let M be a 2‐torsion free Γ‐ring satisfying the condition (*). U be a Lie ideal of M and f be a generalized ‐derivation of M. Thenf(uαmβu)=f(u)αmβu+uαd(m)βu+uαmβd(u),∀u∈U,m∈M and α,β∈Γ.(ii)f(uαmβv+vαmβu)=f(u)αmβv+uαd(m)βv+uαmβd(v)+f(v)αmβu+vαd(m)βu+vαmβd(u),∀u,v∈U,m∈M and α,β∈Γ.d be a ‐derivation of M, then we define Φα=d(uαm)−d(u)αm−uαd(m),∀u∈U,m∈M and α∈Γ.‐derivation of M, then(i)α =−Φα, ∀u∈U,m∈M and α∈Γ.Φ(ii)α=Φα+Φα,∀u,v∈U,m∈Mand α∈Γ.Φ(iii)α=Φα+Φα,∀u∈U,m,n∈M and α∈Γ.Φ(iv)α+β=Φα+Φβ,∀u∈U,m∈M and α,β∈Γ.ΦThe proofs are obvious by using the Definition 3.f is a generalized ‐derivation of M and d is a ‐derivation of M, then we define Ψα=f(uαm)−f(u)αm−uαd(m),∀u∈U,m∈M and α∈Γ.‐derivation of M, then(i)Ψα=−Ψα,∀u∈U,m∈Mand α∈Γ.(ii)Ψα=Ψα+Ψα,∀u,v∈U,m∈Mand α∈Γ.(iii)Ψα=Ψα+Ψα,∀u∈U,m,n∈M and α∈Γ.(iv)Ψα+β=Ψα+Ψβ,∀u∈U,m∈M and α,β∈Γ.The proofs are obvious by using the Definition 4.), Lemma 2.11) Let U be a Lie ideal of a 2‐torsion free prime Γ‐ring M satisfying the condition (*) and U is not contained in Z(M). If a,b∈M (resp. b∈U and a∈M) such that a αUβb=0,∀α,β∈Γ, then a=0 or b=0., Theorem 2.13) Let M be a 2‐torsion free prime Γ‐ring satisfying the condition (*), U be an admissible Lie ideal of M and f be a generalized ‐derivation of M, then Ψα=0,∀u,v∈U and α∈Γ., Theorem 2.14) Let M be a 2‐torsion free prime Γ‐ring satisfying the condition (*), U a square closed Lie ideal of M and f be a generalized ‐derivation of M, then f(uαm)=f(u)αm+uαd(m),∀u∈Um∈M and α∈Γ.‐derivations in Γ‐rings.In this section, we introduce generalized higher ‐derivation of M if there exists an higher ‐derivation D=(di)i∈N of M such that for each α,β∈Γ.Let N and U1 are as in Example 1. If fn :R→R be a generalized higher ‐derivation. Then there exists a higher  derivation dn :R→R such that Let Dn :N→N by Dn)=(dn(x),dn(x)). Then Dn is a higher ‐derivation on N.If we define a mapping Fn :N→N be the additive mapping defined by Fn)=(fn(x),fn(x)). Then by the similar calculation as in Example 1, we can show that, Fn is a generalized higher ‐derivation on N.Let Let M be a 2‐torsion free Γ‐ring satisfying the condition (*), U be a Lie ideal of M and F=(fi)i∈Nbe a generalized higher ‐derivation of M. Thenand α,β∈Γ.x=uα((2u)βm+mβ(2u))+((2u)βm+mβ(2u))αu.Let m and s by (2u)βm+mβ(2u) and (2u)αm+mα(2u) respectively in Replacing Thus we haveU,M)‐ derivation and using the condition (*) On the other hand by the definition of higher  and (2) we get Let M be a 2‐torsion free Γ‐ring satisfying the condition (*), U be a Lie ideal of M and F=(fi)i∈Nbe a generalized higher ‐derivation of M. Thenand α,β∈Γ.u gives us Linearizing of On the other hand α,β∈Γ.Now comparing above two expressions, we get M be a 2‐torsion free Γ‐ring satisfying the condition (*) and U be a Lie ideal of M. Let F=(fi)i∈N be a generalized higher ‐derivation of M. For every fixed n∈N, we define D=(di)i∈N be a higher ‐derivation of M. For every fixed n∈N, we define Let n∈N if and only if n∈N. Also n∈N if and only if n∈N.Let M be a 2‐torsion free Γ‐ring satisfying the condition (*) and U be a Lie ideal of M. For every u∈U,m∈M,α∈Γ and n∈N, thenandU,M)‐derivation of M and generalized higher ‐derivation of M.The proofs are obvious by the Definition 6, higher (Let M be a 2‐torsion free prime Γ‐ring satisfying the condition (*), U be an admissible Lie ideal of M and F=(fi)i∈Nbe a generalized higher ‐derivation of M. Thenand n∈N.We have n∈N, that m∈N and m<n.Now we assume, by induction on x=4(uαvβwγvαu+vαuβwγuαv).Let Then by using Lemma 6, we have D=(di)i∈N is a higher ‐derivation of M. On the other hand, by Lemma 5 and fn(x) and using Now comparing the two expressions of M we get Using Lemma 7 and 2‐torsion freeness of D=(di)i∈N is a higher ‐derivation of M, thus we have U is noncentral, thus we get n∈N.Since Now we prove the main result.Let M be a 2‐torsion free prime Γ‐ring satisfying the condition (*), U be an admissible Lie ideal of M and F=(fi)i∈Nbe a generalized higher ‐derivation of M. Thenand n∈N.We have n∈N, that m<n.Now we assume, by induction on F=(fi)i∈N is a generalized higher ‐derivation of M, we have Now since D=(di)i∈N is a higher ‐derivation of M, thus we haveSince F=(fi)i∈N is a generalized higher ‐derivation of M, thus we have Since Since On the other hand, by using Equation  and LemmBy comparing (5) and (6) and using the condition (*), we getu, gives usLinearizing of (7) with respect to v by vαv in (8) and since Replacing n∈N.Hence by Lemma 4 and since n∈N.Thus by the Remark 2, we have