We study the local indistinguishability problem of quantum states. By introducing an easily calculated quantity, non-commutativity, we present an criterion which is both necessary and sufficient for the local indistinguishability of a complete set of pure orthogonal product states. A constructive distinguishing procedure to obtain the concrete local measurements and classical communications is given. The non-commutativity of ensembles can be also used to characterize the quantumness for classical-quantum or quantum-classical correlated states. Nonlocality is one of the most important features in quantum mechanics. Quantum entanglement was firstly introduced to characterize and quantify the nonlocality, as it acts as a crucial resource in many quantum information processing tasksABρ. Although a set of POPS can always be distinguished globally, it may not be distinguished locally by local operations and classical communications (LOCC). This is called quantum nonlocality without entanglementABρ is not an indicator of local indistinguishability for a set of states that constitute the pure-state decomposition of ABρ491011121314A well known quantum phenomenon that exhibits quantum nonlocality is the local indistinguishability for some quantum states. State discrimination or distinguishing is essentially primitive for many quantum information tasks, such as quantum cryptography7It is natural to ask what kind of quantity or quantumness accounts for the quantum nonlocality. In this article, instead of quantum correlation measures, such as quantum discord, quantum deficit, etc., we investigate local indistinguishability for a set of POPS from the point of quantumness of ensemble. We first introduce an easily calculated quantity, non-commutativity, to quantify the quantumness of a quantum ensemble. Based on the non-commutativity, we present a necessary and sufficient criterion for the local indistinguishability. We also give a constructive distinguishing procedure to judge the local indistinguishability for any given set of POPS by using the criterion. Moreover, by proving the uniqueness of the expression of for semi-classical quantum correlated states, we show that our definition for quantumness of ensembles can be used to characterize the quantumness for semi-classical states.ψ〉 and |ϕ〉 with equal probability can be viewed as a set of binary signals in some communication scheme. If |ψ〉 and |ϕ〉 are orthogonal, the ensemble becomes classical since different states of classical information can be thought of merely as orthogonal quantum statesx = |〈ψ|ϕ〉|, 0 < x < 1, induces quantumness for the ensemble. In fact the ensemble is most ‘quantum’ when A quantum ensemble containing only two pure states |A1, A2, …, nA be a set of operators. We define the total non-commutativity for this set, A, B] = AB − BA, ||A|| is the trace norm of the operator A, ε = {ip, iρ}, where iρ are density operators with probability ip. Denote iA = iρip, then N is the measure of quantumness for the ensemble ε. Here for the problem of local distinguishability of states ρ1, ρ2, …, nρ, the prior probabilities ip are irrelevant. One only needs to concern the quantity N(ε) = N to judge the local distinguishability of the set of states ρ1, ρ2, …, nρ. The prior probabilities ip do make sense when one concerns the quantumness of a state given by the ensemble ε = {ip, iρ} .Taking the above observation into account, we introduce non-commutativity to characterize the quantumness for a quantum ensemble. Let N has the following properties, which make it a well defined measure for quantumness of an ensemble:N is non negative; N is unitary invariant, N(ε) = N(UεU†), where UεU† = {ip, iUUρ†} with U being any unitary matrix; A1 = |ψ〉 〈ψ| and A2 = |ϕ〉 〈ϕ|, without considering their prior probabilities, N is zero only when the overlap x = |〈ϕ|ψ〉| = 0 or 1. N gets to a maximum of 1 when For an ensemble only containing two pure states N({|ia〉}) + N({|ib〉}) ≤ N. One can easily verify the inequality from the definition of N. The equality holds if {|ia〉} and {|ib〉} are either mutually orthogonal or identical. The sum of two sets' non-commutativity is equal to or less than the non-commutativity of the sum of the two sets: The non-commutativity ψ1〉 = |1〉|1〉, st〉 = |s〉 ⊗ |t〉, |s〉 and |t〉 are the computational basis. Both {|iψ〉} and {|st〉} are pure-state decompositions of ρ. However, the set {|st〉} can be locally distinguished while {|iψ〉} cannotρ having zero quantum correlation is not an indicator for the local distinguishability of the states in ρ's pure-state decomposition. On the other hand, if we change the probability of |iψ〉, ρ is no longer an identity and can have nonzero quantum correlation, but the nine states |iψ〉 still remain locally indistinguishable. Usually, the local indistinguishability of a set of pure states has no simple relations with the properties of the related density operatorA set of quantum states corresponds to a quantum ensemble. Nevertheless, one density operator may have many quantum ensemble decompositions. For instance, consider the density operator in a 3 ⊗ 3 system, ε = {ip, |iψ〉} classical if N(ε) = 0, and quantum if N(ε) > 0. If N(ε) = 0, from the properties of non-commutativity, {|iψ〉} must be either mutually orthogonal or identical and the states form a set of classical signals from informatics point of view. If N(ε) > 0, among {|iψ〉} there must be at least one pair of states that are neither orthogonal nor identical. Consider a set of bipartite POPS {|iψ〉 = |ia〉 ⊗ |ib〉}, where |iψ〉 are all mutually orthogonal, |ia〉 and |ib〉 are associated to partite A and B respectively. Obviously, this set forms a classical ensemble ε = {ip, |iψ〉} since N(ε) = 0, where ip is the probability with respect to |iψ〉. Correspondingly one has ensembles Aε = {ip, |ia〉} and Bε = {ip, |ib〉} respectively. If N(Aε) = N(Bε) = 0, we call ε a classical-classical ensemble. If N(Aε) = 0, N(Bε) > 0, we call ε a classical-quantum ensemble. Analogously we can define quantum-classical and quantum-quantum ensembles.To study the local indistinguishability, and its relations with the quantumness of a quantum ensemble in terms of non-commutativity, in the following we call an ensemble finite rounds of LOCC protocols. If there are at least two states in a set which can not be distinguished by finite rounds of LOCC protocols, we say that the set can not be distinguished locally. The arguably more operational “asymptotic local operations and classical communications discrimination problem16A set of states is said to be reliably distinguished locally or distinguished locally if all the states in the set can be distinguished by Theorem 1 If a set of bipartite pure orthogonal product states cannot be locally distinguished, the ensemble composed of these states must be a quantum-quantum ensemble.Proof. Consider a set of bipartite POPS, not necessarily complete, {|iψ〉 = |ia〉 ⊗ |ib〉} with 〈iψ|jψ〉 = 0, ∀i ≠ j. Suppose these states form a classical-quantum or classical-classical ensemble ε = {ip, |ia〉 ⊗ |ib〉} with non zero ip. Since N({|ia〉}) = 0, {|ia〉} must be either mutually orthogonal or identical. To distinguish {|iψ〉} locally, we can first take projective measurement on A side to distinguish those orthogonal states in {|ia〉}. For those identical states |ia〉, for example, |a1〉 = |a2〉, we can take a projective measurement on B to distinguish state |a1〉 from state |a2〉, since |b1〉 and |b2〉 must be mutually orthogonal to ensure that 〈ψ1|ψ2〉 = 0. In this way we can distinguish the states in the set ε reliably. The analysis is similar if ε is a quantum-classical ensemble. Therefore if a set of bipartite POPS forms a classical-classical, quantum-classical or classical-quantum ensemble, the set can be locally distinguished. And if the set of POPS states cannot be locally distinguished, the ensemble composed of these states must be a quantum-quantum one. □n systems, any set of POPS is locally distinguishableSince it is impossible to form a quantum-quantum ensemble for a set of POPS in a 2 ⊗ 2 system, Theorem 1 is both necessary and sufficient for 2 ⊗ 2 systems. For higher dimensional cases, there are quantum-quantum ensembles whose states can be locally distinguished. In fact, for 2 ⊗ ia〉} and {|a′j〉} are orthogonal if and only if 〈ia|a′j〉 = 0, ∀i, j. If a set of states cannot be divided into subsets such that those subsets are mutually orthogonal, then we call the set a single set. Consider a set of states ε = {|a1〉 = |0〉, |a2〉 = |0 + 1〉, |a3〉 = |2〉}. This set can be divided into two single subsets ε1 = {|a1〉, |a2〉} and ε2 = {|a3〉} which are orthogonal and each subset cannot be split further. We call this partition the direct sum decomposition and denote it as ε = ε1 ⊕ ε2. A single set means that the set cannot be decomposed into such direct sums. If all the subsets are single sets in a direct sum decomposition, we say that the decomposition is a single set decomposition. It can verified that the single set decomposition of a non-single set is unique. For a single set, adding some states in the vector space spanned by itself will keep the set a single one. Nevertheless, taking away some states from a single set, the set could become a non-single one. In the following, by a set of states' decomposition we mean the single set decomposition.We say that two sets {|Lemma For a set of states ε = {|ia〉}, m = dim(span{|ia〉}), the following statements are equivalent: (a) ε is a single set. (b) There are m linear independent states ε satisfying the following relations: ε.See Methods for the proof of the Lemma. From the Lemma we haveTheorem 2 For a complete set of m ⊗ n POPS, ε = {|iψ〉 = |ia〉 ⊗ |ib〉} with 〈iψ|jψ〉 = 0, ∀i ≠ j, the set ε cannot be completely locally distinguished if and only if there exist subsets Proof. Recall that a complete set of POPS can be locally distinguished if and only if the states can be distinguished by local nondestructive projective measurement and classical communicationε cannot be completely locally distinguished by local nondestructive projective measurement and classical communication, there must exist a subset ε such that From the view of accessible information, the more quantum an ensemble is, the less information one can get from the ensemble1iψ〉 = |ia〉 ⊗ |ib〉} be a complete set of POPS, corresponding to two sets Aε = {|ia〉} and Bε = {|ib〉}. To verify the inequality (3), one needs to find all the subsets involved. This can be done in the following way.Aε and Bε into subsets, Decompose the sets sψ〉 = |sa〉 ⊗ |sb〉}ij corresponds to two new subsets Find the overlapped states between these subsets, Theorem 2 gives a constructive distinguishing procedure to judge the local distinguishability for a complete set of POPS. Let {|n rounds for the new subsets until each of those new subsets has only one element or both A and B parts cannot be decomposed further. At last we have subset {|kψ〉 = |ka〉 ⊗ |kb〉}stij,…,  corresponding to two new sets iψ〉} can be locally distinguished.We repeat the above process Aε = {|a1〉 = |1〉, |a2,3〉 = |0〉, |a4,5〉 = |2〉, a10〉 = |0〉, |a11〉 = |1〉, |a12〉 = |2〉} and Bε = {|b1〉 = |1〉, b6,7〉 = |0〉, |b8,9〉 = |2〉, |b10,11,12〉 = |3〉}. The distinguishing process is as follows.As an example, there is a complete set of POPS for 3 ⊗ 4 system, Aε) linear independent states in Aε satisfying Aε is a single set. Hence the first measurement should not be applied to A side. We denote B side, we have decomposition BM = 1(|0〉〈0| + |1〉〈1| + |2〉〈2|) + 2|3〉〈3| to distinguish Round 1: (i) There are 3 = dim(span (ii) Find the overlapped states by classical communication, a4〉, |a6〉, |a8〉 in ψ1〉, …, |ψ9〉}01 is the subset described in Theorem 2. In fact they are the nine locally indistinguishable states inψ10〉, |ψ11〉, |ψ12〉}02 further since we have decomposition AM = 1|0〉〈0| + 2|1〉〈1| + 3|2〉〈2|. However, Round 2: (i) Do decomposition for the above four new sets. Note (ii) Find overlapped states by classical communication, ψ1〉, …, |ψ9〉}01, {|ψ10〉}10,02, {|ψ11〉}20,02, {|ψ12〉}30,02, and the set cannot be completely distinguished due to the subset {|ψ1〉, …, |ψ9〉}01.Finally we can divide the set (4) into four parts, {|ε = {ip, iρ} is fixed (see Methods). Denote iX = iρip. We have the non-commutativity describing the quantumness for the state qcρ, N(qcρ) satisfies the following properties, which actually makes it a candidate for quantum-correlation measures for semi-classical statesN(qcρ) describes the quantumness of semi-classical states. It is different from those quantum correlation measures, such as quantum discord, quantum deficit, etc., which are based on the measurements and their estimations involve extremely complicated optimization process. Interestingly, while the quantity N(qcρ) can be easily computed, it shows similar behavior to other quantum correlation measures for semi-classical states. Hence, instead of those quantum correlation measures, one can use the non-commutativity to characterize the quantum correlations of semi-classical states qcρ. Moreover, non-commutativity provides a tool to explore the relation between the ensemble quantumness and those quantum correlation measures. The non-commutativity defined by (1) can be also used to describe the quantumness for bipartite semi-classical states. For a given semi-classical state, We have proposed non-commutativity as a quantumness measure for an ensemble. It has been shown that the local ensemble quantumness, instead of quantum correlation measure, like quantum discord, quantum deficit, etc., which is a function of a density operator, accounts for the local indistinguishability for a complete set of POPS. It implies that the quantumness of local ensembles must satisfy certain conditions so that the states in the ensemble cannot be locally distinguished.A constructive distinguishing procedure to obtain the concrete local measurements and classical communications has been presented to judge the local indistinguishability for a complete set of POPS. Our approach to judge the local indistinguishability can also be directly extended to distinguish a complete set of multipartite POPS or a non-complete set of POPS, when the local operations are restricted within nondestructive projective measurements.Due to that one semi-classical state corresponds to one quantum ensemble, the non-commutativity has been shown to be able to characterize the ensemble quantumness for classical-quantum or quantum-classical systems.n dimensional states A = |ϕ〉〈ϕ| and B = |ψ〉〈ψ|. Denote 〈ϕ|ψ〉 = iθxe with x ∈ , θ ∈  = iθxe|ϕ〉〈ψ| − xeiθ−|ψ〉〈ϕ|. Under the base {|ϕ〉 = |ϕ0〉, |ϕ1〉, …, |nϕ−1〉}, one gets A, B] can then be expressed as iψj, = 〈iϕ|ψ〉〈ψ|jϕ〉 and iψj,. The eigenvalues of the above matrix are ψ〉 and |ϕ〉, |||| is 0 when |〈ϕ|ψ〉| = 0 or 1, and 1 when Consider two pure ε, say, ε such that it is not equal to or orthogonal to ε is a single set. Next choose the third vector ε such that it is independent of ε can be direct sum decomposed into two parts. Continuing with the above process, we can finally get m linear independent states ε satisfying inequality (2).(a) ⇒ (b) First choose any vector in M, an Hermitian operator such that all |ia〉 in ε are the eigenvectors of M. If m linear independent M. Therefore M becomes an identity  and it cannot be used to distinguish the states in ε.(b) ⇒ (c) A nondestructive projective measurement is described by an observable, ε = ε1 ⊕ ε2. Then one can use a corresponding nondestructive projective measurement M = aI1 ⊕ bI2, where a ≠ b, I1, I2 are identity operators, to distinguish the states in ε1 from ε2. Hence if M cannot do any in distinguishing the states in ε, ε must be a single set, that is, statement (a) holds.(c) ⇒ (a) Suppose the set is not a single one and has a decomposition iα〉} is an orthogonal base. Suppose there exists another orthogonal base {|iβ〉} such that ρ = σ, where U|iα〉 = |iβ〉, (U)ij = iju = 〈iα|jβ〉, σ can be reexpressed as iσiq. Set ρ = σ implies that s ≠ t, which means that both {|iα〉} and {|iβ〉} are the eigenvectors of klQ, and iσiq} of state σ can be divided into degenerate part  and non-degenerate part . For non-degenerate part, taking over all k, l, one will find that the intersection of the eigenspaces belonging to the eigenvalues iα〉 = |iβ〉 and then iρip = iσiq. For degenerate part, without loss of generality, assume q1σ1 = q2σ2, then ρ = σ means p1ρ1 ⊗ |α1〉〈α1| + p2ρ2 ⊗ |α2〉〈α2| = q1σ1 ⊗ (|β1〉〈β1| + |β2〉〈β2|). Therefore p1ρ1 = p2ρ2 = q1σ1 = q2σ2. Finally we get iρip = iσiq, ∀i.Let T.M., M.Z., Y.W. and S.M. wrote the main manuscript text. All authors reviewed the manuscript.


We consider the optimal dividends problem for a company whose cash reserves follow a general Lévy process with certain positive jumps and arbitrary negative jumps. The objective is to find a policy which maximizes the expected discounted dividends until the time of ruin. Under appropriate conditions, we use some recent results in the theory of potential analysis of subordinators to obtain the convexity properties of probability of ruin. We present conditions under which the optimal dividend strategy, among all admissible ones, takes the form of a barrier strategy. In the literatures of actuarial science and finance, the optimal dividend problem is one of the key topics. For companies paying dividends to shareholders, a commonly encountered problem is to find a dividend strategy that maximizes the expected total discounted dividends until ruin. The pioneer work can be traced to de Finetti  who consAnalysis of optimal dividends for Lévy risk processes is of particular interest which have undergone an intensive development. For example, Avram et al.  considerThe paper is organized as follows. In X = {Xt}t≥0 be a real-valued Lévy process on a filtered probability space  where 𝔽 = (ℱt)t≥0 is generated by the process X and satisfies the usual conditions of right continuity and completeness. Denote by Px the law of X when X0 = x. Let Ex be the expectation associated with Px. For notational convenience, we write P and E when X0 = 0. Write the Lévy triplet of X as , where a, σ ≥ 0 are real constants and Π is a positive measure on ∖{0} which satisfies the integrability condition dx) = π(x)dx, then we call π the Lévy density. The characteristic exponent of X is given by A is the indicator of set A. Furthermore, define the Laplace exponent of X byσ = 0 and ∫−11 | x | Π(dx) < ∞. If Π{} = 0, then the Lévy process X with no positive jumps is called the spectrally negative Lévy process; if Π{} = 0, then the Lévy process X with no negative jumps is called the spectrally positive Lévy process. It is usual to assume that P(limt→∞Xt = +∞) = 1 which says nothing other than Ψ′(0+) > 0. For more information on Lévy processes we refer to the excellent book by Kyprianou , then the process Ltb can be explicitly represented by U0b = x > b, then Vb(x) the dividend value function if barrier strategy ξb is applied; that is,Vb is the solution to X withδ > 0, the equation Ψ(z) = δ has a unique solution on , say ρ(δ). A typical example is that the Lévy measure of the positive jumps has the following gamma distribution Γ; that is, r is a positive number and γ is an even number.We denote by Vb can be expressed asψρ(δ) given by ψρ(δ)(η) = Ψ(η + ρ(δ)) − δ. Note that the process Following similar reasoning to Yuen and Yin , Vb canY = {Yt}t≥0, with Y0 = 0, where Yt = −Xt, t ≥ 0. It is easy to see that the Lévy triplet of Y is , where ΠY(dx) = πX(−x)dx. Let Y, respectively. Following Klüppelberg et al.  that L−1 = {Lt−1 : t ≥ 0} is the inverse local time such that Lt−1 = inf{s ≥ 0 : Ls > t}, where we take the infimum of the empty set as ∞. Define an increasing process H by {Ht = YLt−1 : t ≥ 0}, that is, the process of new maxima indexed by local time at the maximum. The processes L−1 and H are both defective subordinators, and we call them the ascending ladder time and ladder height process of Y, respectively. It is understood that Ht = ∞ when Lt−1 = ∞. Throughout the paper, we denote the nondefective versions of L, L−1, and H by ℒ, ℒ−1, and ℋ, respectively. In fact, the pair  is a bivariate subordinator. Define Y drifts to −∞, the decreasing ladder height process is not defective. Associated with the ascending and descending ladder processes are the bivariate renewal functions U and k is its joint Laplace exponent such that q ≥ 0 is the killing rate of H so that q > 0 if and only if limt→∞Yt = −∞, c ≥ 0 is the drift of H, and ΠH is its jump measure. Denote the marginal measure of U by U is called the potential/renewal measure. As for the descending ladder process, + and Π− for the restrictions of Π(du) and Π(−du) to . Furthermore, for u > 0, define In this section, we recap some basic facts about ladder processes and potential measure. Consider the dual process g et al. , we now ϕ : → is called a Bernstein function if it admits a representation a ≥ 0 is the killing term, b ≥ 0 is the drift, and μ is the Lévy measure concentrated on  satisfying ∫0∞(1∧x)μ(dx) < ∞. A function ψ is called a special Bernstein function if the function ψ(λ) = λ/ϕ(λ) is again a Bernstein function. Let ϕ to be a special subordinator is that μ is log-convex on . A function ϕ :  → ℝ is called a complete Bernstein function if there exists a Bernstein function η such that ℒ stands for the Laplace transform. It is known that every complete Bernstein function is a Bernstein function and that the following three conditions are equivalent: (i)ϕ  is a complete Bernstein function;(ii)ψ(λ) = λ/ϕ(λ) is a complete Bernstein function;(iii)ϕ is a Bernstein function whose Lévy measure μ is given by where ν is a measure on  satisfying We next introduce the notions of a special Bernstein function and complete Bernstein function and two useful results. Recall that a function ondraček  that  is summu et al.  and Hawku et al.  is givenH be a subordinator whose Lévy density, say μ(x), x > 0, is log-convex. Then, the restriction of its potential measure to  has a nonincreasing and convex density. Furthermore, if the drift of H is strictly positive, then the density is in C1.Let H is a subordinator with Laplace exponent ϕ and potential measure U. Then, U has a density u which is completely monotone on  if and only if the tail of the Lévy measure is completely monotone.Suppose that μ is a completely monotone function if and only if μ has a completely monotone density. Thus, we have the following two equivalent statements: ϕ is a complete Bernstein function if and only if U has a density u which is completely monotone on ; or, equivalently, U has a density u which is completely monotone on  if and only if μ has a completely monotone density.Note that the tail of the Lévy measure ψ(x) = αU, where α−1 = U = ∫0∞P(Ht < ∞)dt, with U given in → and Vb∗′(x) = Vx′(x) = 1 for x > b∗;(iii)δ)Vb∗(x) = 0 for x ∈ .(Γ − Suppose that x→∞h′(x) = ∞, we have (i). For (ii), Vb∗′(x) = h′(x)/h′(b∗) for x ∈ ; it follows from the definition of b∗ that Vb∗(x) ≥ 1 for x ∈ ; Vb∗′(x) = Vx′(x) = 1 for x > b∗ because of Vb∗(x) = x − b∗ + Vb∗(b∗); and Vx′(x) = 1 since Vx(x) = h(x)/h′(x). Finally, (iii) is due to (Γ − δ)h(x) = 0 for x ∈  and (As limb∗) and .h is sufficiently smooth and is convex in the interval . Then, for x > b∗, (i)Vb∗′′(x) = 0 ≤ Vx′′(x−) if σ ≠ 0;(ii)Vb∗′(y) ≥ Vx′(y), y ∈ ;(iii)Vb∗(x) ≥ Vx(x);(iv)δ)Vb∗(x) ≤ 0. = 0 is clear. Also, since h ∈ C2 and is convex in the interval , we have Vx′′(x−) = limy↑xVx′′(y) = limy↑xh′′(y)/h′(x) ≥ 0. Thus, (i) is proved.If y ∈ , by the definition of b∗, we have y ∈ , by the convexity of h on , we have For Vb∗(b∗) = h(b∗)/h′(b∗) ≥ h(b∗)/h′(x) = Vx(b∗) and that (Vb∗ − Vx) is nondecreasing on  because of (ii). Thus, Vb∗(x) ≥ Vx(x); that is, (iii) holds.Note that x > b∗, (Γ − δ)Vx(x−) = limy↑x(Γ − δ)Vx(y) = 0. For x ≤ b∗, we have I1 ≤ 0, and I4 ≥ 0. For I2 + I3, we have J1 ≤ 0. For y > 0, we obtain J2 = 0. These prove (iv).For We now present the proofs of Theorems X by υ = υ = dtΠ(dy). Then, the Lévy decomposition [B = {Bt}t≥0 is a standard Brownian motion and Mt is a martingale with M0 = 0.Define the jump measure of ν is smooth enough for an application of the appropriate version of Itô's formula and the change of variables formula. In fact, if X is of bounded variation, then ν ∈ C1 and we are allowed to use the change of variables formula [X has a Gaussian exponent, then ν ∈ C2 and we are allowed to use Itô's formula [X has unbounded variation and σ = 0, then ν is twice continuously differentiable almost everywhere but is not in C2 and we can use Meyer-Itô's formula [tn,  n ≥ 1}, we get under Pxν implies that ν(x) − ν(y) + (x − y)ν′(y) ≤ 0 for any x ≤ y. Taking expectations on both sides of Vb(x) ≤ 0 for x ∈ ∖{b} and Vb′(x) ≥ 1 for x > 0. Similar to  = 0 and that Us−ξ + ΔXs ≥ b on {ΔLsξ > 0, ΔXs > 0}. Consequently, Vb′(Us−ξ + ΔXs) = 1, and hence tn,  n ≥ 1}, we haven → ∞ in  is strictly convex on . Then, Vb∗ is concave on  because of (Vb(x) ≥ V∗(x). Consequently, Vb(x) = V∗(x) and the proof is complete.If cause of . From Leπ− is log-convex on , it follows from h(x) is strictly convex on . Then, applying δ)Vb∗(x) ≤ 0 for all x > b∗. The result follows from If


AX − XB = C are investigated. For convenience, these equations were called generalized Sylvester-quaternion equations, which include the Sylvester equation as special cases. By the real matrix representations of complex quaternions, the necessary and sufficient conditions for the solvability and the general expressions of the solutions are obtained.Some complex quaternionic equations in the type  Mathematics, as with most subjects in science and engineering, has a long and varied history. In this connection one highly significant development which occurred during the nineteenth century was the quaternions, which are the elements of noncommutative algebra. Quaternions have many important applications in many applied fields, such as computer science, quantum physics, statistic, signal, and color image processing, in rigid mechanics, quantum mechanics, control theory, and field theory; see, for example, .ax + xb = c. In [ax = xb and ax = xb and αxβ + γxδ = ρ linear quaternionic equation with one unknown, αxβ + γxδ = ρ, is solved. In [ax + xb = c is studied. In [α(xα) = (αx)α = αxα = ρ, with 0 ≠ α ∈ O; second, they presented a method which allows to reduce any octonionic equation with the left and right coefficients to a real system of eight equations and finally reached the solutions of this linear octonionic equation from this real system. In [In recent years, quaternionic equations have been investigated by many authors. For example, the author of the paper  classifi = c. In , the linshed. In , the solated. In , the αxβlved. In , Bolat alved. In , the quadied. In , Bolat astem. In , Flaut astem. In –14.In this paper, we aim to obtain the solutions of some linear equations with two terms and one unknown by the method of matrix representations of complex quaternions over the complex quaternion field and to investigate the solutions of some complex quaternionic linear equations.The paper is organized as follows. In The following notations, definitions, propositions, lemmas, and theorems will be used to develop the proposed work. We now start the definitions of the quaternion and complex quaternion and their basic properties that will be used in the sequel.a + bi, where i is the imaginary unit with the defining property i2 = −1. The set of all complex numbers is usually denoted by C. From here, it can be easily said that the set of complex numbers is an extension of the set of real numbers, usually denoted by R. That is, R ⊂ C.It is well known that a complex number is a number consisting of a real and imaginary part. It can be written in the form HR is an algebra over the field R, and this algebra is called the real quaternion algebra and the set {1, i, j, k} is a basis in HR. The elements in HR take the form a = a0 + a1i + a2j + a3k, where a0, a1, a2, a3 ∈ R, which can simply be written as a = Rea + Ima, where Rea = a0 and Ima = a1i + a2j + a3k. The conjugate of a is defined as a, b ∈ HR. The norm of a is defined to be In the literature, firstly, the set of quaternions introduced asa, b ∈ HR be quaternions. Then a and b are similar if and only if a0 = b0 and a12 + a22 + a32 = b12 + b22 + b32, that is, Re(a) = Re(b) and |Ima|2 = |Imb|2.Let Q = c0e0 + c1e1 + c2e2 + c3e3, where cn ∈ C, n ∈ {0,1, 2,3},βmn and et being uniquely determined by em and en. We denote by HC the set of the complex quaternions and HC is an algebra over the field C and this algebra is called the complex quaternion algebra. The set {1, e1, e2, e3} is a basis in HC.A complex quaternion is an element of the form Q = c0e0 + c1e1 + c2e2 + c3e3 ∈ HC,  cm ∈ C, m ∈ {0,1, 2,3}, can be written asam, bm ∈ R,  m ∈ {0,1, 2,3}, and i2 = −1. Therefore, we can write a complex quaternion as the form Q = a + ib, where a = a0 + a1e1 + a2e2 + a3e3 and b = b0 + b1e1 + b2e2 + b3e3 are in HR. The conjugate of the complex quaternion Q is the element HR is denoted by H. For the quaternion a, b ∈ H, if a* is defined asH, the mapa = a0e0 + a1e1 + a2e2 + a3e3 ∈ H, is an isomorphism between H and the algebra of the matrices:λ(a) ∈ M4(R) has as columns the coefficients in R of the basis {1, e1, e2, e3} for the elements {a, ae1, ae2, ae3}. The matrix λ(a) is called the left matrix representation of the element a ∈ H.The element a ∈ H the right matrix representation:a = a0e0 + a1e1 + a2e2 + a3e3 ∈ H.Analogously with the left matrix representation, we have for the element ρ(a) ∈ M4(R) has as columns the coefficients in R of the basis {1, e1, e2, e3} for the elements {a, e1a, e2a, e3a}.We remark that the matrix x, y ∈ H and r ∈ R, one hasλ(x + y) = λ(x) + λ(y),λ(xy) = λ(x)λ(y),λ(rx) = rλ(x), λ(1) = I4,ρ(x + y) = ρ(x) + ρ(y),ρ(xy) = ρ(y)ρ(x),ρ(rx) = rρ(x), ρ(1) = I4,λ(x−1) = (λ(x))−1,ρ(x−1) = (ρ(x))−1, where x−1 is the inverse of x nonzero quaternion.For x ∈ H, let M1×4(R) be the vector representation of the element x. Therefore for all a, b, x ∈ H the following relations are fulfilled:ρ(b)λ(a) = λ(a)ρ(b),λ(x)) = det(ρ(x)) = (n(x))2, where n(x) = a02 + a12 + a22 + a32 and it is the weak norm of x. det,  Θ(Q) ∈ M8(R); see [The matrix(R); see .a, x ∈ H be two quaternions; then, the following relations are true.(i)a*i = ia, where i2 = −1. (ii)ai = ia*, where i2 = −1. (iii)a* = iai, where i2 = −1. −(iv)xa)* = x*a*.((v)For X, A ∈ HC, X = x + iy, A = a + ib,Let X, A ∈ HC, X = x + iy, A = a + ib be given. ThenLet X, A ∈ HC, X = x + iy, A = a + ib be given. ThenLet X ∈ HC,  X = x + iy be given. ThenX, where x, y ∈ H and x and y.Let X, A, B ∈ HC, X = x + iy,  x, y ∈ H be given. Then where 1 = I4 ∈ M4(R) is the identity matrix and 0 = O4 ∈ M4(R) is the zero matrix;; where ;α2 = I4;;;, where Let In this section, the complex quaternionic equations in the typeR. In order to symbolically solve it, we need to examine some operation properties on the matrix Γ(A) − Ψ(B).According to (ii) and (v) cases in A = a + ib, B = c + id ∈ HC be given, and denote δ = Γ(A) − Ψ(B). Then(i)δ iss = A0 − B0.the determinant of (ii)A0 ≠ B0, or |ImA| ≠ |ImB|, then δ is nonsingular and its inverse can be expressed asif (iii)A0 = B0 and |ImA| = |ImB|, then δ is singular and has a generalized inverse as follows:if Let A,  B ∈ HC, there are nonzero P, Q ∈ HC such thatith case in It is a known result that, for all The results in A0 = B0 and |ImA| = |ImB|, it is easily seen thatImA)2 = (ImB)2 = −|ImA|2, we can easily deduce the following equality: δ3 = −4|Im A|2δ. So, the proof of iiith case in Finally, under the conditions that Based on A = a + ib ∈ HC be given and A ∉ R. Then the general solution of the equationP ∈ HC is arbitrary.Let δ| = 0,  in According to (ii) and (v) cases in [Γ(A)−Ψ(A[Γ(A)−ΨA and B are similar, if and only ifthe linear equation(ii)P ∈ HC is arbitrary.in that case, the general solution of  is(36)XLet δ| = 0, which is equivalent, by HC. Now substituting δ− in According to (ii) and (v) cases in A = a + ib, B = c + id ∈ HC,  a, b, c, d ∈ H, be given with A and B  being not similar; that is, ReA ≠ ReB or |ImA | ≠ |ImB|. Then  = Γ(A) − Ψ(B) is nonsingular by Under the assumption of this theorem, A = a + ib ∈ HC be given. Then the general solution of the equationP ∈ HC is arbitrary.Let δ| = 0,  in According to (ii) and (v) cases in [Γ(A)−Ψ(A[Γ(A)−Ψ = −(ImA)C, and then  and (v) cases in and then . In that[Γ(A)−Ψ(B we find .Starting from known results and referring to the real matrix representations of the complex quaternions, in this paper we have investigated solutions of some linear equations with two terms and one unknown by the method of matrix representations of complex quaternions over the complex quaternion field.The methods and results developed in this paper can also extend to complex octonionic equations. We will present them in another paper.


H of a given group G is said to be autopermutable,if HHα = HαH for all α ∈ Aut(G). We alsocall H a self-autopermutable subgroup of G, whenHHα = HαH implies that Hα = H. Moreover, Gis said to be EAP-group, if every subgroup of G isautopermutable. One notes that if α runs over the inner automorphisms of the group, one obtains the notions of conjugate-permutability, self-conjugate-permutability,and ECP-groups, which were studied by Foguel in 1997, Liand Meng in 2007, and Xu and Zhang in 2005, respectively. In the present paper, we determine the structure of a finite EAP-group when its centre is of index 4 in G. We also show that self-autopermutability and characteristic properties are equivalent for nilpotent groups.A subgroup  H be a subgroup of a given group G. Then we call H to be autopermutable, if HHα = HαH for all α ∈ Aut(G). The subgroup H is said to be self-autopermutable, if HHα = HαH implies that Hα = H. Moreover, we call the group G to be an EAP-group if every subgroup of G is autopermutable. Clearly, if α runs over the inner automorphisms of the group, we obtain the notions of conjugate-permutability  =  = c3d,  = 1〉;〈a, b, c, d, e∣a2 = b2 = c2 = d2 = e4 = 1, a, b, c ∈ Z(G),  = b〉;〈a, b, c, d, e∣a2 = b2 = c2 = d4 = e4 = 1, a, b, c ∈ Z(G),  = a〉.〈Let G is a 2-group with cyclic centre of index 4.We remind that a nonabelian group is said to be Hamiltonian, if all of its subgroups are normal. The following result gives our claim, when G be a finite 2-group with cyclic centre of index 4. Then G is an EAP-group if and only if G≅Q8 or 〈a, b∣an+12 = b2 = 1, ab = an+12〉, for all n ≥ 3.Let G to be Q8. Since Q8 is Hamiltonian group, the result follows easily. Now assume G = 〈a, b∣an+12 = b2 = 1, ab = an+12〉, n ≥ 3. One can easily check that G contains exactly three proper subgroups of orders 2i, for 1 ≤ i ≤ n + 1. We also observe that the subgroups of orders 2 are autopermutable and as the subgroups of orders 2n+1 are normal, they are also autopermutable. Now, one can check that there are exactly two cyclic and one noncyclic subgroups of orders 2i, 2 ≤ i ≤ n, so that one of the cyclic subgroups is central and hence all the subgroups of G satisfy the required property.Consider the group G is an EAP-group, Z(G) = 〈x∣xn2 = 1〉, G/Z(G) = {Z(G), aZ(G), bZ(G), abZ(G)}, where a2, b2 ∈ Z(G) and so |a|, |b| ≤ 2n+1. In case n = 1, then the group G is either D8 or Q8. As explained before, D8 cannot be an EAP-group and hence G≅Q8. Now suppose n > 1 and the elements a and b are both of order 2. Then every element y ∈ G has the following form (as G is nilpotent of class 2):α given by α(y) = biajxk is an automorphism of G, which sends a into b. Thus HHα ≠ HαH for the subgroup H = 〈b〉, which contradicts the assumption. Now, if |a | , |b | <2n+1 we may replace a and b by the elements axi and bxj, both of which are of order 2. This reduces to the previous case. Therefore we must have a or b of order 2n+1. Then G has a cyclic subgroup of order 2n+1 and so G is of order 2n+2  (n > 1) with the centre of index 4. Hence, by  =  = c3d,  = 1〉;G = 〈a, b, c, d, e∣a2 = b2 = c2 = d2 = e4 = 1, a, b, c ∈ Z(G),  = b〉;G = 〈a, b, c, d, e∣a2 = b2 = c2 = d4 = e4 = 1, a, b, c ∈ Z(G),  = a〉.Let G be an EAP-group and G/Z(G) = {Z(G), aZ(G), bZ(G), abZ(G)}, where a2, b2 ∈ Z(G). Assume that Z(G) is not an elementary abelian 2-group. Since Z(G) is the direct product of its cyclic subgroups, by the same argument as in Z(G) is an elementary abelian 2-group. Clearly G must be a group of order either 16 or 32. The structure of such groups is given as follows in  =  = c3d,  = 1〉.The sufficient condition is obvious. We only need to prove the necessity condition. Let llows in . If |G| D8 is not an EAP-group, hence the group of form (i) cannot be an EAP-group. For the group of form (ii) we can consider H = 〈b〉 and α ∈ Aut(G) which sends a and b into a and ab, respectively. Clearly, HHα ≠ HαH and hence G cannot be an EAP-group. Thus when |a | = |b|, then G is of the form given in either (iii) or (iv).As G| = 32. Then such groups in the list of small groups with elementary abelian centres of index 4 are only of the following forms: G = 〈a, b, c∣a4 = b4 = c2 = 1,  = c,  =  = 1〉;G = 〈a, b, c, d, e∣a2 = b2 = c2 = d4 = e4 = 1, a, b, c ∈ Z(G),  = b〉;G = 〈a, b, c, d, e∣a2 = b2 = c2 = d2 = e2 = 1, c, d ∈ Z(G),  =  = ab,  = 1〉;G = 〈a, b, c, d, e∣a2 = b2 = c2 = d2 = e4 = 1, a, b, c ∈ Z(G),  = b〉;G = 〈a, b, c, d, e∣a2 = b2 = c2 = d4 = e4 = 1, a, b, c ∈ Z(G),  = a〉.Assume |H = 〈b〉 and α ∈ Aut(G), which sends a, b, and c into a, ab, and c, respectively. In case the group G is of form (ii), we consider H = 〈e〉 and α ∈ Aut(G) which sends a, b, c, d, and e into a, ab, ac, be, and ed, respectively. Also if the group is considered to be of form (iii), one may consider H = 〈e〉 and α ∈ Aut(G) which sends a, b, c, d, and e into e, b, c, d, and a, respectively. Now, one can easily check that in these cases HHα ≠ HαH and so G cannot be an EAP-group. Hence, when |G | = 32, then G is of either form (iv) or form (v). The proof is complete.For the group of form (i) we may consider the cyclic subgroup G is a 2-group. If G is not a 2-group, then we may write G = S1 × S2 ⋯ ×Sk, in such a way that S1 is a Sylow 2-subgroup and Si is an abelian Sylow pi-subgroup, where pi is an odd prime number, for 2 ≤ i ≤ k. Clearly, Aut(G)≅Aut(S1) × Aut(S2)×⋯×Aut(Sk) and for any subgroup H of G, H≅H1 × H2 × ⋯×Hk, where Hi ≤ Si for 1 ≤ i ≤ k. Thus H is an autopermutable subgroup of G if H1 is an autopermutable subgroup of S1. This completes the proof.The necessity condition is obvious and Theorems H of a given group G to be weakly characteristic, when Hα ≤ NG(H) implies that Hα = H for all α ∈ Aut(G). Also, given the subgroups H and K, then H satisfies the subcharacteriser condition, if H⊴K implies that NG)Aut((K) ≤ NG)Aut((H), where NG)Aut((K) = {α ∈ Aut(G); Kα = K} ≤ Aut(G). Clearly, if one considers the inner automorphisms of the group then weakly normal and normaliser condition properties are obtained.We call a subgroup p-subgroups of a given group.The following result of  shows thH be a p-subgroup of a group G. Then the following properties are equivalent: H is a self-conjugate-permutable subgroup;H is a weakly normal subgroup;H satisfies the subnormaliser condition.Let In this section, it is shown that self-autopermutable subgroups in nilpotent groups are always characteristic.H be a subgroup of a group G. H is self-autopermutable, then H is weakly characteristic in G.If H is weakly characteristic, then H satisfies the subcharacteriser condition in G.If Let Hα ≤ NG(H), as H⊴NG(H), we have HHα = HαH. Applying the condition that H is self-autopermutable subgroup of the group G, we get Hα = H. By definition, H is weakly characteristic.(i) If K ≤ G, such that H⊴K. We have Hα ≤ Kα = K ≤ NG(H) for every α ∈ NG)Aut((K). Since H is weakly characteristic in G, we have Hα = H. Thus α ∈ NG)Aut((H) and the result is obtained.(ii) Let The following theorem is one of the main results in this section.H be a subgroup of a nilpotent finite group G. If H satisfies the subcharacteriser condition then H is characteristic in G.Let G≅P1 × P2 × ⋯×Pt, where Pi is a Sylow pi-subgroup of G, for 1 ≤ i ≤ t. We may also write H≅H1 × H2 × ⋯×Ht, with Hi ≤ Pi, 1 ≤ i ≤ t. Since H satisfies the subcharacteriser condition in G, one can easily see that Hi satisfies the subcharacteriser condition in Pi. Therefore Hi⊴Pi implies that Hi is characteristic in Pi, which proves the result.Write Finally, we show that self-autopermutability, weakly characteristic, and subcharacteriser conditions are equivalent, for every subgroup of a nilpotent group.H be a subgroup of a finite nilpotent group G. Then H is a self-autopermutable;H is a weakly characteristic;H satisfies the subcharacteriser condition in G.Let The result follows by


M be a lattice module over the multiplicative lattice L. A nonzero L-lattice module M is called second if for each a ∈ L, a1M = 1M or a1M = 0M. A nonzero L-lattice module M is called secondary if for each a ∈ L, a1M = 1M or an1M = 0M for some n > 0. Our objective is to investigative properties of second and secondary lattice modules.Let  L is a complete lattice in which there is defined as a commutative, associative multiplication which distributes over arbitrary joins and has the compact greatest element 1L (least element 0L) as a multiplicative identity (zero). An element a ∈ L is said to be proper if a < 1L. An element p < 1L in L is said to be prime if ab ≤ p implies either a ≤ p or b ≤ p. If 0L is prime, then L is said to be a domain. For a ∈ L, we define p < 1L in L is said to be primary if ab ≤ p implies either a ≤ p or A multiplicative lattice a, b belong to L, (a:Lb) is the join of all c ∈ L such that cb ≤ a. An element e of L is called meet principal if a⋀be = ((a:Le)⋀b)e for all a, b ∈ L. An element e of L is called join principal if ((ae⋁b):Le) = a⋁(b:Le) for all a, b ∈ L. e ∈ L is said to be principal if e is both meet principal and join principal. e ∈ L is said to be weak meet (join) principal if a⋀e = e(a:Le)  (a⋁(0L:Le) = (ea:Le)) for all a ∈ L. An element a of a multiplicative lattice L is called compact if a ≤ ⋁bα implies a ≤ bα1⋁bα2⋁⋯⋁bαn for some subsets {α1, α2,…, αn}. If each element of L is a join of principal (compact) elements of L, then L is called a PG-lattice (CG-lattice). If L is a CG-lattice and p is a primary element, then If M be a complete lattice. Recall that M is a lattice module over the multiplicative lattice L or simply an L-module in case there is a multiplication between elements of L and M, denoted by lB for l ∈ L and B ∈ M, which satisfies the following properties:lb)B = l(bB);(αlα)(⋁βBβ) = ⋁α,βlαBβ; is the join of all a ∈ L such that aK ≤ N. Particularly, (0M:L1M) is denoted by ann(M). If a ∈ L and N ∈ M, then (N:Ma) is the join of all H ∈ M such that aH ≤ N. An element N of M is called meet principal if (b∧(B:LN))N = bN∧B for all b ∈ L and for all B ∈ M. An element N of M is called join principal if b∨(B:LN) = ((bN∨B):LN) for all b ∈ L and for all B ∈ M. N is said to be principal if it is both meet principal and join principal. In special cases, an element N of M is called weak meet principal  if (B:LN)N = B∧N  ((bN:LN) = b∨(0M:LN)) for all B ∈ M  . N is said to be weak principal if N is both weak meet principal and weak join principal.Let M be an L-module. An element N in M is called compact if N ≤ ⋁αBα implies N ≤ Bα1∨Bα2∨⋯∨Bαn for some subsets {α1, α2,…, αn}. The greatest element of M will be denoted by 1M. If each element of M is a join of principal (compact) elements of M, then M is called a PG-lattice module (CG-lattice module).Let M be an L-module. An element N ∈ M is said to be proper if N < 1M. For all elements N of M,  is a set of all K ∈ M such that N ≤ K ≤ 1M and  is an L-lattice module with a · K = aK∨N for all a ∈ L and K ∈ M such that N ≤ K.Let For various characterizations of lattice modules, the reader is referred to –9.L-lattice module M is called second if for each a ∈ L, a1M = 1M or a1M = 0M.A nonzero L-lattice module M is called secondary if for each a ∈ L, a1M = 1M or an1M = 0M for some n > 0.A nonzero Z be the integers, let Q be the rational numbers, and let Q be Z-module. Suppose L = L(Z) is the set of all ideals of Z and M = L(Q) is the set of all submodules of Q. Thus, M as L-lattice module is a second module, since for every integer n ∈ Z, (nZ)Q = Q or (nZ)Q = 0.Let Every second lattice module is a secondary lattice. But the converse is not true. For this, we can give the following example.Z be the integers and let Z4 be Z-module. Suppose that L = L(Z) is the set of all ideals of Z and M = L(Z4) is the set of all submodules of Z4. Thus, M as L-lattice module is a secondary lattice module, which is not a second lattice module.Let Z be the integers and L = L(Z) the set of all ideals of Z. Thus, L as L-lattice module is neither a second lattice module nor a secondary lattice module.Let L be a CG-lattice and let M be a nonzero L-lattice module. If for each compact a ∈ L, a1M = 1M or a1M = 0M, then M is a second L-lattice module.Let r ∈ L. Since L is a CG-lattice, then we have r = ⋁ici such that ci's are compact elements of L. Then, we obtain r1M = (⋁ici)1M = ⋁ici1M. We have two cases.Let Case 1. If ci1M = 0M for each compact ci ∈ L, then we have r1M = (⋁ici)1M = ⋁ici1M = 0M. Case 2. If ci1M = 1M for some compact ci ∈ L, then we have r1M = (⋁ici)1M = ⋁ici1M = 1M.r1M = 1M or r1M = 0M for each r ∈ L. Consequently, M is second.Hence, M is a second L-lattice module, then ann(M) = (0M:L1M) = p is a prime element of L. In this case, M is called p-second lattice module.If M is a second L-lattice module. Clearly, ann(M) = p is a proper element of L. Let ab ≤ p and assume that b≰p; that is, b1M ≠ 0M. But M is a second L-lattice module; then b1M = 1M. Since b1M = 1M and ab1M = 0M, then a1M = 0M, which implies that a ≤ p.Suppose that M is a secondary L-lattice module, then ann(M) is a primary element of L.If M is a secondary L-lattice module. Let ab ≤ ann(M) and a ≤ ann(M). Since ab ≤ ann(M) and ab1M = 0M and (b)n1M ≠ 0M for each n > 0. Since M is secondary, we have b1M = 1M. Then ab1M = a1M = 0M, which implies a ≤ ann(M).Suppose that L be a CG-lattice. If M is a secondary L-lattice module, then L. In this case, M is called p-secondary lattice module.Let M be a secondary lattice. Then ann(M) is a primary element of L by Let L be a lattice domain and let M be a nonzero L-module. Then M is a second lattice module with ann(M) = 0L if and only if M is a secondary lattice module with Let M is a second lattice module, then M is a secondary lattice module. Since L is domain, then ⇒: Since M is a secondary lattice module with a ∈ L and assume that a1M ≠ 1M. Since M is a secondary lattice module, then there exists a positive integer n such that an1M = 0M; that is, a1M = 0M. Hence, we obtain M is a second lattice. Clearly, ann(M) = 0L.⇐: Suppose that M be a nonzero L-lattice module. An element 0M ≠ N < 1M is said to be pure element, if aN = a1M⋀N for all a ∈ L.Let L be a CG-lattice, let M be a nonzero L-lattice module, and let N be a pure element of M. If M is a p-secondary lattice module, then  and  are both p-secondary lattice modules.Let M is a p-secondary lattice module. Let s ∈ L. Since M is a secondary lattice module, then either s1M = 1M and in this case s · 1N,1M] is a secondary lattice module.Suppose that r be compact and r is compact, there exists a positive integer n such that rn · 1N,1M] is a secondary lattice module.Let c be compact and c is compact, there exists a positive integer k such that ck · N = ckN = 0M,N] and  are both p-second lattice modules.Let M is a p-second lattice module. Let a ∈ L. Since N is pure, we have aN = N⋀a1M. As M is a second lattice module, then either a1M = 0M or a1M = 1M. This implies that either aN = 0M or aN = N. Hence,  is a second lattice module. Now, we show that ann(M) = ann. Clearly, ann(M) ≤ ann. Let r ≤ ann. Thus, we have rN = 0M. Now we assume that r≰ann(M). Then we obtain r1M = 1M, since M is a second lattice module. This implies that 0M = rN = r1M⋀N = 1M⋀N = N, a contradiction. Therefore, r ≤ ann(M).⇒: Suppose that t ∈ L. Since M is a second lattice module, either t1M = 1M and in this case t · 1N,1M] is a second lattice module. It remains to show that ann(M) = ann. Clearly ann(M) ≤ ann. Let s ≤ ann. Thus, s · 1N,1M] and  are both second lattice modules with ann = ann = p. Let r ∈ L. We have two cases.⇐: Suppose that  is a second lattice module. Hence, we have N = rN = N⋀r1M, that is, N ≤ r1M, since N is pure. Because  is a second lattice module and r≰ann; we obtain r · 1N,1M][, that is; s · 1M = N. Thus, s1M⋁N = N, and so s1M ≤ N. Now, we assume that s≰ann(M). Then, we have s1M = 1M, since M is second. Hence, 1M = s1M ≤ N, a contradiction. Consequently, we have s ≤ ann(M).Now we show that L-module M is called a multiplication lattice module if for every element N ∈ M, there exists an element a ∈ L, such that N = a1M.An N of an L-module M is called prime element if N ≠ 1M and whenever r ∈ L and X ∈ M with rX ≤ N, then X ≤ N or r ≤ (N:L1M).A element N of an L-module M is called semiprime element if N ≠ 1M and whenever r ∈ L and X ∈ M with r2X ≤ N, then rX ≤ N.A element N be a proper element of an L-module M. Then N is a semiprime element if and only if whenever r ∈ L, X ∈ M and k is a positive integer with rkX ≤ N, then rX ≤ N.Let We know that a prime element is semiprime, but the converse is not true in general. The following proposition shows that the converse is true when the module is secondary and multiplication.M be a multiplication and secondary L-lattice module. For all element N of M such that 1M ≠ N ∈ M, N is a semiprime element of M if and only if N is a prime element of M.Let N is a semiprime element of M and let rX ≤ N, where r ∈ L, X ∈ M. Since M is a secondary lattice module, then either rn1M = 0M for some positive integer n or r1M = 1M. ⇒: Suppose that Case  1. If rn1M = 0M, then rn1M ≤ N. Since N is a semiprime element, we have r1M ≤ N.Case  2. If r1M = 1M, then we have X = rX, since M is a multiplication lattice module. Then we have rX = X ≤ N. N is a prime element of M.Therefore, ⇐: It is obvious.M be an L-lattice module and let N be a proper element of M. N is called a primary element of M, if whenever a ∈ L, X ∈ M such that aX ≤ N, then X ≤ N or M is nonzero and 0M is primary, then M is said to be primary lattice module.Let L-lattice module M is said to be simple lattice module if M = {0M, 1M}.An Every multiplication secondary lattice module is a primary lattice module.M be a multiplication secondary module and rX = 0M for some r ∈ L, X ∈ M. Now, we assume that M is a secondary module, then we have r1M = 1M. Because M is a multiplication, then we have rX = X. Consequently, we obtain X = 0M.Let Every multiplication second lattice module is a simple lattice module.M be a multiplication and second module. Since M is a multiplication, for every N ∈ M, there exists a ∈ L such that N = a1M. Then we obtain a1M = 1M or a1M = 0M, since M is second. Thus, we have N = 1M or N = 0M for every N ∈ M; that is, M is simple.Let L be a domain and let M be a nonzero L-lattice module. If r1M = 1M for every 0L ≠ r ∈ L, then M is said to be divisible.Let L-lattice module M is said to be torsion if there exists 0L ≠ r ∈ L such that r1M = 0M.A nonzero L be a domain. Let M be a secondary L-lattice module. Then either M is a divisible module or M is a torsion module.Let M is a secondary module over a domain L. If M is not divisible, then there exists 0L ≠ r ∈ L such that r1M ≠ 1M. Since M is a secondary lattice module, then there exists a positive integer n such that rn1M = 0M. Since 0L ≠ r and L is a domain, then we have rn ≠ 0L. Consequently, there exists 0L ≠ rn = s ∈ L such that s1M = 0M. Therefore, M is a torsion lattice module.Suppose that


C-approximating posets, the concept of countably QC-approximating posets is introduced. With the countably QC-approximating property, some characterizations of generalized completely distributive lattices and generalized countably approximating posets are given. The main results are as follows: (1) a complete lattice is generalized completely distributive if and only if it is countably QC-approximating and weakly generalized countably approximating; (2) a poset L having countably directed joins is generalized countably approximating if and only if the lattice σc(L)op of all σ-Scott-closed subsets of L is weakly generalized countably approximating.As a generalization of countably  L is continuous if and only if the lattice σ(L)op of all Scott-closed subsets of L is completely distributive. Gierz et al. in [C-continuous lattices. And they showed that for any poset L, σ(L)op is a C-continuous lattice and that L is continuous if and only if σ(L)op is continuous.The notion of continuous lattices as a model for the semantics of programming languages was introduced by Scott in . Later, t al. in  introduct al. in  introduct al. in  proved tt al. in  introducσ-Scott topology. Yang and Liu in [C-approximating posets and showed that the lattice of all σ-Scott-closed subsets of a poset is a countably C-approximating lattice and that a complete lattice is completely distributive if and only if it is countably approximating and countably C-approximating.On the other hand, Lee in  introducd Liu in  introducd Liu in , 10, Maod Liu in  introducC-approximating posets to the concept of countably QC-approximating posets. With the countably QC-approximating property, we present some characterizations of GCD lattices and generalized countably approximating posets.In this paper, we generalize the concept of countably L, ≤) be a poset. Then L with the dual order is also a poset and denoted by Lop. A principal ideal  is a set of the form ↓x = {y ∈ L∣y ≤ x} . For X⊆L, we write ↓X = {y ∈ L∣∃   x ∈ X, y ≤ x}, ↑X = {y ∈ L∣∃   x ∈ X, x ≤ y}. A subset X is a(n) lower set  if X =  ↓X . The supremum of X is denoted by ∨X or supX. A subset D of L is directed if every finite subset of D has an upper bound in D. A subset D of L is countably directed if every countable subset of D has an upper bound in D. Clearly every (countably) directed set is nonempty, and every countably directed set is directed but not vice versa. A poset L is a directed complete partially ordered set  if every directed subset of L has a supremum. A poset is said to have countably directed joins if every countably directed subset has a supremum.We quickly recall some basic notions and results . Let  to denote the power set of X and Pfin(X) to denote the set of all nonempty finite subsets of X. For a poset L, define a preorder ≤  on P(L)∖{∅} by G ≤ H if and only if ↑H⊆↑G for all G, H⊆L. That is, G ≤ H if and only if for each y ∈ H there is an element x ∈ G with x ≤ y. We say that a nonempty family F of subsets of L is (countably) directed if it is (countably) directed in the Smyth preorder. More precisely, F is directed if for all F1, F2 ∈ F, there exists F ∈ F such that F1, F2 ≤ F; that is, F⊆↑F1∩↑F2.For a set c on points of L to the nonempty subsets of L, one obtains the concept of weakly generalized countably approximating posets.Generalizing the relation ≪L be a poset having countably directed joins. A binary relation ≪c on P(L)∖{∅} is defined as follows. A≪cB if and only if for any countably directed set D⊆L, ∨  D ∈ ↑B implies D ∩↑A ≠ ∅. We write F≪cx for F≪c{x} and y≪cH for {y}≪cH. If for each x ∈ L, ↑x = ∩{↑F∣F ∈ ω(x)}, where ω(x) = {F∣F ∈ Pfin(L) and F≪cx}, then L is called a weakly generalized countably approximating poset. A weakly generalized countably approximating poset which is also a complete lattice is called a weakly generalized countably approximating lattice.Let L with the condition that for each x ∈ L, ω(x) is countably directed is called a generalized countably approximating poset (lattice) in [A weakly generalized countably approximating poset (lattice) tice) in .As a generalization of completely distributive lattice, the following concept of GCD lattices was introduced in .L be a poset. A binary relation ⊲ on P(L) is defined as follows. A⊲B if and only if whenever S is a subset of L for which ∨S exists, ∨S ∈ ↑B implies S ∩↑A ≠ ∅. A complete lattice L is called a generalized completely distributive lattice or shortly a GCD lattice, if and only if for all x ∈ L, ↑x = ∩{↑F∣F ∈ Pfin(L) and F⊲x}.Let U of a poset L is Scott-open if ↑U = U and for any directed set D⊆L, supD ∈ U implies U∩D ≠ ∅. All the Scott-open sets of L form a topology, called the Scott topology and denoted by σ(L). The complement of a Scott-open set is called a Scott-closed set. The collection of all Scott-closed sets of L is denoted by σ(L)op. The topology on L generated by {L∖↓x∣x ∈ L} as a subbase is called the upper topology and denoted by ν(L).A subset σ-Scott-open sets.Replacing directed sets with countably directed sets in L be a poset. A subset U of L is called σ-Scott-open if ↑U = U and for any countably directed set D⊆L, supD ∈ U implies U∩D ≠ ∅. All the σ-Scott-open sets of L form a topology, called the σ-Scott topology and denoted by σc(L). The complement of a σ-Scott-open set is called a σ-Scott-closed set. The collection of all σ-Scott-closed sets of L is denoted by σc(L)op.Let L, the σ-Scott topology σc(L) is closed under countably intersections and the Scott topology σ(L) is coarser than σc(L); that is, σ(L)⊆σc(L).(1) For a poset σ-Scott-closed if and only if it is a lower set and closed under countably directed joins.(2) A subset of a poset is σ-Scott-closed subsets for a poset, Mao and Xu in [C-approximating posets.To study the order structure of the lattice of all nd Xu in  introducL be a poset and x, y ∈ L. We say that x is σ-beneath y, denoted by x≺σy, if for any nonempty σ-Scott-closed set F⊆L for which ∨F exists, ∨F ≥ y always implies that x ∈ F. Poset L is said to be countably C-approximating if for each x ∈ L, x = ∨↓σ≺x, where ↓σ≺x = {y ∈ L∣y≺σx}. A complete lattice which is also countably C-approximating is called a countably C-approximating lattice.Let L, the lattice σc(L)op is countably C-approximating.For a poset L be a poset and C ∈ σc(σc(L)op)op. It is straightforward to check that ⋁σc(L)opC = ∪C. For each F ∈ σc(L)op, we have that F = ⋁σc(L)op{↓x∣x ∈ F}. Suppose C ∈ σc(σc(L)op)op with ⋁σc(L)opC⊇F. Then for each x ∈ F, since ⋁σc(L)opC = ∪C⊇F, there exists A ∈ C such that x ∈ A. Noticing that A ∈ σc(L)op is a lower set, we have ↓x⊆A ∈ C. It follows from C ∈ σc(σc(L)op)op being a lower set that ↓x ∈ C. Thus by x≺σF holds in σc(L)op. Hence, F = ⋁σc(L)op{↓x∣x ∈ F}⊆∨↓σ≺F⊆F. So, F = ∨↓σ≺F and by the arbitrariness of F ∈ σc(L)op, we conclude that σc(L)op is countably C-approximating.Let QC-approximating posets. Firstly, we generalize the relation ≺σ on points of a poset L to the nonempty subsets of L.In this section, we introduce the concept of countably L, the σ-beneath relation ≺σ on nonempty subsets of L is defined as follows: A≺σB if and only if whenever S is a nonempty σ-Scott-closed subset of L for which ∨S exists, ∨S ∈ ↑B implies S ∩↑A ≠ ∅. We write F≺σx for F≺σ{x}. Set c(x) = {F∣F ∈ Pfin(L)  and  F≺σx}.For a poset The next proposition is basic and the proof is omitted.L be a poset. ThenG, H⊆L, G≺σH⇒G ≤ H;∀G, H⊆L, G≺σH⇔∀h ∈ H, G≺σh;∀E, F, G, H⊆L, E ≤ G≺σH ≤ F⇒E≺σF;∀x, y ∈ L, {x}≺σ{y}⇔x≺σy.∀Let σ, we have the concept of countably QC-approximating posets.With the relation ≺L is said to be countably quasi-C-approximating, shortly countably QC-approximating, if for all x ∈ L, ↑x = ∩{↑F∣F ∈ c(x)}. A countably QC-approximating poset which is also a complete lattice is called a countably QC-approximating lattice.A poset C-approximating posets are countably QC-approximating.Countably L be a countably C-approximating poset. Then for all x ∈ L,F∣F ∈ c(x)} = ↑x. By L is countably QC-approximating.Let By L, the lattice σc(L)op is countably QC-approximating.For any poset QC-approximating lattices and GCD lattices.In the sequel, we explore relationships between countably Every GCD lattice is weakly generalized countably approximating.L be a GCD lattice. For all x ∈ L and F ∈ Pfin(L), F⊲x implies F≪cx. Then ↑x⊆∩{↑F∣F ∈ ω(x)}⊆∩{↑F∣F ∈ Pfin(L) and F⊲x} = ↑x. So ↑x = ∩{↑F∣F ∈ ω(x)}. By L is weakly generalized countably approximating.Let QC-approximating.Every GCD lattice is countably L be a GCD lattice. For each x ∈ L and F ∈ Pfin(L), F⊲x implies F≺σx. Then ↑x⊆∩{↑F∣F ∈ c(x)}⊆∩{↑F∣F ∈ Pfin(L) and F⊲x} = ↑x. Thus ↑x = ∩{↑F∣F ∈ c(x)}. By L is countably QC-approximating.Let The following theorem characterizes GCD lattices.L be a complete lattice. Then the following statements are equivalent:L is a GCD lattice;L is countably QC-approximating and weakly generalized countably approximating.Let (1)⇒(2): follows from Propositions L is countably QC-approximating and weakly generalized countably approximating. Then for each x ∈ L, by the weakly generalized countably approximating property of L, we have ↑x = ∩{↑F∣F ∈ ω(x)}. Now for each F ∈ ω(x), we show that ↑F = ∩{↑F′∣F′ ∈ Pfin(L) and F′≺σF}. To this end, it suffices to show that ∩{↑F′∣F′ ∈ Pfin(L) and F′≺σF}⊆↑F. Suppose t ∈ ∩{↑F′∣F′ ∈ Pfin(L) and F′≺σF} and t ∉ ↑F. Then for any yF ∈ F, t ∉ ↑yF. By the countably QC-approximating property of L, there exists FyF ∈ c(yF) such that FyF≺σyF and t ∉ ↑FyF. Let t ∈ ∩{↑F′∣F′ ∈ Pfin(L) and F′≺σF}. Thus ↑x = ∩{↑F∣F ∈ ω(x)} = ∩{↑F′∣F′ ∈ Pfin(L), ∃F ∈ Pfin(L) such that F′≺σF≪cx}.(2)⇒(1): suppose that F′≺σF≪cx, we will show that F′⊲x. For any A⊆L with ∨A ≥ x, let G = {∨E∣E is a countable subset of A}. Then G is a countably directed set and ∨G = ∨A ∈ ↑x. Since F≪cx, there exists a countable subset E⊆A such that ∨E = ∨↓E ∈ ↑F. By E is σ-Scott-closed. It follows from F′≺σF that ↓E∩↑F′ ≠ ∅. This implies A∩↑F′ ≠ ∅, showing that F′⊲x. Thus, ↑x⊆∩{↑W∣W ∈ Pfin(L), W⊲x}⊆∩{↑F′∣F′ ∈ Pfin(L), ∃ F ∈ Pfin(L), F′≺σF≪cx} = ↑x. So, ↑x = ∩{↑W∣W ∈ Pfin(L) and W⊲x}. Therefore, L is a GCD lattice.Suppose L is called a hypercontinuous poset x} is directed and x = sup{y ∈ L∣y≺ν(L)x}, where y≺ν(L)x⇔x ∈ intν(L)↑y. A hypercontinuous poset which is also a complete lattice is called a hypercontinuous lattice.Recall that a poset set (see ) if for L be a complete lattice. Then L is a GCD lattice if and only if Lop is a hypercontinuous lattice.Let L, both L and Lop are continuous, and ν(L) = σ(L). It follows from  is hypercontinuous.Let σc(L)op is a GCD lattice. The following theorem gives comprehensive characterizations of generalized countably approximating posets.So, in view of L be a poset having countably directed joins. Then the following statements are equivalent:L is a generalized countably approximating poset;σc(L) is a hypercontinuous lattice;σc(L)op is a GCD lattice;σc(L)op is a weakly generalized countably approximating lattice.Let (i)⇔(ii) by (ii)⇔(iii) by (iii)⇔(iv) follows from


Z with its two crisp boundaries named upper-bound set and lower-bound set. In this paper, the concept of similarity degree between two interval sets is defined at first, and then the similarity degrees between an interval set and its two approximations  and lower approximation set Z)) are presented, respectively. The disadvantages of using upper-approximation set Z) or lower-approximation set Z) as approximation sets of the uncertain set (uncertain concept) Z are analyzed, and a new method for looking for a better approximation set of the interval set Z is proposed. The conclusion that the approximation set R0.5(Z) is an optimal approximation set of interval set Z is drawn and proved successfully. The change rules of R0.5(Z) with different binary relations are analyzed in detail. Finally, a kind of crisp approximation set of the interval set Z is constructed. We hope this research work will promote the development of both the interval set model and granular computing theory.The interval set is a special set, which describes uncertainty of an uncertain concept or set  Since the twenty-first century, researchers have done more and more research on uncertain problems . It is aU and is induced by an equivalence relation. In the knowledge space, two certain sets named upper approximation set and lower approximation set are used to describe the target concept X as its two boundaries. If knowledge granularity in knowledge space is coarser, then the border region of described target concept is wider and approximate accuracy is relatively lower. On the contrary, if knowledge granularity in knowledge space is finer, then the border region is narrower and approximate accuracy is relatively higher.Rough set theory is a mathematical tool to handle the uncertain information, which is imprecise, inconsistent, or incomplete. The basic thought of rough set is to obtain concepts and rules through classification of relational database and discover knowledge by the classification induced by equivalence relations; then approximation sets of the target concept are obtained with many equivalence classes. Rough set is a useful tool to handle uncertain problems, as well as fuzzy set theory, probability theory, and evidence theory. Because rough set theory has novel ideas and its calculation is easy and simple, it has been an important technology in intelligent information processing –9. The kThe interval set theory is an effective method for describing ambiguous information –12 and cZ is built in a certain knowledge space induced by many conditional attributes, and we find that this approximation set may have better similarity degree with the target concept Z than that of Z is obtained by cut-set with some threshold. And then, the decision-making rules can be obtained through the approximation set instead of Z in current knowledge granularity space. In addition, the change rules of similarity between a target concept Z and its approximation sets are analyzed in detail.In this paper, an approximation set of the target concept The method used is getting the approximation of interval sets with a special approximation degree. With this method, we can use certain sets to describe an interval set in Pawlak's space. Our motivation is to get a mathematical theory model, which can be helpful to promote interval sets development in knowledge acquisition.Z with the different knowledge granularity spaces are discussed in The rest of this paper is organized as follows. In In order to introduce the approximation set of interval set more easily, many basic concepts will be reviewed at first.U be a finite set which is called universal set, and then let 2U be the power set of  U and let interval set Z be a subset of 2U. In mathematical form, interval set Z is defined as Z =  = {Z ∈ 2U∣Zl⊆Z⊆Zu}. If Zl = Zu, Z is a usual classical set.An interval set is a new collection, and it is described by two sets named upper bound and lower bound. The interval set can be defined as follows. Let U be all papers submitted to a conference. After being reviewed, there are 3 kinds of results. The first kind of results is the set of papers certainly accepted and represented by Zl. The second kind of results is the set of papers that need to be further reviewed and represented by Zu − Zl. The last kind of results is the set of papers rejected and represented by U − Zu. Although every paper just can be rejected or accepted, no one knows the final result before further evaluation. Through reviewing, the set of papers accepted by the conference is described as .In order to better explain the interval set, there is an example , 18 as fR⊆A, let us define one unclear binary relationship IND(R) = {∣ ∈ U2,  ∀b ∈ R → b(x) = b(y)}.For any attribute set S = 〈U, A, V, f〉. U is the domain, and A = C ∪ D is the set of all attributes. Subset C is a set of conditional attributes, and D is a set of decision-making attributes. V = ∪r∈AVr is the set of attribute values. Vr describes the range of attribute values r where r ∈ A. f : U × A → V is an information function which describes attribute values of object x in U.A knowledge expression system can be described as S = 〈U, A, V, f〉. For any X⊆U and R⊆A, upper approximation set X on R are defined as follows:U/IND(R) = {X∣ = b(y)))} is the classification of equivalence relation R on U. Upper approximation set and lower approximation set of rough set X on R can be defined in another form as follows:x]R ∈ U/IND(R) and [x]R is an equivalence class of x on relation R. U according to knowledge R; U according to knowledge R. Let  boundary region of target concept X on relation R. Let  positive region of target concept X on relation R. Let  negative region of target concept X on relation R. BNR(X) is a set of objects which just possibly belong to target concept X.A knowledge-expression system is described as A and B be two subsets of domain U, which means A⊆U,  B⊆U. Defining a mapping S : U × U → , that is,  → S, S is the similarity degree between A and B, if S satisfies the following conditions.A, B⊆U,  0 ⩽ S ⩽ 1 (boundedness).For any A, B⊆U,  S = S (symmetry).For any A, B⊆U,  S = 1;  S = 0 if and only if A∩B = ϕ.For any Let Any formula satisfying (1), (2), and (3) is a similarity degree formula between two sets. Zhang et al.  gave outZ =  = {Z ∈ 2U∣Zl⊆Z⊆Zu} be an interval set and let N =  = {N ∈ 2U∣Nl⊆N⊆Nu} be also an interval set. Similarity degree between two interval sets can be defined as follows:S accords with Let Z =  = {Z ∈ 2U∣Zl⊆Z⊆Zu} be an interval set. Let R be an equivalence relation on domain U. Upper approximation set of this interval set Z is defined as Z is defined as Let Zu and inner circle standing for a set Zl represent an interval set Z, and each block represents an equivalence class. The black region represents Zu and inner circle standing for a set Zl represent an interval set Z, and each block represents an equivalence class. The black region represents Figures Z, then the similarity degree between Z and Z, then the similarity degree between Z and If Z? In this paper, the better approximation sets of target concept will be proposed. Let U be a nonempty set of objects. Let Z⊆U, x ∈ Z, and the membership degree of x belonging to set Z is defined asμZR(x) ⩽ 1.If the knowledge space keeps unchanged, is there a better approximation set of the target concept U be a nonempty set of objects, and let knowledge space be U/IND(R). Let Z⊆U,  x ∈ Z, and the membership degree belonging to set Z isRλ(Z) = {x ∈ Z∣μZR(x)⩾λ,  1⩾λ > 0}, then Rλ(Z) is called λ-approximation set of set Z.Let Z =  = {Z ∈ 2U∣Zl⊆Z⊆Zu} and Rλ(Z) = ; then Rλ(Z) is called λ-approximation set of the interval set Z.Let Zu and inner circle standing for a set Zl represent an interval set Z, and each block represents an equivalence class. The black region represents R0.5(Zl), and the whole colored region (black and gray region) represents R0.5(Zu).a, b, c, and d be all real numbers. If 0 < a < b,  0 < c < d, then a/b < (a + d)/(b + c).Let a, b, c, and d be all real numbers. In the numbers, 0 < a < b,  0 < c < d. If a/b⩾c/d, then a/b ⩽ (a − c)/(b − d). If  a/b ⩽ c/d, then a/b⩾(a − c)/(b − d).Let R0.5(Z) and Z, Theorems In order to better understand the similarity degree between U be a finite domain, let Z be an interval set on U, and let R be an equivalence relation on U. Then, Let U/R = {{x1, x2}, {x3, x4}, {x5, x6}}, Zu = {x1, x2, x3, x4, x5}, Zl = {x2, x3, x4}. Then, R0.5(Zu) = {x1, x2, x3, x4, x5, x6}, R0.5(Zl) = {x1, x2, x3, x4}.For example, let S) = (1/2) + (3/(2 × 4)) = 7/8, And then we can have According to x ∈ R0.5(Zl), we have μZlR(x)⩾0.5. That is,(1) There we first proveR is an equivalence relation on U, the classifications induced by R can be denoted as [x1]R, [x2]R,…, [xn]R. Then, R0.5(Zl) = {x∣μZlR(x)⩾0.5} = {x∣μZlR(x) = 1}∪{x∣0.5 ⩽ μXR(x) < 1}. Obviously, x∣0.5 ⩽ μZlR(x) < 1} = [xi1]R ∪ [xi2]R ∪ ⋯∪[xik]R. So, Zl ∪ R0.5(Zl) = Zl ∪ ([xi1]R − Zl)∪([xi2]R − Zl)∪⋯∪([xik]R − Zl) and the intersection set between any two elements in  Zl, ([xi1]R − Zl), ([xi2]R − Zl),…, ([xik]R − Zl) is empty, we have that |Zl ∪ R0.5(Zl)| = |Zl | +|([xi1]R − Zl)|+|([xi2]R − Zl)|+⋯ + |([xik]R − Zl)|. So,Because xi1]R∩Zl | ⩾|[xi1]R − Zl|. In the same way, according to |[xi2]R∩Zl | ⩾|[xi2]R − Zl | ,…, |[xik]R∩Zl | ⩾|[xik]R − Zl| and Because(2) In a similar way with (1), we can have the inequalityFrom (1) and (2), we have Z and its approximation set R0.5(Z) is better than the similarity degree between Z and its lower approximation set U be a finite domain, let Z be an interval set on U, and let R be an equivalence relation on U. IfLet U/R = {{x1}, {x2, x3, x4, x5, x6}, {x7, x8, x9, x10}},  Zu = {x1, x2, x7}, Zl = {x1, x2}. Then, R0.5(Zu) = {x1}, R0.5(Zl) = {x1},For example, let S) = (1/(2 × 2)) + (1/(2 × 3)) = 5/12,And then we can have According to xj1]R, [xj2]R,…, [xjs]R are empty sets. Becausexj1]R∩Zl ≠ ϕ. In the same way, we have [xj2]R∩Zl ≠ ϕ,…, [xjs]R∩Zl ≠ ϕ. Then we have Zl∩R0.5(Zl) = Zl − ([xj1]R ∩ Zl)−([xj2]R ∩ Zl)−⋯ − ([xjs]R ∩ Zl) = Zl − (Zl − R0.5(Zl)). Because the intersection sets between any two elements in [xj1]R∩Zl, [xj2]R∩Zl,…, [xjs]R∩Zl are empty sets, we have Zl∩R0.5(Zl) = Zl − (Zl − R0.5(Zl)), |Zl∩R0.5(Zl)| = |Zl | −|Zl − R0.5(Zl)|, and ∪⋯∪([xjs]R − Zl)). Because the intersection sets between any two elements in ([xj1]R − Zl), ([xj2]R − Zl),…, ([xjs]R − Zl) are empty sets, (1) Let For(2) In a similar way with (1), we can easily obtain the conclusion thatAccording to (1) and (2), the inequality Z and its approximation set R0.5(Z) is better than the similarity degree between Z and its lower approximation set U be a finite domain, Z an interval set on U, and R an equivalence relation on U. If 1⩾λ > 0.5, then Let U/R = {{x1}, {x2, x3}, {x4, x5, x6}},  Zu = {x1, x2, x3, x4, x5}, Zl = {x1, x2, x3}. Then, R0.5(Zu) = {x1, x2, x3, x4, x5, x6}, R0.75(Zu) = {x1, x2, x3},R0.5(Zl) = {x1, x2, x3}, R0.75(Zl) = {x1, x2, x3}.For example, let S) = (3/(2 × 3)) + (5/(2 × 6)) = 11/12, S) = (3/(2 × 3)) + (3/(2 × 5)) = 4/5, And then we can have x ∈ R0.5(Zl), then μZlR(x)⩾0.5, which meansR0.5(Zl) = {x∣μZlR(x)⩾0.5} = {x∣μZlR(x) = 1} ∪ {x∣0.5 ⩽ μZlR(x) < 1}, we can easily get x∣0.5 ⩽ μZlR(x) < 1} = [xi1]R ∪ [xi2]R ∪ ⋯∪[xik]R and Rλ(Zl) = {x∣μZlR(x)⩾λ > 0.5} = {x∣μZlR(x) = 1}∪{x∣0.5 < λ ⩽ μZlR(x) < 1}, and then we can get Rλ(Zl)⊆R0.5(Zl). To simplify the proof, let {x∣0.5 < λ ⩽ μZlR(x) < 1} = [xi1]R ∪ [xi2]R ∪ ⋯∪[xiq]R and q ⩽ k in this paper. So, Zl∩[xi2]R | +⋯+|Zl∩[xik]R|. And(1) For all Zl ∩ [xiq+1]R | +⋯+|Zl ∩ [xik]R | ⩾|([xiq+1]R − Zl)|+⋯ + |([xik]R − Zl)|, according to And because |(2) The inequalityAccording to (1) and (2), the inequality Based on Theorems U be a finite domain, Z an interval set on U, and R an equivalence relation on U. Z⊆2U. IfLet U be a finite domain, Z an interval set on U, and R an equivalence relation on U. if 0.5 ⩽ λ1 < λ2 ⩽ 1, then S)⩾S).Let U/R = {{x1}, {x2, x3}, {x4, x5, x6}, {x7, x8, x9, x10}}, Zu = {x1, x2, x3, x5, x6, x7}, Zl = {x1, x2, x3, x5, x6}. Then, R0.6(Zu) = {x1, x2, x3, x4, x5, x6}, R0.8(Zu) = {x1, x2, x3}, R0.6(Zl) = {x1, x2, x3, x4, x5, x6}, R0.8(Zl) = {x1, x2, x3}.For example, S) = (5/(2 × 6)) + (5/(2 × 7)) = 65/84, S) = (3/(2 × 5)) + (3/(2 × 7)) = 18/35, S) > S). This example is in accordance with the theorem.And then we can have x ∈ Rλ1(Zl), we haveR is an equivalence relation on U, all the classifications induced by R can be denoted by [x1]R, [x2]R,…, [xn]R. We have Rλ1(Zl) = {x∣μZlR(x)⩾λ1} = {x∣μZlR(x) = 1}∪{x∣λ1 ⩽ μZlR(x) < 1} and x∣λ1 ⩽ μZlR(x) < 1} = [xi1]R ∪ [xi2]R ∪ ⋯∪[xiu]R. We can also get Rλ2(Zl) = {x∣μZlR(x)⩾λ2} = {x∣μZlR(x) = 1}∪{x∣λ2 ⩽ μZlR(x) < 1}. Let {x∣λ2 ⩽ μZlR(x) < 1} = [xi1]R ∪ [xi2]R ∪ ⋯∪[xiv]R, where 0.5 ⩽ λ1 < λ2 ⩽ 1 and v ⩽ u. So, ∪[xiu]R), xi1]R, [xi2]R,…, [xiu]R are empty sets, we haveZl ∪ Rλ1(Zl) = Zl ∪ ([xi1]R − Zl)∪([xi2]R − Zl)∪⋯∪([xiv]R − Zl)∪⋯∪([xiu]R − Zl) and because the intersection sets between any two elements in Zl, ([xi1]R − Zl), ([xi2]R − Zl),…, ([xiv]R − Zl),…, ([xiu]R − Zl) are empty sets, we easily havexiv+1]R∩Zl | ⩾|[xiv+1]R − Zl|. According to |[xiv+2]R∩Zl | ⩾|[xiv+2]R − Zl | ,…, |[xiu]R∩Zl | ⩾|[xiu]R − Zl|, we have |Zl∩[xiv+1]R| + ⋯+|Zl∩[xiu]R | ⩾|([xiv+1]R − Zl)|+⋯+|([xiu]R − Zl)|. And based on (1) For any (2) In the same way as  (1), the inequalityS)⩾S) is held. So, the proof of According to (1) and (2), the inequality Z and its approximation set Rλ(Z) is a monotonically decreasing function with the parameter λ, and the similarity degree reaches its maximum value when λ = 0.5.Theorems S) in Pawlak's approximation spaces with different knowledge granularities. In this paper, we focus on discussing how the similarity degree between Z and R0.5(Z) changes when the granules are divided into more subgranules in Pawlak's approximation space. In other words, it is an important issue concerning how S) changes with different knowledge granularities in Pawlak's approximation space.In different Pawlak's approximation spaces with different knowledge granularities, the change rules of the uncertainty of rough set are a key issue , 22. Manx1]R, [x2]R,…, [xn]R be classifications of U under equivalence relation R. Let [x1]R′, [x2]R′,…, [xn]R′ be classifications of U under equivalence relation R′. If R′⊆R, then [xi]R′⊆[xi]R (1 ≤ i ≤ n). And then, U/R′ is called a refinement of U/R, which is written as xj ∈ U, then [xj]R′ ⊂ [xj]R. And then, U/R′ is called a strict refinement of U/R, which is written as U/R′≺U/R.Let R′⊆[x]R is always satisfied, and ∃y ∈ U,  [y]R′ ⊂ [y]R. And then, there must be two or more granules in U/R′ whose union is [y]R. To simplify the proof, we suppose that there is just only one granule which is divided into two subgranules, denoted by [xit1]R′ and [xit2]R′ in U/R′, and other granules keep unchanged.Next, we will analyze the relationship between There are 9 cases, and only 6 cases are possible.U be a finite domain, Z an interval set on U, and R and R′ two equivalence relations on U. Let [xit]R be one granule which is divided into two subgranules marked as [xit1]R′ and [xit2]R′. IfS) ⩽ S).Let (1)xit]R is contained in both positive region of Zl and positive region of Zu. In this case, obviously S) = S) is held.[(2)xit]R is contained in both positive region of Zl and negative region of Zu. In this case, obviously, S) = S) is held.[(3)xit]R is contained in both negative region of Zl and negative region of Zu. In this case, obviously, S) = S) is held.[(4)xit]R is contained in both negative region of Zl and boundary region of Zu. In this case,R ∪ [xi2]R ∪ ⋯∪[xim]R where m⩾k. When [xit]R is in boundary region of Zu, we should further discuss this situation. To simplify the proof, we suppose that there is just only one granule marked as [xit]R in U/R which is divided into two subgranules marked as [xit1]R′ and [xit2]R′ in U/R′. And the other granules keep unchanged.(a)k < t ⩽ m, then [xit]R⊄R0.5(Zu).If (1)xit1]R′⊆R0.5′(Zu), [xit2]R′⊄R0.5′(Zu). From the proof of If [Because [xit1]R′ ∪ [xit2]R′ = [xit]R, [xit1]R′⊆R0.5′(Zu), and [xit2]R′⊄R0.5′(Zu), we havexi11]R′⊆R0.5′(Zu), |[xit1]R′∩Zu | /(|[xit1]R′∩Zu | +|[xit1]R′ − Zu|)⩾0.5, which means |[xit1]R′∩Zu | ⩾|[xit1]R′ − Zu|. According to (2)xit1]R′⊄R0.5′(Zu),  [xit2]R′⊄R0.5′(Zu), thenIf [Because [xit]R⊄R0.5(Zu), the case that [xi11]R′⊆R0.5′(Zu) and [xi12]R′⊆R0.5′(Zu) is impossible.(b)t ⩽ k, then [xit]R⊆R0.5(Zu).If 1 ⩽ (1)xit1]R′⊆R0.5′(Zu) and [xit2]R′⊆R0.5′(Zu), then we can easily haveIf [(2)xit1]R′⊆R0.5′(Zu) and [xit2]R′⊄R0.5′(Zu), thenIf [Because [xit2]R′⊄R0.5′(Zu), we have the following.(i)xit2]R′∩Zu = ϕ, then we have |Zu∩[xit1]R | = |Zu∩[xit]R| and |([xit1]R′ − Zu)|<|([xit]R − Zu)|. Therefore,If [(ii)xit1]R′⊆Zu, then |[xit1]R′∩Zu | = |[xit1]R′|. Therefore,If [Because(iii)xit1]R′⊆BNR′(Zu) and [xit2]R′⊆BNR′(Zu), because [xit1]R′⊆R0.5′(Zu) and [xit2]R′⊄R0.5′(Zu), we haveIf [BecauseLet S) ⩽ S) when(5)xit]R is contained in boundary region of Zl and positive region of Zu. In this case,R is contained in boundary region of Zl and boundary region of Zu.∩Zl|/|[xit2] − Zl| and S)⩾|[xit2]∩Zu|/|[xit2] − Zu|, we easily have S) ⩽ S).According to the proofs of (4) and (5), if  From (1), (2), (3), (4), (5), and (6), Z and its approximation set R0.5(Z) is a monotonically increasing function when the knowledge granules in U/R are divided into many finer subgranules in U/R′, where U/R′ is a refinement of U/R.Z. In this paper, the approximation set R0.5(Z) of target concept Z in current knowledge space is proposed from a new viewpoint and related properties are analyzed in detail.With the development of uncertain artificial intelligence, the interval set theory attracts more and more researchers and gradually develops into a complete theory system. The interval set theory has been successfully applied to many fields, such as machine learning, knowledge acquisition, decision-making analysis, expert system, decision support system, inductive inference, conflict resolution, pattern recognition, fuzzy control, and medical diagnostics systems. It is an important tool of granular computing as well as the rough set which is one of the three main tools of granular computing , 26. In R0.5(Z) of the interval set Z is defined and the change rules of S) in different knowledge granularity spaces are analyzed. These researches show that R0.5(Z) is a better approximation set of Z than both In this paper, the interval set is transformed into a fuzzy set at first, and then the uncertain elements in boundary region are classified by cut-set with some threshold. Next, the approximation set ormation . The fuzormation –32. Receormation , the casormation , and incormation –38. In t


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