() @ Applied General Topology c© Universidad Politécnica de Valencia Volume 14, no. 1, 2013 pp. 85-95 Strongly path-confluent mappings Abdo Qahis and Mohd. Salmi Md. Noorani Abstract In this paper, we introduce a new class of path-confluent mapping, called strongly path-confluent maps. We discuss and study some char- acterizations and some basic properties of this class of mappings. Rela- tions between this class and some other existing classes of mappings are also obtained. Also we study some operations on this class of mappings, such as: composition property, composition factor property, component restriction property and path-component restriction property. 2010 MSC: 54C05, 54C08, 54C10. Keywords: continuum, connectedness, components, path-components, quasi-components, confluent maps, path-confluent maps, quasi- confluent maps, strongly confluent maps, strongly path- confluent maps. 1. Introduction and preliminaries Throughout this work a space will always mean a topological space on which no separation axiom is assumed unless explicitly stated and all mappings are assumed to be continuous. In this paper, we obtain a new kind of path-confluent mapping called, strongly path-confluent. A subset K of a space X is said to be a continuum if K is connected and compact. Using this idea of a con- tinuum, Charatonik introduced and studied the concept of confluent mapping between topological spaces [1] as follows: a mapping f : X −→ Y is said to be confluent provided that for each subcontinuum K of Y and for each com- ponent C of f−1(K) we have f(C) = K. Motivated by Charatonik’s work, we have introduced the notions of quasi-confluent and path-confluent mappings in [9, 10] and studied their basic properties. Recall that space X is said to be connected between two subsets A and B if there is no closed-open set F such that A ⊂ F and B ∩ F = φ. The connectedness of a space X between points 86 A. Qahis and M. S. M. Noorani is an equivalence relation on X. The equivalence classes of this relation are called quasi-components of the space X, that is, a quasi-component of a space X containing a point p ∈ X is the set of all points x ∈ X such that the space X is connected between {p} and {x}. In other words, a quasi-component of a space X containing a point p ∈ X is the intersection of all closed and open subsets of X containing p (see [7]). Moreover a mapping f : X −→ Y is said to be quasi-confluent provided that for each continuum K subset of Y and for each quasi-component QC of f−1(K) we have f(QC) = K. The notion of path-component of a point p ∈ X is a maximal path-connected subset of X containing p and it is denoted by PC(X, p) (see [5]). By using this notion we introduced the appropriate definition for path-confluent mapping as follows: A mapping f : X −→ Y is said to be path-confluent provided that for each continuum K subset of Y and for each path-component PC of f−1(K) we have f(PC) = K. In this paper we are interested in further generalization of the work of Chara- tonik in the context of path-components and connectedness. In Section 2 we introduce the notion of strongly path-confluent mapping and study some char- acterizations and some basic properties of this class of mappings. Also we study its relation with other known classes of generalized confluent mappings, namely the classes of confluent, quasi-confluent, path-confluent, and strongly confluent mappings. In Section 3 we study the composition property and com- position factor property for this class. In Section 4 we study the notion of path-component restriction property for the class of strongly path-confluent mappings. We denote by C, QC(or QT ), and PC(or PT ) the components, quasi- components, and path-components of any topological spaces X at any point x ∈ X, respectively, and the symbol N is used for the set of natural numbers. Now we recall some known notions, definitions which will be used in this work. Definition 1.1 ([4]). A mapping f : X −→ Y is said to be strongly confluent provided for each connected non-empty subset K of Y and for each component C of f−1(K) we have f(C) = K. Definition 1.2 ([11]). A mapping f : X −→ Y is a local homeomorphism if for each point x ∈ X there exists an open neighborhood U of x such that f(U) is an open neighborhood of f(x) and the restriction mapping f | U : U −→ f(U) is a homeomorphism. Definition 1.3 ([2]). A class M of mappings between topological spaces is said to have the component restriction property provided that for each mapping f : X −→ Y ∈ M and for each B subset of Y , if A ⊂ X is the union of some components of f−1(B), the restriction mapping f | A : A −→ f(A) ∈ M. Strongly path-confluent mappings 87 2. Strongly path-confluent mappings In this section we introduce the concept of strongly path-confluent mapping and discuss some of its interesting properties and its relations with other known mappings. Definition 2.1. A mapping f : X −→ Y is said to be strongly path-confluent provided that for each connected K subset of Y and for each path-component PC of f−1 (K) we have f(PC) = K. Proposition 2.2. The following statements are true: (1) every strongly path-confluent mapping is path-confluent; (2) every strongly path-confluent mapping is confluent; (3) every strongly path-confluent mapping is strongly confluent; (4) every strongly path-confluent mapping is quasi-confluent. Proof. The proof comes directly from the definitions. � The following diagram follows immediately from the definitions in which none of these implications is reversible as shown by the following example. - - ? ? � � � � � �� H H H H H Hj � � � � ��* H H H H H HYquasi-confluent path-confluent strongly confluent confluent strongly path-confluent Diagram 2.1 Example 2.3. Let X = {(x, y) : x = 0, and y ∈ [−1, 1]} ∪ {(x, y) : y = sin π x : x ∈ (0, 1]} ⊂ R2 with usual topology and Y = X/R, where R the equivalence relation in X, given by R = {((0, y), (0−y)) : y ∈ [−1, 1]}∪{(p, p) : p ∈ X}. Then the natural projection f : X −→ Y is: (1) confluent; (2) quasi-confluent; (3) strongly confluent. But, it is neither path-confluent nor strongly path-confluent mappings. Proposition 2.4. Let f : X −→ Y be a mapping. If Y is totally disconnected space, then the following items are equivalent: (1) f is strongly path-confluent; (2) f is path-confluent. Proof. (1) =⇒ (2): Follows immediately from the Proposition 2.2. (2) =⇒ (1): The proof comes directly from the fact that in totally disconnected space the classes of connected and continua are the same. � 88 A. Qahis and M. S. M. Noorani Recall that a space X is called almost discrete if every open subset of X is closed; equivalently, if every closed subset of X is open (see [6, 7]). Proposition 2.5. Let f : X −→ Y be a mapping of locally path-connected space X into totally disconnected space Y . If X is almost discrete and Y is Hausdorff, then the following items are equivalent: (1) f is strongly path-confluent; (2) f is strongly confluent; (3) f is path-confluent; (4) f is confluent; (5) f is quasi-confluent. Proof. The proof of the implication (1) =⇒ (2) is obvious. (2) =⇒ (3): Suppose that mapping f : X −→ Y is a strongly confluent. Let K be any continuum subset of Y and PC be any path-component of f−1(K). Then, K is connected subset of Y . Since Y is totally disconnected and Haus- dorff space, then K is closed singleton set. Since, X is locally path-connected and almost discrete space, then the set f−1(K) is locally path-connected. Hence, its components and path-component are the same. Thus, f(PC) = K by assumption. Therefore f is path-confluent mapping. The proofs of the implications (3) =⇒ (4) and (4) =⇒ (5) are obvious. (5) =⇒ (1): Assume that mapping f : X −→ Y is a quasi-confluent. Let K be any connected subset of Y and PC be any path-component of f−1(K). Since Y is totally disconnected and Hausdorff space, then K is closed single- ton set. Thus, K is continuum in Y . Since, X is locally path-connected and almost discrete space, then the set f−1(K) is locally path-connected. Hence, its quasi-components and path-components are the same. Thus f(PC) = K by assumption. Therefore f is path-confluent mapping. � Proposition 2.6. Let f : X −→ Y be a mapping of space X into compact space Y . If every connected subset of Y is closed, then the following items are equivalent: (1) f is strongly path-confluent; (2) f is path-confluent. Proof. (1) =⇒ (2) :Obvious. (2) =⇒ (1): Let K ⊆ Y be an arbitrary connected, and PC be an arbitrary path-component of f−1(K). Then K ⊆ Y is closed by the assumption. So, K is compact subset of Y . Thus, K is continuum subset of Y . Then f(K) = PC by assumption. Therefore, f is strongly path-confluent. � Proposition 2.7. If f : X −→ Y is a mapping of space X into compact Hausdorff space Y , then the following statements are equivalent: (1) f is path-confluent; (2) for each closed connected K ⊆ Y , the path-components of f−1(K) are mapped into K under f. Strongly path-confluent mappings 89 Proof. (1) =⇒ (2): Let K ⊆ Y be any closed connected, and PC be any path- component f−1(K). Since, Y is compact space, then K is compact. Thus, K is continuum in Y . So, f(PC) = K by the path-confluence of f. (2) =⇒ (1): Let K be any continuum in Y , and PC be any path-component of f−1(K). Since, Y is Hausdorff space, then K is closed. So, K is closed connected subset of Y . Thus, f(PC) = K. Hence, f is path-confluent mapping. � Recall that a connected space X is said to be σ-connected provided that it can not be decomposed into countably many mutually separated non-empty subsets, (see[3, 8]). Also a space X is said to be hereditarily σ-connected provided that it is connected and each connected subset of it is σ-connected (see[4]). The following theorem shows the hereditarily σ-connected property is strongly path-confluent property. Theorem 2.8. A surjective strongly path-confluent mapping preserves heredi- tarily σ-connected spaces. Proof. Assume that mapping f : X −→ Y be a surjective strongly path- confluent such that X is a hereditarily σ-connected space, and suppose on the contrary that Y is not hereditarily σ-connected. Let K be any connected subset of Y such that K = ∞⋃ i=1 Ai, where Ai and Aj are non-empty mutually separated sets for i 6= j and i, j ∈ N. Then f−1(Ai) and f −1(Aj) are non-empty mutually separated for i 6= j and i, j ∈ N. Let PC be a path-component of f−1(K). Since, f is strongly path- confluent, then we have f(PC) = K. So, we infer that PC ∩ f−1(Ai) 6= φ for i ∈ N Since, PC ⊂ f−1(K) = f−1( ∞⋃ i=1 Ai) = ∞⋃ i=1 f−1(Ai) then, PC = ∞⋃ i=1 (f−1(Ai) ∩ PC), where f−1(Ai)∩PC and f −1(Aj)∩PC are non-empty mutually separated sets for i 6= j and i, j ∈ N. But this contradicts the fact that PC is σ-connected set. Thus, K is σ-connected. Hence, Y is hereditarily σ-connected space. � Theorem 2.9. Let f : X −→ Y be a mapping between topological spaces X and Y such that Y = Y1 ∪ Y2 is a decomposition of Y into connected subsets. If the following properties hold: (1) either Y1 ∩ Y2 6= φ or Y1, and Y2 are separated; (2) the intersection of any two connected subsets of Y is connected; (3) f |f−1(Yi) is strongly path-confluent mapping for i = 1, 2. Then f is strongly path-confluent. 90 A. Qahis and M. S. M. Noorani Proof. Let K be an arbitrary connected subset of Y , and PC be any path- component of f−1(K). Assume that Y = Y1 ∪ Y2, if Y1 ∩ Cl(Y2) = φ = Cl(Y1) ∩ Y2, then either K ⊆ Y1 or K ⊆ Y2. So that by condition (3) we infer that f(PC) = K. Therefore, f is strongly path-confluent mapping. Suppose that Y1 ∩ Y2 6= φ and that K − Y1 6= φ 6= K − Y2. Let x ∈ PC such that f(x) ∈ Y1. Since, K∩Y1 is a connected subset of Y1, if PC1 is a path-component of x in f−1(K ∩ Y1), then by the condition (3), we have (2.1) f(PC1) = K ∩ Y1, and PC1 ⊆ PC. Also, let x′ ∈ PC1 ∩ f −1(y) such that y ∈ K ∩ Y1 ∩ Y2 6= φ. Since, K ∩ Y2 is a connected subset of Y2, if PC2 is a path-component of x ′ in f−1(K ∩ Y2), then by the condition (3), we have (2.2) f(PC2) = K ∩ Y2, and PC2 ⊆ PC. By (2.1) and (2.2) we obtain K = K ∩ (Y1 ∪ Y2) = (K ∩ Y1) ∪ (K ∩ Y2) = f(PC1) ∪ f(PC2) ⊆ f(PC). But, we always have f(PC) ⊆ K. So, f(PC) = K. Hence, f is strongly path-confluent mapping. � The following corollary is a generalization of the Theorem 2.9. Corollary 2.10. If f : X −→ Y be a mapping, and Y1, Y2, ..., Yk are connected subsets of Y such that Y = Y1 ∪ ... ∪ Yk. If the following properties hold: (1) either Yi ∩ Yj 6= φ or Yi, Yj are separated, for each i 6= j and i, j ∈ {1, ..., k}; (2) the intersection of any two connected subsets of Y is connected; (3) f |f−1(Yi) is strongly path-confluent mapping for i ∈ {1, ..., k}. Then f is strongly path-confluent. 3. The composition properties We say that a class M of mappings has the composition property provided that for any two mappings f : X −→ Y and g : Y −→ Z belonging to M then their composition gof belongs to M. Also, we say that a class M of map- pings has the composition factor property provided that for any two mappings f : X −→ Y and g : Y −→ Z such that gof belongs to M, then g belongs to M. Before we prove the main results in this section, we need to introduce the following lemma. Lemma 3.1. Let f : X −→ Y and g : Y −→ Z be two mappings. If f is a strongly path-confluent and h = gof, then for each connected K subset of Z, and each path-component PC of h−1(K), we have f(PC) is a path-component of g−1(K). Strongly path-confluent mappings 91 Proof. Let K be any connected in Z, and PC be any path-component of h−1(K). Since PC ⊆ h−1(K) = (gof)−1(K) = f−1(g−1(K)), then PC ⊆ f−1(g−1(K)). So, f(PC) ⊆ g−1(K). It is obviously that f(PC) ⊆ PT for some path-component PT of g−1(K). Then, PC ⊆ f−1(PT ). Since, f(PC) ⊆ PT ⊆ g−1(K), then PC ⊆ f−1(PT ) ⊆ f−1(g−1(K)) = (gof)−1(K) = h−1(K). Thus, PC is the path-component of f−1(PT ). Now, let Q be any connected in PT . So, f−1(Q) ⊆ f−1(PT ). Since PC is a path-component of f−1(PT ), then, PC ∩ f−1(Q) is the path-component of f−1(Q). But, f is strongly path- confluent mapping. Thus, f(PC ∩ f−1(Q)) = Q = f(PC) ∩ Q. Which implies that Q ⊆ f(PC) ⊆ g−1(K). Hence, f(PC) = PT . Therefore, f(PC) is a path-component of g−1(K). � The following theorem shows that the class of strongly path-confluent map- pings has the composition property. Theorem 3.2. Let f : X −→ Y and g : Y −→ Z be two strongly path-confluent mappings. Then h = gof is strongly path-confluent mapping. Proof. Let K ⊆ Z be a connected and PC be any path-component of h−1(K). Since, f is a strongly path-confluent mapping, then f(PC) is a path-component of g−1(K) by the Lemma 3.1. Then by the strongly path-confluence of g, we infer that h(PC) = g(f(PC)) = K. Hence, h = gof is strongly path-confluent mapping. � Proposition 3.3. Let f : X −→ Y and g : Y −→ Z be two mappings. If f is a homeomorphism and g is a strongly path-confluent mapping, then gof is a strongly path-confluent mapping. Proof. Let K be a connected subset of Z, and PC be the path-component of the inverse image (gof)−1(K). We want to prove that gof(PC) = K. Obviously PC ⊆ (gof)−1(K) = f−1g−1(K). So, f(PC) ⊆ g−1(K). Since, f is a homeomorphism, then f(PC) is the path-component of g−1(K). Since, g is strongly path-confluent, then g(f(PC)) = K. Therefore, gof is strongly path-confluent mapping. � The following theorem shows that the class of strongly path-confluent map- pings has the composition factor property. Theorem 3.4. Let f : X −→ Y and g : Y −→ Z be two mappings. If h = gof is strongly path-confluent mapping, then g is also strongly path-confluent mapping. Proof. Let K be a connected subset of Z and PC be any path-component of g−1(K). Then (3.1) g(PC) ⊆ K. 92 A. Qahis and M. S. M. Noorani On the other hand, since, h is a strongly path-confluent we have that for each x ∈ f−1(PC) h(PC(h−1(K), x)) = K. Then f(PC(h−1(K), x)) ⊆ PC. So, we get (3.2) K = gf(PC(h−1(K), x)) ⊆ g(PC). Then from (3.1) and (3.2), we get g(PC) = K. Hence, g is strongly path- confluent. � Remark 3.5. If h = gof is strongly path-confluent mapping, then f is not necessarily strongly path-confluent mapping as shown by the following example. Example 3.6. Let X = {a, b, c, d, e}, Y = {ℓ, m, n, o} and Z = {p, q, r} with topologies τ = {φ, X, {a}, {c, d}, {a, c, d}, {c, d, e}, {c, d, a, e}}, σ = {φ, Y, {n}, {n, o}}, and γ = {φ, Z}} defined on X, Y and Z, respectively. Let f : X −→ Y be a mapping defined by: f(a) = ℓ, f(b) = m, f(c) = f(d) = n, and f(e) = o. Also let g : Y −→ Z be a mapping defined by: g(ℓ) = g(m) = p, g(n) = r, and g(o) = q. Assume that h = g ◦ f : X −→ Z which is defined by: h(a) = h(b) = p, h(c) = h(d) = r, and h(e) = q. Then the mappings h and g are strongly path-confluent, but f is not strongly path-confluent. Note that, if we take the subcontinuum K = {ℓ, n} of Y . Then the path-components of f−1(K) = {a, c, d} are PC = {a} and PT = {c, d} and f(PC) = ℓ 6= K and f(PT ) = n 6= K. Hence, f is not strongly path-confluent mapping. Now, the following theorem clarifies under certain conditions, the mapping f will be strongly path-confluent. Let a mapping f : X −→ Y be given. Recall that a subset A ⊂ X is said to be an inverse set under f provided that A = f−1(f(A)), (see [1]). Theorem 3.7. Let f : X −→ Y and g : Y −→ Z be two mappings. If h = g ◦f is strongly path-confluent mapping, and if every set in Y is an inverse set, then f is strongly path-confluent mapping. Proof. Let K be any connected set in Y , and let PC be any path-component of f−1(K). Then, g(K) will be connected in Z. Since g−1(g(K)) ⊂ Y , and since, every set in Y is an inverse set, then K is an inverse set and, thus g−1(g(K)) = K. Then f−1g−1(g(K)) = f−1(K), which implies that h−1(g(K)) = (gof)−1(g(K)) = f−1(K). That is, PC is a path-component of h−1(g(K)). Since h is a strongly path-confluent mapping. So, h(PC) = g(K). Therefore, f(PC) = K. Hence, f is a strongly path-confluent mapping. � Strongly path-confluent mappings 93 Corollary 3.8. Let f : X −→ Y and g : Y −→ Z be two mappings. If h = g ◦ f is strongly path-confluent mapping, and g is a homeomorphism then f is strongly path-confluent mapping. Proof. Since, g is a homeomorphism, then every set in Y is an inverse set. Then, by Theorem 3.7, the mapping f is strongly path-confluent. � 4. The path-component restriction property In this section we study the path-component restriction property for the class of strongly path-confluent mappings. Definition 4.1 ([10]). A class M of mappings between topological spaces is said to have the path-component restriction property provided that for each mapping f : X −→ Y ∈ M and for each B subset of Y , if A ⊂ X is the union of some path-components of f−1(B), the restriction mapping f | A : A −→ f(A) ∈ M. The following theorem shows that the class of strongly path-confluent map- pings has the path-component restriction property. Theorem 4.2. Let f : X −→ Y be a strongly path-confluent mapping and let B ⊆ Y and A ⊂ X is the union of some path-components of f−1(B). Then the restriction mapping f | A : A −→ f(A) is a strongly path-confluent. Proof. Assume that the mapping f : X −→ Y is a strongly path-confluent. Take B ⊆ Y , and A is the union of some path-components of f−1(B). Suppose that K be any connected subset of f(A), and let PC and PT be the path- components of (f |A) −1(K) and f−1(K), respectively. Since, (f |A) −1(K) = A ∩ f−1(K), then (4.1) PC ⊂ PT. Since, PC ⊂ A, then φ 6= PC = A ⋂ PC ⊂ A ⋂ PT . But, K ⊂ f(A) ⊂ B. So, f−1(K) ⊂ f−1(f(A)) ⊂ f−1(B). According to the assumption on A, we get PT ⊂ f−1(K) ⊂ A, whence PT ⊂ (f |A) −1(K) ⊂ f−1(K). Which implies that (4.2) PT ⊂ PC. Then from (4.1) and (4.2), we get PC = PT , and f | A(PC) = f(PC) = f(PT ) = K. Therefore the restriction mapping f | A is a strongly path- confluent. � The following corollary is a particular case of the Theorem 4.2. Corollary 4.3. Let f : X −→ Y be a strongly path-confluent mapping. Let A ⊂ X be an inverse set under f, then the restriction mapping f | A : A −→ f(A) is also strongly path-confluent. 94 A. Qahis and M. S. M. Noorani Proposition 4.4. Let f : X −→ Y be a strongly path-confluent mapping. If f is local homeomorphism, then restriction mapping f | U : U −→ f(U) is also strongly path-confluent for every open subset U of X. Proof. Since, f is local homeomorphism, then the restriction mapping f | U : U −→ f(U) is a homeomorphism. Which means that U = f−1(f(U)). Hence, U ⊆ X is an inverse set. By the Corollary 4.3, we infer that f | U is strongly path-confluent mapping. � The following proposition shows that if the mapping f : X −→ Y has the component restriction property, then it also has the path-component restriction property, which means that the path-component restriction property is weaker notion than the notion of component restriction property. Proposition 4.5. Let M denote the class of mappings. If M has the compo- nent restriction property, then it has also the path-component restriction prop- erty. Proof. Assume that class M of mappings has the component restriction prop- erty and let f : X −→ Y ∈ M. Take B be a subset of Y , and A ⊂ X is the union of some components of f−1(B). Then the restriction mapping f | A : A −→ f(A) ∈ M. We need to show that A ⊂ X is the union of some path-components of the set f−1(B). Now, let A = ∪ α∈∆ Cα, for some compo- nents Cα of the inverse set f −1(B), where ∆ be the index set. Since, each com- ponent is a disjoint union of path-components, then we can put Cα = ∪ β∈I PCβ with ∩ β∈I PCβ = φ, for some path-components PCβ of the inverse set f −1(B), where I be the index set. Hence, we get A = ∪ α∈∆ Cα = ∪ α∈∆ ( ∪ β∈I PCβ). Then by the Definition 4.1, the class M of mappings has the path-component restriction property. � Proposition 4.6. The classes of strongly path-confluent mappings has the component restriction property. Proof. Let M be the class of strongly path-confluent mappings, and let f : X −→ Y ∈ M. Take B ⊆ Y , and A is the union of some components of f−1(B). Let K ⊂ f(A) be a subcontinuum, and C and PC be the component and path-component of (f | A) −1(K) = A ∩ f−1(K). Thus, C is contained in a component T of f−1(K). Let PT be the path-component of f−1(K). Obviously, PC ⊂ C and PT ⊂ T . So, PC ⊂ C ⊂ T . Since C ⊂ A, it follows that φ 6= C = A ∩ C ⊂ A ∩ T . Further K ⊂ f(A) ⊂ B, implies that T ⊂ f−1(B). According to the assumptions on A, we infer that T ⊂ A, whence PT ⊂ T ⊆ (f | A) −1(K). Which implies that T = C. Thus PC = PT and consequently (f | A)(PC) = f(PC) = f(PT ) = K by Theorem 4.2. Therefore, f | A is strongly path-confluent mapping. � Strongly path-confluent mappings 95 Acknowledgements. The authors would like to acknowledge the finan- cial support received from Universiti Kebangsaan Malaysia under the research grant UKM TOPDOWN-ST-06-FRGS0001- 2012. The authors also wish to gratefully acknowledge all those who have generously given of their time to referee our paper. References [1] J. J. Charatonik, Confluent mappings and unicoherence of continua, Fund. Math. 56 (1964), 213–220. [2] J. J. Charatonik, Component restriction property for classes of mappings, Mathematica Pannonica 14, no. 1. (2003), 135–143. [3] J. Grispolakis, A. Lelek and E. D. Tymchatyn, Connected subsets of nitely Suslinian continua, Colloq. Math. 35 (1976), 31–44. [4] J. 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(Received May 2012 – Accepted October 2012) Abdo Qahis (cahis82@gmail.com) School of mathematical Sciences, Faculty of Science and Technology,University Kebangsaan Malaysia, 43600 UKM, Selangor Darul Ehsan, Malaysia Mohd. Salmi Md. Noorani (msn@ukm.my) School of mathematical Sciences, Faculty of Science and Technology,University Kebangsaan Malaysia, 43600 UKM, Selangor Darul Ehsan, Malaysia Strongly path-confluent mappings. By A. Qahis and M. S. M. Noorani