RATIO MATHEMATICA 23 (2012), 39–50 ISSN: 1592-7415 Hv-semigroups as noise pollution models in urban areas Achilles Dramalidis, Thomas Vougiouklis Democritus University of Thrace, School of Education, 681 00 Alexandroupolis, Greece adramali@psed.duth.gr, tvougiou@eled.duth.gr Abstract Poor urban planning may give rise to noise pollution, since side-by- side industrial and residential buildings can result in noise pollution in the residential area. In this paper we represent the noise pollution with an Hv-semigroup. More specific, we introduce the concept of right reproductive Hv-semigroup which seems to be a useful tool to study the noise pollution problem in urban areas. Key words: Hv-structures, Hv-group, Hv-semigroup MSC2010: 20N20. 1 Introduction The Hv-structures, introduced in the Fourth AHA Congress [13], where the known axioms were replaced by weaker ones. More precisely in axioms like associativity, commutativity and so on, the equality was replaced by the non-empty intersection. In (H, ·) we abbreviate by WASS, the weak associativity : (xy)z ∩x(yz) 6= ∅, ∀x,y,z ∈ H and by COW, the weak commutativity : xy ∩yx 6= ∅, ∀x,y ∈ H Recall some basic definitions [14]: Definition 1.1 Let H be a non-empty set and · : H×H → P(H)−{∅} be a hyperoperation defined on H. Also, we have abbreviated the ”hyperoperation” by ”hope”[16]. The (H, ·) is called Hv-semigroup if it is WASS and it is called A. Dramalidis, T. Vougiouklis Hv-group if it is Hv-semigroup and the reproduction axiom x · H = H · x = H,∀x ∈ H, is valid. The hyperstructure (R, +, ·) is called Hv-ring if both hopes (+) and (·) are WASS, the reproduction axiom is valid for (+) and (·) is weak distributive with respect to (+): x(y + z) ∩ (xy + xz) 6= ∅, (x + y)z ∩ (xz + yz) 6= ∅,∀x,y,z ∈ R. An Hv-ring (R, +, ·) is called dual Hv-ring [7] if the hyperstructure (R, ·, +) is Hv-ring, too. Let (H, ·) be a hypergroupoid. An element e ∈ H is called right unit element if a ∈ a · e,∀a ∈ H and is called left unit element if a ∈ e ·a,∀a ∈ H. The element e ∈ H is called unit element if a ∈ a ·e∩e ·a,∀a ∈ H. An element e ∈ H is called right scalar unit element if a = a · e,∀a ∈ H and is called left scalar unit element if a = e ·a,∀a ∈ H. It is called scalar unit element if a = a · e = e ·a,∀a ∈ H. Let (H, ·) be a hypergroupoid endowed with at least one unit element. An element a′ ∈ H is called an inverse element of the element a ∈ H, if there exists a unit element e ∈ H such that e ∈ a′ ·a∩a·a′. The element a ∈ H is called right simplifiable element (resp. left ) if ∀x,y ∈ H the following is valid: x ·a = y ·a ⇒ x = y (resp. a ·x = a ·y ⇒ x = y). Moreover, let us define here: If x ∈ x ·y (resp. x ∈ y ·x ) ∀y ∈ H, then, x is called left absorbing-like element (resp. right absorbing-like element ).The nth power of an element h, denoted hs, is defined to be the union of all expres- sions of n times of h, in which the parentheses are put in all possible ways. An Hv-group (H, ·) is called cyclic with finite period with respect to h ∈ H, if there exists a positive integer s, such that H = h1 ∪ h2 ∪ ... ∪ hs. The minimum such an s is called period of the generator h. If all generators have the same period, then H is cyclic with period. If there exists h ∈ H and s positive integer, the minimum one, such that H = hs then H is called single- power cyclic and h is a generator with single-power period s. The cyclicity in the infinite case is defined similarly. Thus, for example, the Hv-group (H, ·) is called single-power cyclic with infinite period with generator h if every ele- ment of H belongs to a power of h and there exists s0 ≥ 1, such that ∀s ≥ s0 we have: h1 ∪h2 ∪ ... ∪hs−1 ⊂ hs . An element a ∈ H is called idempotent element if a2 = a. The main tool to study Hv-groups, is the fundamental relation β*. The re- lation β* is defined in Hv-groups, as the smallest equivalence relation on H, 40 Hv-semigroups as noise pollution models in urban areas such that the quotient would be group. It is called the fundamental group and β* is called the fundamental equivalence relation on H. The relation β is defined on an Hv-group in the same way as in a hypergroup. The basic Theorem is the following [14]: Let (H, ·) be an Hv-group and denote by U the set of all finite products of elements of H. We define the relation β on H by setting xβy iff {x,y} ⊂ u where u ∈ U. Then β* is the transitive closure of β. An element s ∈ H is called single if β*(s) = {s}. The set of all single elements is denoted by SH and if SH 6= ∅ then one can find easily the fun- damental classes. A way to find the fundamental classes is given in [6],[13],[14]. For more definitions, results and applications onHv-structures, see also books [4],[5],[14]. 2 The noise problem Noise pollution is displeasing human, animal or machine-created sound that disrupts the activity or balance of human or animal life. The source of most outdoor noise worldwide is not only transportation systems (including motor vehicle noise, aircraft noise and rail noise), but, noise caused by people as well (audio entertainment systems, electric megaphones and loud people) [9]. The fact that noise pollution is also a cause of annoyance, is that, a 2005 study by Spanish researchers found that in urban areas households are willing to pay approximately 4 Euros per decibel per year for noise reduction [2]. Poor urban planning may give rise to noise pollution, since side-by-side industrial and residential buildings can result in noise pollution in the resi- dential area. We set the following problem: The noise pollution that comes from a cer- tain block of flats in urban areas, obviously annoys not only the block of flats itself but possibly neighboring blocks of flats or buildings, as well. If every city is considered as a set Ω with elements the blocks of flats or the buildings, then, the above situation could be described with an algebraic hyperstructure and its properties. In this paper, we present the right repro- ductive Hv-semigroup, as a tool to study the noise pollution problem in urban areas. One can find hyperstructures on related problems in several survey papers like [10], [11],[12] and papers with a wide variety of applications [1], [3], [15]. 41 A. Dramalidis, T. Vougiouklis Definition 2.1 A hypergroupoid (H,*), such that, the weak associativity holds and ∀x ∈ H, H*x=H , is called right reproductive Hv-semigroup. A hypergroupoid H,*), such that, the weak associativity holds and ∀x ∈ H, x*H=H , is called left reproductive Hv-semigroup Now we give the following definition: Definition 2.2 Let Ω 6= ∅ and f : Ω → P(Ω) be a map, then we define a hyperoperation rL: Ω × Ω → P(Ω), on Ω as follows: ∀x,y ∈ Ω, we set xrLy = f(x) ∪{x} We call the above hyperoperation (rL) noise hyperoperation. Remark that the noise hyperoperation, always contains the element x ∈ Ω. That means that the element x ∈ Ω could be considered as the representative of the elements of the set xrLy. So, we symbolize: xrLy = f(x) ∪{x} = [x] If, ∀x ∈ Ω,x ∈ f(x) then the hyperoperation is simplified as xrLy = f(x) = [x] Therefore, the noise hyperoperation (rL) depends only on the left element. That means that if one composes an element x, on the left, with any other element y, on the right, then the result is always the same set [x]. Example 2.1 Consider a set Ω = {x1,x2,x3,x4,x5,x6,x7,x8,x9} and a map f : Ω → P(Ω) such that: f(x1) = {x2},f(x2) = {x2,x3},f(x3) = {x2},f(x4) = {x4} f(x5) = {x5,x6,x7},f(x6) = {x6,x7},f(x7) = {x5}, f(x8) = {x8,x9},f(x9) = ∅. Then, as in the defined above noise hyperoperation: [x1] = f(x1) ∪{x1} = {x1,x2}, [x2] = f(x2) = {x2,x3}, [x3] = f(x3) ∪{x3} = {x2,x3}, [x4] = f(x4) = {x4}, [x5] = f(x5) = {x5,x6,x7}, [x6] = f(x6) = {x6,x7} [x7] = f(x7) ∪{x7} = {x5,x7}, [x8] = f(x8) = {x8,x9}, [x9] = f(x9) ∪{x9} = {x9}. Then, the ”multiplication” table of (rL) is given by: 42 Hv-semigroups as noise pollution models in urban areas rL x1 x2 x3 x4 x5 x6 x7 x8 x9 x1 x1,x2 x1,x2 x1,x2 x1,x2 x1,x2 x1,x2 x1,x2 x1,x2 x1,x2 x2 x2,x3 x2,x3 x2,x3 x2,x3 x2,x3 x2,x3 x2,x2 x2,x3 x2,x3 x3 x2,x3 x2,x3 x2,x3 x2,x3 x2,x3 x2,x3 x2,x2 x2,x3 x2,x3 x4 x4 x4 x4 x4 x4 x4 x4 x4 x4 x5 x5,x6,x7 x5,x6,x7 x5,x6,x7 x5,x6,x7 x5,x6,x7 x5,x6,x7 x5,x6,x7 x5,x6,x7 x5,x6,x7 x6 x6,x7 x6,x7 x6,x7 x6,x7 x6,x7 x6,x7 x6,x7 x6,x7 x6,x7 x7 x5,x7 x5,x7 x5,x7 x5,x7 x5,x7 x5,x7 x5,x7 x5,x7 x5,x7 x8 x8,x9 x8,x9 x8,x9 x8,x9 x8,x9 x8,x9 x8,x9 x8,x9 x8,x9 x9 x9 x9 x9 x9 x9 x9 x9 x9 x9 Example 2.2 Let X 6= ∅ and m : X → [0, 1] be a fuzzy subset of X. We define the hyperoperation (◦) on X as follows: ∀x,y ∈ X, ◦ : X × X → P(X), such that x◦y = {z ∈ X/m(z) = m(x)} Then, consider the map f(x) = {z ∈ X/m(z) = m(x)}. Since x ∈ f(x),∀x ∈ X, as above we have the hyperoperation (rL) on X as follows : ∀x,y ∈ X, rL : X × X → P(X) , such that xrLy = f(x). 3 The right reproductive Hv-semigroup Some properties of (rL) 1. xrLΩ = [x],∀x ∈ Ω 2. [x]rLy = [x]rL[y] ⊃ [x],∀y ∈ Ω 3. x2 = xrLx = [x],∀x ∈ Ω Proposition 3.1 The hypergroupoid (Ω, rL) is an Hv-semigroup. Proof. We have to prove that the weak associativity holds. Indeed, ∀x,y,z ∈ Ω xrL(yrLz) = ⋃ v∈yrLz (xrLv) = ⋃ v∈[y] (xrLv) = [x] (xrLy)rLz = ⋃ w∈xrLy (wrLz) = ⋃ w∈[x] (wrLz) = ⋃ w∈[x] [w] ⊃ [x] Therefore (xrLy)rLz ⊃ xrL(yrLz), so (xrLy)rLz ∩xrL(yrLz) 6= ∅,∀x,y,z ∈ Ω. 2 43 A. Dramalidis, T. Vougiouklis Proposition 3.2 ∀x ∈ Ω, ΩrLx = Ω and xrLΩ = [x]. Proof. ∀x ∈ Ω, ΩrLx = ⋃ ω∈Ω (ωrLx) = ⋃ ω∈Ω [ω] = Ω. On the other hand, ∀x ∈ Ω,xrLΩ = ⋃ ω∈Ω (xrLω) = [x].2 By propositions (3.1) and (3.2), we get that: Proposition 3.3 The hypergoupoid (Ω,rL) is a right reproductive Hv-semigroup. Remark that the right reproductive Hv-semigroup (Ω,rL) is an Hv-group if, ∀x ∈ Ω, we have xrLΩ = Ω. Proposition 3.4 The strong associativity of (rL) is valid iff we have⋃ w∈[x] (wrLz) = [x],∀x,z ∈ Ω Proof. Let (x,y,z) ∈ Ω3, such that (xrLy)rLz = xrL(yrLz), then (xrLy)rLz = xrL(yrLz) ⇒ [x]rLz = xrL[y] ⇒ [x]rLz = [x] ⇒ ⋃ w∈[x] (wrLz) = [x]. Now,let (x,y,z) ∈ Ω3, such that⋃ w∈[x] (wrLz) = [x] then, (xrLy)rLz = [x]rLz = ⋃ w∈[x] (wrLz) = [x] xrL(yrLz) = xrL[y] = [x].2 For the hyperoperation (rL), we shall check conditions such that the strong or the weak commutativity is valid. Proposition 3.5 . If y ∈ [x] and x ∈ [y],∀x,y ∈ Ω, then the weak com- mutativity of (rL) is valid. The strong commutativity of (rL) is valid, iff [x] = [y],∀x,y ∈ Ω. 44 Hv-semigroups as noise pollution models in urban areas Proof. Let y ∈ [x] and x ∈ [y],∀x,y ∈ Ω, then y ∈ [x] and x ∈ [y] ⇒ x,y ∈ [x] and x,y ∈ [y] ⇒ [x] ∩ [y] 6= ∅ ⇒ ⇒ (xrLy) ∩ (yrLx) 6= ∅. The proof for the strong commutativity is straightforward.2 Proposition 3.6 Let (Ω, +,rL) be an Hv-ring. If xrLΩ = Ω,∀x ∈ Ω then the hyperstructure (Ω, +,rL) is a dual Hv-ring, i.e. both (Ω, +,rL) and (Ω,rL, +) are Hv-rings. Proof. From the remark of proposition 3.3, we have that the (Ω,rL) is an Hv-group. For the weak distributivity of (+) with respect to (rL) we have: ∀x,y,z ∈ Ω x + (yrLz) ⊃ x + y and (x + y)rL(x + z) = ⋃ s∈x+y,t∈x+z (srLt) ⊃ ⋃ s∈x+y s = x + y So, [x + (yrLz)] ∩ [(x + y)rL(x + z)] 6= ∅,∀x,y,z ∈ Ω Similarly, the weak distributivity of (+) with respect to (rL) from the right side. 2 4 Special elements Since x ∈ xrLy,∀x,y ∈ Ω, the next proposition is obvious: Proposition 4.1 (a) All the elements of Ω are right unit elements with respect to (rL). (b) All the elements of Ω are left absorbing-like elements with respect to (rL). Proposition 4.2 The left scalar elements of the Hv-semigroup (Ω,rL), are left absorbing elements. Proof. Let u ∈ Ω be a left scalar unit element, then urLx = x,∀x ∈ Ω. But since u ∈ urLx,∀x ∈ Ω, we get that urLx = u,∀x ∈ Ω.2 Proposition 4.3 The right scalar unit elements of the Hv-semigroup (Ω,rL), are idempotent elements. 45 A. Dramalidis, T. Vougiouklis Proof. Let α ∈ Ω be a right scalar unit element, then xrLα = x,∀x ∈ Ω. So, αrLα = α ⇒ α2 = α.2 Proposition 4.4 If there exists x ∈ Ω such that f(x)=x or [x]=x, then x is left absorbing element and every element of (Ω,rL) is right scalar unit of x. Proof. Suppose there exist x ∈ Ω such that [x]=x, then ∀y ∈ Ω : xrLy = [x] = x. That means that x is left absorbing element and every element of (Ω,rL) is right scalar unit of x.2 Since all the elements of the Hv-semigroup (Ω,rL) are right unit elements, let us denote [7] by IlrL (x,y) the set of the left inverses of the element x ∈ Ω, associated with the right unit y ∈ Ω , with respect to the hyperoperation (rL). The set of the right inverses of the element x ∈ Ω , associated with the right unit y ∈ Ω, with respect to the hyperoperation (rL), is denoted by IrrL (x,y). Proposition 4.5 y ∈ IlrL (x,y) Proof. Let x′ ∈ Ω such that x′ ∈ IlrL (x,y) ⇒ y ∈ x ′rLx. But ∀x ∈ Ω the relation y ∈ yrLx is valid. That means that y ∈ IlrL (x,y).2 Proposition 4.6 IrrL (x,y) = Ω if and only if y ∈ [x]. Proof. Let y ∈ Ω be right unit element and x ∈ Ω,then y ∈ [x] ⇔ y ∈ xrLx′,∀x′ ∈ Ω ⇔ x′ ∈ IrrL (x,y),∀x ′Ω ⇔ IrrL (x,y) = Ω.2 Since x ∈ [x],∀x ∈ Ω,the following is obvious. Corollary 4.1 IrrL (x,x) = Ω Remark 4.1 Notice that, according to the example 2.1, the elements x4 and x9 are idempotent elements, since x 2 4 = x4 and x 2 9 = x9. They are, also, left absorbing elements, since x4rLx = x4 and x9rLx = x9,∀x ∈ Ω. Also, taking for example, the element x2 of Ω, notice that I l rL (x,x2) = {x1,x2,x3}, ∀x ∈ Ω. Even more, since x2 ∈ [x1] we get that IrrL (x1,x2) = Ω and I r rL (x1,x1) = Ω. 46 Hv-semigroups as noise pollution models in urban areas 5 Applications As we mentioned above, the noise pollution in urban areas coming from a spot, annoys a certain area in which the noisy spot belongs to. That was the motivation which led to the mathematical expression xrLy = [x], ∀x,y ∈ Ω. That means that if a city is considered as a set Ω with elements its buildings (or spots which could produce noise pollution), then every building (or a spot) x, which is a source of noise pollution, together with any other building (or a spot) y of the city, will affect anyhow the noise pollution area [x], where x ∈ [x] and maybe y. It is clear, that the source of the noise pollution x, could not be seen as the center of a cyclic disk, but as any spot of a certain area which is affected by x. We shall try to explain some of the properties of the noise hyperoperation (rL) developed above, in terms of noise pollution problems in urban areas. The property x ∈ xrLy,∀y ∈ Ω means that the building x, as a source of noise pollution, first of all, annoys the residents of the building x. The property rL[y] = [x] means that the source of noise pollution x together with any region [y] is not only independent on the spots of the region [y] but the noise pollution region remains [x], as well. The property [x]rLy = [x]rL[y] ⊃ [x] means that the noise pollution region that results when either the noise pollution region [x] operates with the spot y or with the region [y], is the same and anyhow this noise pollution region is bigger than [x]. The property xrLy = xrLz means that [x] remains the noise pollution region when x as a source of noise pollution affects any other spot of the city Ω. Continuously, the relation xrLΩ 6= Ω means that, the noise pollution region coming from spot x, can’t affect the whole city Ω. The weak associativity which is expressed by the inclusion on the left parenthesis, i.e. (xrLy)rLz ⊃ xrL(yrLz) actually means that, the noise pollution region coming from the noise pollution region [x] together with any spot, is not only bigger than that one which comes from the noise pollution spot x together with any other region but includes it, as well. An absorbing element, as in the relation αrLx = α, could be considered as a spot surrounded by a wall or a forest, which doesnt annoy any other spot of the city Ω. Since the weak associativity is valid, the concept of transitive closure can be applied here, in order to obtain the fundamental β* classes. The actual meaning of this situation is that the city Ω can be divided, using the noise hyperoperation, in a partition, where every fundamental class does not annoy any other blocks of flats from other fundamental classes. The next example gives an idea: 47 A. Dramalidis, T. Vougiouklis Example 5.1 According to the example 2.1, consider now that Ω is a city where Ω = {x1,x2,x3,x4,x5,x6,x7,x8,x9}. From ”multiplication” table of (rL),we obtain that β*(x1) = {x1,x2,x3}, β*(x4) = {x4},β*(x5) = {x5,x6,x7},β*(x8) = {x8,x9} . So,the fundamental semi-group Ω/β* is: Ω/β* = { {x1,x2,x3},{x4},{x5,x6,x7},{x8,x9} } and the ”multiplication” table is: ◦ x1 x4 x5 x8 x1 x1 x1 x1 x1 x4 x4 x4 x4 x4 x5 x5 x5 x5 x5 x8 x8 x8 x8 x8 In other words and beyond the mathematical content of the present exam- ple, the city Ω was divided into four regions, where every region (fundamental class) does not annoy any other spot belonging to the rest regions. So, one could consider that among the four regions there exists a green park, full of trees, which absorbs the possible noise pollution caused by any of the four regions. Since β*(x4) = {x4}, the element x4 ∈ Ω (spot or building of the city) is a single element and that means that it doesn’t annoy any other spot of the city Ω , so it can be considered as the remotest spot of the city. References [1] N. Antampoufis, s1-Hv-groups, s1-hypergroups and the ∂-operation, Pro- ceedings 10th AHA Congress 2008, Brno - Czech Republic, (2009), 99 112. [2] J. Barreiro, M. Snchez, M. Viladrich-Grau, How much are people willing to pay for silence? A contingent valuation study, Applied Economics, (2005), 37 (11). [3] J. Chvalina, S. Hoskova, Modelling of join spaces with proximities by first-order linear partial differential operators, Italian Journal of Pure and Applied Math., N.21 (2007), 177-190. [4] P. Corsini, Prolegomena of Hypergroup Theory, Aviani Editore, 1993 48 Hv-semigroups as noise pollution models in urban areas [5] P. Corsini, V. Leoreanu, Applications of Hyperstructure Theory, Kluwer Academic Publishers, Boston / Dordrecht / London, 2003. [6] B. Davvaz, A brief survey of the theory of Hv-structures, 8 th AHA Congress, Spanidis Press (2003), 39-70. [7] A. Dramalidis, Dual Hv-rings (MR1413019), Rivista di Matematica Pura ed Applicata, Italy, v. 17, (1996), 55-62. [8] A. Dramalidis, Geometrical Hv-structures, Proc. of the ”Structures el- ements of hyperstructures”, Alexandroupolis, Greece, Spanidis Press, (2005), 41-51. [9] J. M. Field, Effect of personal and situational variables upon noise an- noyance in residential areas, Journal of the Acoustical Society of Amer- ica, (1993), 93: 2753-2763. [10] A. Maturo, Alternative Fuzzy Operations and Applications to Social Sci- ences. International Journal of Intelligent Systems, DOI: 10.1002/ int. 20383. ISSN 1098-111X, (2009), Vol. 24, pp. 1243-1264. [11] A. Maturo, B. Ferri, On some applications of fuzzy sets and commuta- tive hypergroups to evaluation in architecture and town-planning, Ratio Mathematica, 13, (1999), 51-60. [12] A. Maturo, A. Ventre, Multilinear Decision Making, Consensus and As- sociated Hyperstructures, 10th AHA Congress, Brno, Czech Rep. 2008, (2009),181-190. [13] T. Vougiouklis, The fundamental relation in hyperrings. The general hyperfield, Proc. of the 4th AHA, Xanthi (1990), World Scientific, 1991, 203-211. [14] T. Vougiouklis, Hyperstructures and their representations, Monographs, Hadronic Press, USA, 1994. [15] T. Vougiouklis, P. Kambaki, Algebraic Models in Applied Research, Jor- dan Journal of Mathematics and Statistics, (JJMS) 2008 1(1), 78-87. [16] T. Vougiouklis, On the hyperstructures with ∂-hopes, 10th AHA Congress, Brno, Czech Republic 2008, (2009), 281-296. 49 A. Dramalidis, T. Vougiouklis 50