Approach of the value of a rent when non-central moments of the capitalization factor are known: an R application with interest rates following normal and beta distributions Ratio Mathematica ISSN: 1592-7415 Vol. 33, 2017, pp. 5-20 eISSN: 2282-8214 5 On Some Applications of the Vougiouklis Hyperstructures to Probability Theory Antonio Maturo1, Fabrizio Maturo2 3doi: 10.23755/rm.v33i0.372 Abstract Some important concepts about algebraic hyperstructures, especially from a geometric point of view, are recalled. Many applications of the Hv structures, introduced by Vougiouklis in 1990, to the de Finetti subjective probability theory are considered. We show how the wealth of probabilistic meanings of Hv-structures confirms the importance of the theoretical results obtained by Vougiouklis. Such results can be very meaningful also in many application fields, such as decision theory, highly dependent on subjective probability. Keywords: algebraic hyperstructures; subjective probability; Hv structures, join spaces. 2010 AMS subject classification: 20N20; 60A05; 52A10. 1 Department of Architecture, University of Chieti-Pescara, Italy; antomato75@gmail.com 2 Department of Business Administration, University of Chieti-Pescara, Italy; f.maturo@unich.it 3 ©Antonio Maturo and Fabrizio Maturo. Received: 15-10-2017. Accepted: 26-12-2017. Published: 31-12-2017. Antonio Maturo and Fabrizio Maturo 6 1 Introduction The theory of the algebraic hyperstructures was born with the paper (Marty, 1934) at the VIII Congress of Scandinavian Mathematicians and it was developed in the last 40 years. In the book "Prolegomena of hypergroup theory" (Corsini, 1993) all the fundamental results on the algebraic hyperstructures, until 1992, have been presented. A complete bibliography is given in the appendix. A review of the results until 2003 is in (Corsini, Leoreanu, 2003). Perhaps the most important motivation for the study of algebraic hyperstructures comes from the basic text "Join Geometries" by Prenowitz and Jantosciak (1979), which in addition to giving an original and general approach to the study of Geometry, introduces an interdisciplinary vision of Geometry and Algebra, showing how the Euclidean Spaces can be drawn as Join Spaces, i.e. commutative hypergroups that satisfy an axiom called "incidence property". Moreover, various other geometries, such as the Projective Geometries (Beutelspacher, Rosembaum, 1998), are also Join Spaces. Considering, for example, the Affine Geometries, it is seen that associative property is not satisfied in many important geometric spaces. This and other important geometric and algebraic issues have led to the study of weak associative hyperstructures. The theory of such hyperstructures, called Hv- structures, was carried out by Thomas Vougiouklis, who introduced the concept of Hv-structures in the work “The fundamental relation in hyperrings. The general hyperfield” (1991),. presented at the 4th AHA Conference, Xanthi, Greece, 1990. Subsequently Vougiouklis found many fundamental results on the Hv-structures in numerous works (e.g. Vougiouklis, 1991, 1992, 1994a, 1994b; Spartalis, Vougiouklis, 1994). A collection of all the results on the subject until 1994 is in the important book “Hyperstructures and their representations” (Vougiouklis, 1994c). Subsequent insights into Hv-structures were made by Vougiouklis in many subsequent works (1996a, 1996b, 1996c, 1997, 1999a, 1999b, 2003a, 2003b, 2008, 2014), also in collaboration with other authors (Dramalidis, Vougiouklis, 2009, 2012; Vougiouklis et al., 1997; Nikolaidou, Vougiouklis, 2012). From the Hv-structures of Vougiouklis, the idea in the Chieti-Pescara research group was conceived to interpret some important structures of subjective probability as algebraic structures. Some paper on this topic are (Doria, Maturo, 1995, 1996; Maturo, 1997a, 1997b, 1997c, 2000a, 2000b, 2001a, 2001b, 2003c, 2008, 2010). The study of applications of hyperstructures to the treatment of uncertainty and decision-making problems in Architecture and Social Sciences begins with On Some Applications of the Vougiouklis Hyperstructures to Probability Theory 7 a series of lectures held at the Faculty of Architecture in Pescara by Giuseppe Tallini in 1993, on hyperstructures seen from a geometric point of view, and was developed at various AHA conferences (Algebraic Hyperstructures and Applications) as well as various seminars and conferences with Piergiulio Corsini from 1994 to 2014. For example, in December 1994 and October 1995, two conferences on "Hyperstructures and their Applications in Cryptography, Geometry and Uncertainty Treatment" were organized by Corsini, Eugeni and Maturo, respectively in Chieti and Pescara, with which it was initiated the systematic study of the applications of hyperstructures to the treatment of uncertainty and Architecture. In (Corsini, 1994), it is proved that the fuzzy sets are particular hypergroups. This fact leads us to examine properties of fuzzy partitions from a point of view of the theory of hypergroups. In particular, crisp and fuzzy partitions given by a clustering could be well represented by hypergroups. Some results on this topic and applications in Architecture are in the papers of Ferri and Maturo (1997, 1998, 1999a, 1999b, 2001a, 2001b). Applications of hyperstructures in Architecture are also in (Antampoufis et al., 2011; Maturo, Tofan, 2001). Moreover, the results on fuzzy regression by Fabrizio Maturo, Sarka Hoskova-Mayerova (2016) can be translate as results on hyperstructures. A new research trend concerns the applications of hypergroupoid to Social Sciences. Vougiouklis, in some of his papers (e.g. 2009, 2011), propose hyperstructures as models in social sciences; Hoskova-Mayerova and Maturo analyze social relations and social group behaviors with fuzzy sets and Hv- structures (2013, 2014), and introduce some generalization of the Moreno indices. 2 Fundamental Definitions on Hyperstructures Let us recall some of the main definitions on the hyperstructures that will be applied in this paper to represent concepts of Logic and Subjective Probability. For further details on hyperstructure theory, see, for example, (Corsini, 1993; Corsini, Leoreanu, 2003; Vougiouklis, 1994c). Definition 2.1 Let H be a non-empty set and let *(H) be the family of non-empty subsets of H. A hyperoperation on H is a function  HH  *(H), such that to every ordered pair (a, b) of elements of H associates a non-empty subset of H, noted ab. The pair (H, ) is called hypergroupoid with support H and hyperoperation . Antonio Maturo and Fabrizio Maturo 8 If A and B are non-empty subsets of H, we put AB = {ab: aA, bB}. Moreover, a, bH, we put, aB = {a}B and Ab = A{b}. Definition 2.2 A hypergroupoid (H, ) is said to be: • a semihypergroup, if a, y, cH, a(bc) = (ab)c (associativity); • a quasihypergroup, if aH, aH = H = Ha (riproducibility); • a hypergroup if it is both a semihypergroup and a quasihypergroup; • commutative, if a, bH, ab = ba; • idempotent, if aH, aa={a}. • weak associative, if a, b, cH, a(bc)  (ab)c  ; • weak commutative, if a, bH, ab  ba ≠. The weak associative hypergroupoid, called also Hv-semigroup by Vougiouklis (1991), appear to be particularly significant in the Theory of Subjective Probability, and all results found by Vougiouklis in later papers (e.g.1992, 1994a, 1994b), should have important logic and probabilistic meanings. Vougiouklis (1991) introduced also the notation “Hv-group” for the weak associative quasihypergroups. A Hv-semigroup is said to be left directed if a, b, cH, a(bc)  (ab)c, and right directed if a(bc)  (ab)c. Let (H, ) a hypergroupoid. Using a geometric language, a singleton {a}, aH, is said to be a block of order 1 (briefly 1-block) generated by a. Every hyperproduct ab, a, bH, is a block of order 2 (2-block), called block generated by (a, b). For every a1, a2, a3 H, the hyperproducts a1  (a2  a3) and (a1  a2)  a3 are the 3-blocks generated by (a1, a2, a3). For recurrence, for every a1, a2, …, anH, n > 2, the blocks generated by (a1, a2, …, an) are the hyperproducts AB, with A block of order s < n, generated by (a1, …, as), and B block of order n-s generated by (as+1, …, an). In general, for every n > 1, a block of order n (or n-block) is a hyperproduct AB, with A block of order s < n, B block of order n - s. For every nN, let n be the set of all the blocks of order n, and let 0 = {n, nN}. Then for every nN0, a geometric space (H, n) is associated to the hypergroupoid (H, ). A polygonal with length m of (H, n) is a n-tuple (A1, A2, …, Am) of blocks of n such that Ai  Ai+1 ≠ . Let n be the set of all the polygonals of (H, n). The relation n and n* are defined as: a, bH, a n b   An: {a, b} A, a, bH, a n* b   Pn: {a, b} P. n is reflexive and symmetric, n* is the transitive closure of n. For n = 0 we have the classical relations  and * considered in many papers, e.g. (Freni,1991; Corsini, 1993; Gutan, 1997; Vougiouklis 1999b), On Some Applications of the Vougiouklis Hyperstructures to Probability Theory 9 A more restrictive condition than the weak associativity is the “strong weak associativity”, called also feeble associativity. Definition 2.3 A hypergroupoid (H, ) is said to be feeble associative if, for every a1, a2, …, anH, the intersection of all the blocks generated by (a1, a2, …, an) is not empty. If (H, ) is a commutative quasihypergroup, the -division is defined in H, as the hyperoperation /  HH  *(H) that to every pair (a, b) HH associates the nonempty set {xH: abx}. Definition 2.4 A commutative hypergroup (H, ) is said to be a join space if the following “incidence property” hold: a, b, c, d H, a / b  c / d ≠  ad  bc ≠ . (2.1) A join space (H, ) is: • open, if, a, b H, a ≠ b, a  b  {a, b} = ; • closed, if, a, b H, {a, b}  a  b; • -idempotent, if, aH, a  a = {a}; • / -idempotent, if, aH, a / a = {a}. A join space (H, ) is said to be a join geometry if it is -idempotent and / -idempotent. We have the following theorem. Theorem 2.1 Let H = Rn and  the hyperoperation that to every (a, b)HH associates the open segment with extremes a and b if a ≠ b, and a  a = {a}. (H, ) is a join geometry, called Euclidean join geometry. Let (H, ) be a join geometry. We can note that it is open. Using a notation like that of Euclidean join geometry, in this paper the elements of H are called points and a block ab, with a ≠ b, is called (open) “-segment” with extremes a and b or simply “segment” if only the hyperoperation  is considered in the context. The concept of join space leads to a unified vision of Algebra and Geometry, that can be very useful from the point of view of advanced didactics (Di Gennaro, Maturo, 2002). Also, as some of our papers show, join geometries have important applications in subjective probability. Moreover, we can introduce general uncertainty measures in join geometries such that in the Euclidean join geometries reduce to the de Finetti coherent probability (Maturo, 2003a, 2003b, 2006, 2008; Maturo et al., 2010). Antonio Maturo and Fabrizio Maturo 10 3 Subjective Probability and Hyperstructures Let us recall the concept of coherent probabilty and its geometric representation with the notation given in (Maturo, 2006). The coherence of an assessment of probabilities p = (p1, p2, …, pn) on a n-tuple E = (E1, E2, …, En) of events is defined by an hypothetical bet with a n-tuple of wins S = (S1, S2, …, Sn) (de Finetti, 1970; Coletti, Scozzafava, 2002; Maturo 2006). For every i {1, 2, …, n} an individual A, called the better, pays the stake piSi to an individual B, called the bank, and, if the event Ei occurs, A receives from B the win Si. If Si < 0 the verse of the bet on Ei is inverted, i. e. B pays the stake and A pays the win. The total random gain GA of A is given by the formula: GA, p, S = (|E1| – p1) S1 + (|E2| – p2) S2 + … + (|En| – pn) Sn. (3.1) where |Ei| = 1 if the event Ei is verified and |Ei| = 0 if the event Ei is not verified. The atoms associated with the set of events E = {E1, E2, …, En} are the intersections A1A2…An, where Ai{Ei, -Ei}, different from the impossible event . Let At(E) be the set of the atoms. Then GA(p, S) can be interpret as the function: GA, p, S: a = A1A2…An  At(E)  (|E1| – p1) S1 + (|E2| – p2) S2 + … + (|En| – pn) Sn. (3.2) Definition 3.1 The probability assessment p = (p1, p2, …, pn) on the n- tuple E = (E1, E2, …, En) of events is said to be coherent if, for every S = (S1, S2, …, Sn)  R n, there are a, bAt(E) such that GA, p, S(a)  0 and GA, p, S(b)  0. We note that the previous definition implies a hyperoperation. Let  be an algebra of events containing the set E. Then  also contains At(E) and we can define the hyperoperation  on : : (A, B) At(A, B). (3.3) The above considerations show that it may be important, in a probabilistic context, to know the properties of the algebraic hyperstructure (, ), introduced in (Doria, Maturo, 2006), and called hypergroupoid of atoms. The coatoms associated with E are the nonimpossible complementary events of the atoms. Let Co(E) be the set of coatoms, and k be the number of atoms. For k = 1, At(E) = {}, where  is the certain event and Co(E) is empty. For k = 2, At(E) = Co(E) and for k >2 the sets At(E) and Co(E) are disjoint and with the some number of elements. For every A, B, C  , we have (we write X Y to denote X  Y): (AB)C = ({X C, X (-C), XAt(A, B)}  {Y C, Y (-C), YCo(A, B)})-{}, On Some Applications of the Vougiouklis Hyperstructures to Probability Theory 11 A(BC) = ({A Z, (-A) Z, ZAt(B, C)}  {A T, (-A) T, TCo(B, C)}-{}, At{A, B, C} = {X C, X (-C), XAt(A, B)}-{} = {A Z, (-A) Z, ZAt(B, C)}-{}. Then: At{A, B, C}  (AB)C  A(BC). Therefore, the following theorem applies: Theorem 3.1 Let  be an algebra of events, and  the hyperoperation defined by (3.3). Then (, ) is a commutative Hv-semigroup. The algebra associated with the set of events E = {E1, E2, …, En}, denoted with Alg(E) is the set containing the impossible event  and all the unions of the elements of At(E), i.e. XAlg(E) iff Y(At(E)) such that X is the union of the elements of Y. If |At(E)|=s, then |Alg(E)| = 2s. Let  be an algebra of events. We can introduce the following hyperoperation on : : (A, B) Alg(A, B) (3.4) The hyperoperation  is commutative, and, since {A, B}  Alg(A, B), (, ) is a quasihypergroup. Moreover At(A, B)  Alg(A, B) and so  is an extension of the operation  and we have: At{A, B, C}  (A  B)  C  A  (B  C). Theorem 3.2 Let  be an algebra of events, and  the hyperoperation defined by (3.4). Then (, ) is a commutative Hv-group. Suppose A, B, C are logically independent events, then |At(A, B| = 4, |Alg(A, B| = 24 = 16, |At(A, B, C)| = 8, Alg(A, B, C)| = 28 = 256. Moreover Alg(A, B) contains ,  and other 7 elements with their complements. If X is one of these elements, then X  C contains , , C, -C and other 12 elements. Then (A  B)  C has 712+4= 88 elements and 168 elements are in Alg(A, B, C) but not in (A  B)  C. So, in general, we can write: At{A, B, C}  Co{A, B, C}  (A  B)  C, A  (B  C)  Alg(A, B, C). Let (H, ) be a join geometry. From the associative and commutative properties, for every a1, a2, …, anH there is only a block a1 a2 …an generated by (a1, a2, …, an) and this block depend only by on the set {a1, a2, …, an} and not on the order of the elements. By the idempotence we can reduce to the case in which a1, a2, …, an are distinct. Definition 3.2 For every A  H, A ≠ , the convex hull of A, in (H, ), is the set [A] = {xH: nN,  a1, a2, …, anA : x  a1 a2 …an}. If A is finite then [A] is said to be the polytope generated by A. Antonio Maturo and Fabrizio Maturo 12 Let E = (E1, E2, …, En) be a n-tuple of events set and let At(E) the set of atoms associated to E. For every a = A1A2…An At(E), let xi(a) =1 if Ai = Ei and xi(a) = 0 if Ai = - Ei. The atom a is identified with the point (x1(a), x2(a), …, xn(a))R n. From definition 3.1, the following theorem applies (Maturo, 2006, 2008, 2009). Theorem 3.3 Let (Rn, ) the Euclidean join geometry. The probability assessment p = (p1, p2, …, pn) on the n-tuple E = (E1, E2, …, En) of events is coherent iff p[At(E)]. The theorem 3.3 opens the way to introduce measures of uncertainty that are different from the probability and coherent with respect non-Euclidean join geometries. We can introduce many possible join geometries. The following is an example. Example 3.1 Let H = Rn and  the hyperoperation that to every (a = (a1, a2, …, an), b = (b1, b2, …, bn))HH associated the Cartesian product of the open segments Ir with extremes ar and br belonging to (R, ). We can prove that (H, ) is a join geometry, called the Cartesian join geometry. Some applications of the Cartesian join geometry to problems of Architecture and Town-Planning are in (Ferri, Maturo, 2001a, 2001b). In a general join geometry with support Rn we can introduce the following definition: Definition 3.3 Let (Rn, ) be a join geometry. The measure assessment m = (m1, m2, …, mn) on the n-tuple E = (E1, E2, …, En) of events is said to be coherent with respect to (Rn, ) iff m[At(E)]. For example,  can be the hyperoperation that to every (a, b)HH associates a particular curve with extremes a and b, and the polytope [At(E)] is a deformation of the Euclidean polytope, obtained by replacing the segments with curves. It can have important meanings in appropriate contexts of Physics or Social Sciences. In a generic join geometry (Rn, ) can happen that some of the most intuitive properties of the Euclidean join geometry fall. To avoid this, you should restrict yourself to join geometries where some additional properties apply. Important is the following: Ordering condition. If a, b, c, are distinct elements of Rn, at most one of the following formulas occurs: abc, bac, cab. A join geometry (Rn, ) with the order condition is said to be an ordered join geometry. On Some Applications of the Vougiouklis Hyperstructures to Probability Theory 13 4 Conditional Events, Conditional Probability and Hyperstructures The “axiomatic probability” by Kolmogorov, usually considered as the “true probability” is based on the assessment of a universal set U, whose elements are called the atoms, a -algebra S of subsets of U, whose elements are called the events, and a finite measure p on S, called the probability, such that p(U) = 1. Let S* = S-{}. In the Kolmogorov approach to probability no consideration is given to the logical concept of conditional event E|H, with ES and HS*, but only the conditional probability p(E|H) is defined, only in the case in which p(H) > 0, by the formula: p(E|H) = p(E H)/p(H). (4.1) On the contrary, the “subjective probability” (de Finetti,1970; Dubins, 1975; Coletti, Scozzafava 2002; Maturo, 2003b, 2006, 2008b), don’t consider the events as subsets of a given universal set U, but they are logical propositions that can assume only one of the truth values: true and false. A sharp separation is given among the concepts concerning the three areas of the logic of the certain, the logic of the uncertain and the measure theory. The conditional event E|H is a concept belonging to the logic of the certain and it is a proposition that can assume three values: true if both E and H are verified, false if H is verified but E is not and empty (or undetermined) if H is not verified. The conditional event E|H reduces to the event E if H is the certain event . In the appendix of his fundamental book (1970) de Finetti presents also some different interpretations of the logical concept of three valued proposition. By the point of view of Reichenbach (1942) the value “empty” is replaced by the value “undetermined”. In the following we assume the notation of Reichenbach and we write T for true, F for false and U for undetermined. The set V = {F, U, T} is also ordered by putting F < U < T. A numerical representation of the ordered set V is given by associating 0 to F, 1 to T and the number 1/2 to U. An alternative, in a fuzzy contest, we can associate to U is the fuzzy number u with support and core the interval [0, 1], then the relation 0 < u <1 is a consequence of the usual order relation among the trapezoidal fuzzy numbers. In the subjective probability, the conditional probability p(E|H) of the conditional event E|H is given by an expert and no condition is given about the belonging of the events E and H to a structured set, e.g. like an algebra. The only condition of H ≠ is required, because if H =  we have the totally undetermined conditional event. Antonio Maturo and Fabrizio Maturo 14 If C is a set of conditional events the assessment of a subjective conditional probability to the elements of C must satisfy some coherence conditions. The coherence of an assessment of probabilities p = (p1, p2, …, pn) on a n- tuple K = (E1|H1, E2|H2, …, En|Hn) of conditional events is defined by an hypothetical bet with a n-tuple of wins S = (S1, S2, …, Sn) (de Finetti, 1970; Coletti, Scozzafava, 2002; Maturo, 2006). For every i {1, 2, …, n} an individual A, called the better, pays the stake piSi to an individual B, called the bank, and, • if the event EiHi occurs, A receives from B the win Si; • if the event -Hi occurs, the amount paid piSi is refunded to A; • if the event (-Ei) Hi occurs, no payment is made to A. The total random gain GA of A is given by the formula: GA, p, S = |H1| (|E1| – p1) S1 +…+|Hn|(|En| – pn) Sn. (4.2) where |Ei| = 1 if the event Ei is verified and |Ei| = 0 if the event Ei is not verified, and similarly to H. The atoms associated with the set of conditional events K = {E1|H1, E2|H2, …, En|Hn} are the intersections A1A2…An, where Ai{Ei Hi, -Ei Hi, -Hi}, different from the impossible event . The complement of H = {Hi, i {1, 2, …, n}} is said to be the inactive atom. Let Atc(E) be the set of the atoms associated to K. Then GA, p, S can be interpret as the function: GA, p, S: a = A1A2…An  Atc(E)  (|A1| – p1) S1 + (|A2| – p2) S2 + … + (|An| – pn) Sn (4.3) where |Ai| = 1, 0, pi, if Ai = Ei Hi, -Ei Hi, -Hi, respectively. Definition 4.1 The conditional probability assessment p = (p1, p2, …, pn) on the n-tuple K = (E1|H1, E2|H2, …, En|Hn) of conditional events is said to be quasi-coherent if, for every S = (S1, S2, …, Sn)  R n, there are a, bAtc(E) such that GA, p, S(a)  0 and GA, p, S(b)  0. Moreover, p = (p1, p2, …, pn) is said to be coherent if, for any s  n and for any {i1, i2, …, is}  {1, 2, …, n}, the conditional probability assessment pi1, i2, …, is = (pi1, pi2, …, pis) on (Ei1|Hi1, Ei2|Hi2, …, Eis|His) is quasi-coherent. Let K = (E1|H1, E2|H2, …, En|Hn) be a n-tuple of conditional events and let Atc(K) the set of atoms associated to K = {E1|H1, E2|H2, …, En|Hn}. For every a = A1A2…AnAtc(E), let xi(a) = |Ai|. The atom a is identified with the point (x1(a), x2(a), …, xn(a))R n. From definition 4.1, the following theorems applies: Theorem 4.1 Let (Rn, ) the Euclidean join geometry. The probability assessment p = (p1, p2, …, pn) on the n-tuple K = (E1|H1, E2|H2, …, En|Hn) of conditional events is quasi-coherent iff p[Atc(K)]. On Some Applications of the Vougiouklis Hyperstructures to Probability Theory 15 Theorem 4.2 The probability assessment p = (p1, p2, …, pn) on K = (E1|H1, E2|H2, …, En|Hn) is coherent iff for any s  n and for any {i1, i2, …, is}  {1, 2, …, n}, the conditional probability assessment pi1, i2, …, is = (pi1, pi2, …, pis) on (Ei1|Hi1, Ei2|Hi2, …, Eis|His) belongs to [Atc(Ei1|Hi1, Ei2|Hi2, …, Eis|His)]  Rs. Let  be an algebra of events. An axiomatic formalization of the coherence conditions in the case in which K = {E|H, E, H - {}} is in Dubins (1975). In terms of hyperstructures, conditional events can be defined by the following hyperstructure, introduced in (Doria, Maturo, 1996) and studied in (Maturo, 1997c). Definition 4.2 Let  be an algebra of events. We define on  the hyperoperation: : (E, H)   {E H, H}. We have: E  H  H  E = {E H}; (E  H)  K = {E H K, H K, K}, E  (H  K) = {E H K, H K, E K, K}; E  E = {E}. Then we have the following theorem. Theorem 4.3 The hyperstructure (, ), let us call the hyperstructure of conditional events, is a weak commutative and idempotent Hv-semigroup. Moreover (, ) is right directed, i.e. (E  H)  K  E  (H  K). Any singleton {H} is the conditional event H|H and any set {E, H} with E  H is the conditional event E|H, true if E is verified, false if H is verified but not E, and it is not undetermined if H is not verified. Many other meanings, of the finite subsets of , are considered in (Maturo, 1997c). The coherence conditions of definition 4.1 and theorems 4.1 and 4.2 lead us to associate the n-tuple K = (E1|H1, E2|H2, …, En|Hn) of conditional events with the set all the conditional events A|B with AAt{E1, E2, …, En} and B an union of elements of {H1, H2, …, Hn}. Then, if  is an algebra of events, and    is a set of nonempty events, closed with respect to the union, the following hyperoperation can be introduced: : (E|H, F|K)  ()()  {A|B: AAt{E, F}, B{H, K, HK}}. We can prove the following thorem Theorem 4.4 The hyperstructure (, ) is a commutative Hv- semigroup, called hypergroupoid of conditional atoms and, for  = {}, is isomorphic to (, ). Antonio Maturo and Fabrizio Maturo 16 5 Conclusions and Perspectives of Research We have shown that all logical operations related to subjective probability can reduce to Vougiouklis hyperstructures. (, ) and (, ) are commutative Hv-semigroups, and (, ) is a commutative Hv-group. The hyperoperation  isweak commutative and idempotent and (, ) is a right directed Hv-semigroup. To verify the coherence of a subjective probability assignment p = (p1, p2, …, pn) on the n-tuple E = (E1, E2, …, En) of events, we represent the atoms as points of the space Rn, in which the i-th axis is associated with the event Ei. The assessment p is coherent iff p belongs to the polytope of the join geometry (Rn, ) generate from the atoms. More complex is the coherence check of a conditional probability assessment p = (p1, p2, …, pn) on the n-tuple K = (E1|H1, E2|H2, …, En|Hn) of conditional events, as in this case we must consider polytopes in all the join geometries (Rs, ), s  n associated to subsets of K = {E1|H1, E2|H2, …, En|Hn}. A research perspective is to investigate the properties of the considered Vougiouklis structures, highlighting their meanings from the point of view of logic and subjective probability. A further research perspective is studying the measures that can be obtained by applying the geometric coherence conditions in ordered join geometries other than the Euclidean join geometry. On Some Applications of the Vougiouklis Hyperstructures to Probability Theory 17 References Antampoufis N., Vougiouklis T., Dramalidis A., (2011), Geometrical and circle hyperoperations in urban applications, Ratio Sociologica, Vol 4, N.2, 2011,53-66.- Beutelspacher A., Rosembaum U., (1998), Projective Geometry, Cambridge University Press Coletti G., Scozzafava R., (2002), Probabilistic logic in a coherent setting, Kluwer Academic Publishers, London Corsini P., (1993), Prolegomena of hypergroup theory, Aviani Ed. Udine, (1993). Corsini P., (1994) Join spaces, power sets, fuzzy sets, Proc. of the Fifth International Congress on Algebraic Hyperstructures and Applications, Jasi, Romania, Corsini P., Leoreanu L., (2003), Applications of the Hyperstructure Theory, Kluver Academic Publishers London de Finetti B. (1970), Teoria delle Probabilità, vol. 1 and 2, Einaudi, Torino Di Gennaro F., Maturo A., (2002), La teoria delle iperstrutture: un efficace strumento per una visione unitaria di algebra e geometria, Periodico di Matematiche, Serie VIII, Vol 2, n. 4, 5-16 Doria S., Maturo A., (1995), A hyperstructure of conditional events for artificial intelligence, in Mathematical models for handling partial knowledge in artificial intelligence, Plenum press, New York, pp.201-208, (1995). Doria S., Maturo A., (1996), Hyperstructures and geometric spaces associated to a family of events, in Rivista di Matematica Pura ed Applicata, N 19, 1996, pp.125-137. Dramalidis A., Vougiouklis T., (2003), On a class of geometric fuzzy Hv- structures, 8th AHA, Samothraki, 2002, Spanidis Press, Xanthi, 137-145. Dramalidis A., Vougiouklis T., (2009), Fuzzy Hv-substructures in a two- dimensional Euclidean vector space, 10th AHA, Brno, Czech Republic 2008, (2009), 151-159, and Iranian J. Fuzzy Systems, 6(4), 2009, 1-9. Dramalidis A., Vougiouklis T., (2012), Hv-semigroups as noise pollution models in urban areas, Ratio Mathematica 23, 2012, 39-50. Dubins L. E. (1975), Finitely additive conditional probabilities, conglomerability and disintegrations, The Annals of Probability, 3, 89-99 Ferri B., Maturo A., (1997), Hyperstructures as tool to compare urban projects, in Ratio Mathematica, 12, 1997, 79-89. Ferri B., Maturo A., (1998), An application of the fuzzy set theory to evaluation of urban projects, in New trends in Fuzzy Systems, pp.82-91, Scientific Word, Singapore. Antonio Maturo and Fabrizio Maturo 18 Ferri B., Maturo A., (1999a), Fuzzy Classification and Hyperstructures: An Application to Evaluation of Urban Project, in: Classification and Data Analysis, 55-62, Springer-Verlag, Berlin. Ferri B., Maturo A., (1999b), On Some Applications of Fuzzy Sets and Commutative Hypergroups To Evaluation In Architecture And Town- Planning, Ratio Mathematica, 13, pp. 51-60. Ferri B., Maturo A., (2001a), Mathematical models based on fuzzy classification and commutative hypergroups to solve present problems of evaluation in town planning, in Fuzzy Systems & A. I., Vol. VII, Nos. 1-3, 2001, pp. 7-15. Ferri B., Maturo A., (2001b), Classifications and Hyperstructures in problems of Architecture and Town-Planning, Journal of Interdisciplinary Mathematics, Vol 4(2001), No 1, 25-34. Freni D., (1991), Une note sur le cœur d’un hypergroup et sur la closure transitive * de , in Rivista di Matematica pura e applicata, 8, 153-156 Gutan M, (1997), Properties of hyperproducts and the relation  in quasihypergroups, in Ratio Mathematica, 12, 19-34 Hoskova-Mayerova, S., Maturo, A. (2013). Hyperstructures in Social Sciences, AWER Procedia Information Technology & Computer Science, 3(2013), 547—552, Barcelona, Spain. Hoskova-Mayerova, S., Maturo, A. (2014). An analysis of Social Relations and Social Group behaviors with fuzzy sets and hyperstructures, Proceeding of AHA 2014, accepted in International Journal of Algebraic Hyperstructures and its Applications. Marty F., (1934), Sur une généralization de la notion de group, IV Congres des Mathematiciens Scandinave, Stockholm. Maturo A., (1997a), Probability assessments on hyperstructures of events, in Proceedings of WUPES ’97, Prague, pp. 111-121, (1997). Maturo A., (1997b), Conditional events, conditional probabilities and hyperstructures, in Proceedings EUFIT ’97, September 8-11, pp. 1509/1513, Aachen, Germany. Maturo A., (1997c), On some hyperstructures of conditional events, Proc. of 6th AHA Congress, Prague, 1996, Democritus Un. Press, 115-132 Maturo A. (2000a), Fuzzy events and their probability assessments, Journal of Discrete Mathematical Sciences & Cryptography, Vol. 3, Nos 1-3, 83-94. Maturo A. (2000b), On a non-standard algebraic hyperstructure and its application to the coherent probability assessments, IJPAM, N 7(2000) 33-50. Maturo A., (2001a), Hypergroups and Generalized probabilities, in Advances in Generalized Structures, Approssimate Reasoning and Applications, pp. 13-29, Performantica Press, Iasi, Romania. On Some Applications of the Vougiouklis Hyperstructures to Probability Theory 19 Maturo A., (2001b), Finitely additive conditional probability with values on a hyperstructure, Journal of Information&Optimization Sciences, Vol22 (2001), 227-240 Maturo A., (2003a), Cooperative games, Finite Geometries and Hyperstructures, Ratio Matematica 14, 2003, 57-70 Maturo A. (2003b), Sull'assiomatica di Bruno de Finetti per la previsione e la probabilità coerenti: analisi critica e nuove prospettive, Periodico di Matematiche, Serie VIII, Vol 3, n. 1, 41-54. Maturo A. (2003c), Probability, utility and hyperstructures, 8th AHA, Samothraki, 2002, Spanidis Press, Xanthi, 203-214. Maturo A., (2006), A Geometrical Approach to the Coherent Conditional Probability and its extensions, Scientific Annals of the University Ion Ionescu de la Brad, Tom XLIX, Vol 2., 2006, 243-255. Maturo A., (2008a), Join coherent previsions, Set-valued Mathematics and Applications, Vol 1 No 2 (2008), 135-144. Maturo, A., (2008b), La moderna visione interdisciplinare di Geometria, Logica e Probabilità in Bruno de Finetti, Ratio Sociologica, Vol 1, No2, 39-62. Maturo, A., (2009). Coherent conditional previsions and geometric hypergroupoids, Fuzzy sets, rough sets and multivalued operations and applications, 1, N.1, 51–62. Maturo A., Squillante M., Ventre A.G.S., (2010), Decision making, fuzzy measures an hyperstructures, Advances and Applications in Statistical Sciences, Vol 2, Issue 2, 233-253. Maturo A., Tofan I., (2001), Iperstrutture, strutture fuzzy ed applicazioni, dierre edizioni, San Salvo. Maturo F., Hoskova-Mayerova S., (2016), Fuzzy Regression Models and Alternative Operations for Economic and Social Sciences. In: "Recent Trends in Social Systems: Quantitative Theories and Quantitative Models", series: "Studies in Systems, Decision and Control". Springer International Publishing (Verlag). (2016), 235–248. Nikolaidou P., Vougiouklis T., (2012), Hv-structures and the Bar in questionnaires, Italian J. Pure and Applied Math. N.29, 2012, 341-350. Prenowitz W., Jantosciak J., (1979), Join Geometries, Springer-Verlag UTM, New York, (1979). Reichenbach H., (1942), Philosophic foundation of quantum mechanics, General Publish Compaby, Toronto. Ross T. J., (1997), Fuzzy Logic with engineering applications, MacGraw Hill. Spartalis S., Vougiouklis T., (1994), The fundamental relations on Hv- rings, Rivista Mat. Pura ed Appl., N.14 (1994), 7-20. Antonio Maturo and Fabrizio Maturo 20 Vougiouklis T., (1991), The fundamental relation in hyperrings. The general hyperfield, 4th AHA, Xanthi, Greece (1990), World Scientific (1991), 203-212 Vougiouklis T., (1993), Representations of Hv-structures, Proc. of the Int.Conf. on Group Theory 1992, Timisoara (1993), 159 -184. Vougiouklis T., (1994a), Hv-modulus with external P-hyperoperations, 5 th AΗΑ, Iasi, (1993), Hadronic Press (1994), 191-197, by T.Vougiouklis, A. Dramalidis Vougiouklis T., (1994b), Hv-vector spaces, 5th Int. Congress on A.Η.Α., Iasi, (1993), HadronicPress (1994), 181-190. Vougiouklis T., (1994c), Hyperstructures and their representations, Hadronic Press, U.S.A. Vougiouklis T., (1996a), Construction of Hv-structures with desired fundamental structures, Pr. Int. Workshop Monteroduni: New Frontiers in Hyperstructures and Related Algebras, Hadronic Press (1996),177-188. Vougiouklis T., (1996b), Hv-groups defined on the same set, Discrete Mathematics, Elsevier, V.155, N 1-3, (1996), 259-265. Vougiouklis T., (1996c), On Hv-fields, Proc. of 6 th AHA Congress, Prague, 1996, Democritus Un. Press (1997) 151-159 Vougiouklis T., Spartalis S., Kessoglides M., (1997), Weak hyperstructures on small sets, Ratio Matematica Ν.12(1997), 90-96. Vougiouklis T., (1997), Convolutions on WASS hyperstructures, Discrete Mathematics, Elsevier, V.174, (1997), 347-355 Vougiouklis T., (1999a), On Hv-rings and Hv-representations, Discrete Mathematics, Elsevier, 208/209 (1999), 615-620 Vougiouklis T., (1999b), Fundamental Relations in hyperstructures, Bulletin of the Greek Mathematical Society, V.42 (1999), 113-118 Vougiouklis T., (2003a), Special elements in Hv-structures, Bulletin of the Greek Math. Society V.48 (2003), 103-112. Vougiouklis T., (2003b), Finite Hv-structures and their representations, Rendiconti Seminario Matematico di Messina Serie II, T.XXV, V.9(2003), 245-265. Vougiouklis T. (2008), Hv-structures: Birth and … childhood, J. Basic Science 4, N.1 (2008),119-133 Vougiouklis T. (2009), Hyperstructures as models in social sciences, Ratio Sociologica, V.2, N.2, 2009,21-34. Vougiouklis T. (2011), Bar and Theta Hyperoperations, Ratio Mathematica, 21, 2011, 27-42. Vougiouklis T. (2014), From Hv-rings to Hv-fields, Int. J. Algebraic Hyperstructures Appl. Vol.1, No.1, 2014, 1-13.