On Some Applications of the Vougiouklis Hyperstructures to Probability Theory

Antonio Maturo, Fabrizio Maturo


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.


algebraic hyperstructures; subjective probability; Hv structures; join spaces

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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.

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.

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.

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, 5th 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 6th 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.

DOI: http://dx.doi.org/10.23755/rm.v33i0.372


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