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Applications of Hypergroup Theory in Solving the Rubik's Cube

Yıl 2023, Cilt: 13 Sayı: 3, 2068 - 2092, 01.09.2023

Burcu Nişancı Türkmen Gamze Ela Kukuş

https://doi.org/10.21597/jist.1268544

Öz

The solution of the Rubik's cube was designed with an algebraic structure different from the group algebraic structure existing in the literature with the help of hypergroups. The generalized permutation hypergroup concept has been adopted as the method by which the strategy of the Rubik's cube solution will develop; carried out with the development of methods applied in group theory. Although there are inadequacies in the application of abstract algebra studies all over the world, with this study, the basis of the analysis of the Rubik's cube with the help of hypergroups was obtained; Thus, it has been contributed to national and international academic research to be the application of the studies carried out in the field of abstract algebra in our country. Therefore, with the help of the Rubik's cube, it has been proven that the algebra studies in the literature can be translated into practice.

Anahtar Kelimeler

Group Theory Rubik’s Cube Hypergroup Theory

Destekleyen Kurum

TUBİTAK 2209-A

Proje Numarası

1919B012105753

Kaynakça

  • AMERI, R. (2003), On categories of hypergroups and hypermodules, Journal of Discrete Mathematical Sciences and Crytography, 6:2-3, 121-132.
  • BAKER, M., BOWLER, N. (1991), Matroids over hyperfields. arxiv 2017, arXiv:1601.01204. adresinden alındı.
  • BARLOTTI, A., STRAMBACH, K. (1971), Multigroups and the foundations of Geometry. Rend. Circ. Mat. Palermo XL, 5–68.
  • BOURBAKI, N. (1971). Éléments de Mathématique, Algèbre; Hermann: Paris, France.
  • BRANDELOW, C. (1982), Inside the Rubik’s Cube and Beyond. Birkh¨auser, 12, 17, 19, 23.
  • CHVALINA, J., CHVALINOVA, L. (1996), State hypergroups of Automata. Acta Math. Inform. Univ. Ostrav. 4, 105–120.
  • CHVALINA, J., KˇREHLIK, S., NOVAK, M. (2016), Cartesian composition and the problem of generalizing the MAC condition to quasimultiautomata. An. St. Univ. Ovidius Constanta, 24, 79–100.
  • CHVALINA, J., NOVAK, M., KˇREHLIK, S. (2019), Hyperstructure generalizations of quasi-automata induced by modelling functions and signal processing. AIP Conf. Proc., 2116, 310006. CHVALINA, J., NOVAK, M., SMETANA, B., STAN’EK, D. (2021). Sequences of Groups, Hypergroups and Automata of Linear Ordinary Differential Operators. Mathematics, 9, 319.
  • CHRORANI, M., ZAHEDI, M. M. (2012), Some hypergroups induced by tree automata. Aust. J. Basic Appl. Sci., 6, 680–692.
  • CHRORANI, M. (2018), State hyperstructures of tree automata based on lattice-valued logic. RAIRO Theor. Inf. Appl., 52, 23–42.
  • CONNES, A., CONSANI, C. (2011), The hyperring of adèle classes. J. Number Theory, 131, 159–194.
  • CONNES, A., CONSANI, C. (2010), From monoids to hyperstructures: In search of an absolute arithmetic. arXiv 2010, arXiv:1006.4810. adresinden alındı.
  • CORSINI, P. (1993), Prolegomena of Hypergroup Theory, 2nd ed. Tricesimo Italy, Aviani editore Italy.
  • CORSINI, P., LEOREANU, V. (2003), Applications of Hyperstructures Theory; Kluwer Academic Publishers: Dordrecht, The Netherlands.
  • DANIELS, L. (2014), Group Theory and the Rubik's Cube, A project submitted to the Department of Mathematical Sciences in conformity with the requirements for Math 4301, Lakehead University Thunder Bay, Ontario, Canada.
  • DEMAINE, E.D., DEMAIN, M. L, EISENSTAT, S., LUBIW, A., WINSLOW, A. (2011), Algorithms for Solving Rubik’s cubes. Lecture Notes in Computer Science, 6942, 689-700, 1.
  • DRAMALIDIS, A. (2011), On geometrical hyperstructures of finite order. Ratio Math., 21, 43–58.
  • DRESHER, M., ORE, O. (1938), Theory of multigroups. Am. J. Math., 1938, 60, 705–733.
  • DUMMIT, D. S., FOOTE, R. M. (1999), Abstract Algebra. Prentice Hall., 9.
  • EATON, E.J., ORE, O. (1940), Remarks on multigroups. Am. J. Math. 62, 67–71.
  • EATON, E.J. (1940), Associative Multiplicative Systems. Am. J. Math. 62, 222–232.
  • FRENI, D. (1985), Sur les hypergroupes cambistes. Rend. Ist. Lomb., 119, 175–186.
  • FRENI, D. (1986), Sur la théorie de la dimension dans les hypergroupes. Acta Univ. Carol. Math. Phys. 27, 67–80.
  • FRENI, D. (2004), Strongly Transitive Geometric Spaces: Applications to Hypergroups and Semigroups Theory. Commun. Algebra, 32, 969–988.
  • GALLIAN, J.A. (2010), Contemporary Abstract Algebra. Brooks/Cole, Cengage Learning, 3, 4, 9.
  • GIONFRIDDO, M. (1978), Hypergroups associated with multihomomorphisms between generalised graphs. Convegno su: Sistemi binari e loro appl., Taormina (ME), 161-174.
  • GRIFFITHS, L.W. (1938), On hypergroups, multigroups, and product systems. Am. J. Math. 60, 345–354.
  • HOSKOVA-MAYEROVA, S., MATURO, A. (2018), Algebraic hyperstructures and social relations. Ital. J. Pure Appl. Math., 39, 701–709.
  • HOSKOVA-MAYEROVA, S., MATURO, A. (2015). A. An analysis of social relations and social group behaviors with fuzzy sets and hyperstructures. Int. J. Algebraic Hyperstruct. Appl., 2, 91–99.
  • HOSKOVA-MAYEROVA, S., MATURO, A. (2013). Hyperstructures in social sciences. AWER Procedia Inf. Technol. Comput. Sci., 3, 547–552.
  • JANTOSCIAK, J. (1985), Classical geometries as hypergroups. In Proceedings of the Atti del Convegno su Ipergruppi altre Structure Multivoche et loro Applicazioni, Udine, Italy, 15–18, 93–104.
  • JANTOSCIAK, J. (1994). A brief survey of the theory of join spaces. In Proceedings of the 5th Intern. Congress on Algebraic Hyperstructures and Applications, Iasi, Romania, 4–10 July 1993; Hadronic Press: Palm Harbor, FL, USA, 109–122.
  • JOYNER, D. (2009), Adventures in Group Theory: Rubik’s Cube, Merlin’s Machine, and other Mathematical Toys. John Hopkins University Press, Second Edition, ISBN-13 978-0801890130.
  • JUN, J. (2017), Geometry of hyperfields. arXiv:1707.09348 adresinden alındı.
  • KACPRZYK, M. (2018), Eds, J., Springer International Publishing: Cham, Switzerland, 2018, 103–111.
  • KOTTWITZ, S. (2008), Example: Sudoku 3D cube. Tex example.net, http://www.texample.net/tikz/examples/sudoku-3d-cube/ 13, 14, 15, 23 adresinden alındı.
  • KRASNER, M. (1937), Sur la primitivité des corps B-adiques. Mathematica, 13, 72–191.
  • KRASNER, M. (1940), La loi de Jordan—Holder dans les hypergroupes et les suites generatrices des corps de nombres P—adiqes, (I). Duke Math. J. 6, 120–140, (II) Duke Math. J., 7, 121–135.
  • KRASNER, M. (1941), La caractérisation des hypergroupes de classes et le problème de Schreier dans ces hypergroupes. C. R. Acad. Sci. (Paris), 212, 948–950.
  • KRASNER, M. (1944), Hypergroupes moduliformes et extramoduliformes. Acad. Sci. (Paris), 219, 473–476.
  • KRASNER, M., KUNTZMANN, J. (1947), Remarques sur les hypergroupes. C.R. Acad. Sci. (Paris), 224, 525–527.
  • KRASNER, M. (1957), Approximation des Corps Valués Complets de Caractéristique p6=0 par Ceux de Caractéristique 0, Colloque d’ Algèbre Supérieure (Bruxelles, Decembre 1956), Centre Belge de Recherches Mathématiques, Établissements Ceuterick, Louvain, Librairie Gauthier-Villars, Paris., 129–206.
  • KREHLIK, S. (2020), n-Ary Cartesian Composition of Multiautomata with Internal Link for Autonomous Control of Lane Shifting. Mathematics, 8, 835.
  • KUNTZMANN, J. (1937), Opérations multiformes. Hypergroupes. C. R. Acad. Sci. (Paris), 204, 1787–1788.
  • KUNTZMANN, J. (1937), Homomorphie entre systémes multiformes. C. R. Acad. Sci. (Paris), 205, 208–210.
  • LORSCHEID, O. (2019), Tropical geometry over the tropical hyperfield. arXiv:1907.01037 adresinden alındı.
  • MARTY, F. (1934), Sur une Généralisation de la Notion de Groupe. Huitième Congrès des Mathématiciens Scand. Stockholm, 45–49.
  • MARTY, F. (1935), Rôle de la notion de hypergroupe dans l’ étude de groupes non abéliens. C. R. Acad. Sci. (Paris), 201, 636–638.
  • MARTY, F. (1936), Sur les groupes et hypergroupes attachés à une fraction rationelle. Ann. L’ Ecole Norm. 1936, 3, 83–123.
  • MASSOUROS, C.G. (1989), Hypergroups and convexity. Riv. Mat. Pura Appl., 4, 7–26.
  • MASSOUROS, C.G. (1996), Hypergroups and Geometry. Mem. Acad. Romana Math. Spec., Issue XIX, 185–191.
  • MASSOUROS, C. G. (2015), On connections between vector spaces and hypercompositional structures. Ital. J. Pure Appl. Math., 34, 133–150.
  • MASSOUROS, C.G., MITTAS, J. (1990), Languages—Automata and hypercompositional structures. In Proceedings of the 5th Intern. Congress on Algebraic Hyperstructures and Applications, Xanthi, Greece, 27–30 June 1990; World Scientific: Singapore, 199, 137–147.
  • MASSOUROS, C.G. (1993), Automata, Languages and Hypercompositional Structures. Ph.D. Thesis, National Technical University of Athens, Athens, Greece, 1993.
  • MASSOUROS, C.G. (1994), Automata and hypermoduloids. In Proceedings of the 5th Intern. Congress on Algebraic Hyperstructures and Applications, Iasi, Romania, 4–10 July 1993; Hadronic Press: Palm Harbor, FA, USA, 1994, 251–265.
  • MASSOUROS, C.G. (1994), An automaton during its operation. In Proceedings of the 5th Internation Congress on Algebraic Hyperstructures and Applications, Iasi, Romania, 4–10 July 1993; Hadronic Press: Palm Harbor, FA, USA, 1994; pp. 267–276.
  • MASSOUROS, C.G. (2003), On the attached hypergroups of the order of an automaton. J. Discrete Math. Sci. Cryptogr., 6, 207–215.
  • MASSOUROS, G.G (1994), Hypercompositional structures in the theory of languages and automata. An. Sti. Univ. Al. I. Cuza Iasi Sect. Inform., III, 65–73.
  • MASSOUROS, G.G. (2021), Hypercompositional structures from the computer theory. Ratio Math., 1999, 13, 37–42.
  • MASSOUROS, C.G., MASSOUROS, G.G. (2009), Hypergroups associated with graphs and automata. AIP Conf. Proc.,1168, 164–167.
  • MASSOUROS, C.G. (2016), On path hypercompositions in graphs and automata. MATEC Web Conf. 41, 5003.
  • MASSOUROS, G.G. (2020), Hypercompositional algebra, computer science and geometry. Mathematics 8, 1338.
  • MASSOUROS, C.G. (1988), Free and cyclic hypermodules. Ann. Mat. Pura Appl., 150, 153–166.
  • MATURO, A., HOSKOVA-MaAYEROVA, S., SOITU, D.T., KACPRZYK, J. (2017), Eds.;Studies in Systems, Decision and Control 66; Springer International Publishing: Cham, Switzerland, 2017, 211–221.
  • MITTAS, J. (1975), Espaces vectoriels sur un hypercorps. Introduction des hyperspaces affines et Euclidiens. Math. Balk., 5, 199–211.
  • MITTAS, J., MASSOUROS, C.G. (1989), Hypergroups defined from linear spaces. Bull. Greek Math. Soc., 30, 63–78.
  • NOVAK, M., KREHLIK, S., STANEK (2019), D. n-ary Cartesian composition of automata. Soft Comput., 24, 1837–1849.
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Rubik Küpün Çözümlenmesinde Hiper Grup Teori Uygulamaları

Yıl 2023, Cilt: 13 Sayı: 3, 2068 - 2092, 01.09.2023

Burcu Nişancı Türkmen Gamze Ela Kukuş

https://doi.org/10.21597/jist.1268544

Öz

Rubik küpünün çözümünün hiper gruplar yardımıyla literatürde var olan grup cebirsel yapısından farklı bir cebirsel yapıyla tasarımı sağlanmıştır. Rubik küp çözümünün stratejisinin gelişim kaydedeceği yöntem olarak genelleştirilmiş permütasyon hiper grup kavramı benimsenmiş olup; grup teoride uygulanan yöntemlerin geliştirilmesi ile gerçekleştirilmiştir. Soyut cebir çalışmalarının uygulama bulması konusunda tüm dünyada yetersizlikler mevcut olmasına rağmen bu çalışma ile Rubik küpün hiper gruplar yardımıyla çözümleme esası elde edilmiş; böylece ulusal ve uluslararası akademik araştırmalara, ülkemizdeki soyut cebir alanında yürütülen çalışmaların uygulaması olması hususunda katkı sunulmuştur. Dolayısıyla Rubik küp yardımıyla literatürde var olan cebir çalışmalarının uygulamaya dönüştürülebilir olduğu kanıtlanmıştır.

Anahtar Kelimeler

Hiper Grup Teorisi Rubik Küp Grup Teorisi

Proje Numarası

1919B012105753

Kaynakça

  • AMERI, R. (2003), On categories of hypergroups and hypermodules, Journal of Discrete Mathematical Sciences and Crytography, 6:2-3, 121-132.
  • BAKER, M., BOWLER, N. (1991), Matroids over hyperfields. arxiv 2017, arXiv:1601.01204. adresinden alındı.
  • BARLOTTI, A., STRAMBACH, K. (1971), Multigroups and the foundations of Geometry. Rend. Circ. Mat. Palermo XL, 5–68.
  • BOURBAKI, N. (1971). Éléments de Mathématique, Algèbre; Hermann: Paris, France.
  • BRANDELOW, C. (1982), Inside the Rubik’s Cube and Beyond. Birkh¨auser, 12, 17, 19, 23.
  • CHVALINA, J., CHVALINOVA, L. (1996), State hypergroups of Automata. Acta Math. Inform. Univ. Ostrav. 4, 105–120.
  • CHVALINA, J., KˇREHLIK, S., NOVAK, M. (2016), Cartesian composition and the problem of generalizing the MAC condition to quasimultiautomata. An. St. Univ. Ovidius Constanta, 24, 79–100.
  • CHVALINA, J., NOVAK, M., KˇREHLIK, S. (2019), Hyperstructure generalizations of quasi-automata induced by modelling functions and signal processing. AIP Conf. Proc., 2116, 310006. CHVALINA, J., NOVAK, M., SMETANA, B., STAN’EK, D. (2021). Sequences of Groups, Hypergroups and Automata of Linear Ordinary Differential Operators. Mathematics, 9, 319.
  • CHRORANI, M., ZAHEDI, M. M. (2012), Some hypergroups induced by tree automata. Aust. J. Basic Appl. Sci., 6, 680–692.
  • CHRORANI, M. (2018), State hyperstructures of tree automata based on lattice-valued logic. RAIRO Theor. Inf. Appl., 52, 23–42.
  • CONNES, A., CONSANI, C. (2011), The hyperring of adèle classes. J. Number Theory, 131, 159–194.
  • CONNES, A., CONSANI, C. (2010), From monoids to hyperstructures: In search of an absolute arithmetic. arXiv 2010, arXiv:1006.4810. adresinden alındı.
  • CORSINI, P. (1993), Prolegomena of Hypergroup Theory, 2nd ed. Tricesimo Italy, Aviani editore Italy.
  • CORSINI, P., LEOREANU, V. (2003), Applications of Hyperstructures Theory; Kluwer Academic Publishers: Dordrecht, The Netherlands.
  • DANIELS, L. (2014), Group Theory and the Rubik's Cube, A project submitted to the Department of Mathematical Sciences in conformity with the requirements for Math 4301, Lakehead University Thunder Bay, Ontario, Canada.
  • DEMAINE, E.D., DEMAIN, M. L, EISENSTAT, S., LUBIW, A., WINSLOW, A. (2011), Algorithms for Solving Rubik’s cubes. Lecture Notes in Computer Science, 6942, 689-700, 1.
  • DRAMALIDIS, A. (2011), On geometrical hyperstructures of finite order. Ratio Math., 21, 43–58.
  • DRESHER, M., ORE, O. (1938), Theory of multigroups. Am. J. Math., 1938, 60, 705–733.
  • DUMMIT, D. S., FOOTE, R. M. (1999), Abstract Algebra. Prentice Hall., 9.
  • EATON, E.J., ORE, O. (1940), Remarks on multigroups. Am. J. Math. 62, 67–71.
  • EATON, E.J. (1940), Associative Multiplicative Systems. Am. J. Math. 62, 222–232.
  • FRENI, D. (1985), Sur les hypergroupes cambistes. Rend. Ist. Lomb., 119, 175–186.
  • FRENI, D. (1986), Sur la théorie de la dimension dans les hypergroupes. Acta Univ. Carol. Math. Phys. 27, 67–80.
  • FRENI, D. (2004), Strongly Transitive Geometric Spaces: Applications to Hypergroups and Semigroups Theory. Commun. Algebra, 32, 969–988.
  • GALLIAN, J.A. (2010), Contemporary Abstract Algebra. Brooks/Cole, Cengage Learning, 3, 4, 9.
  • GIONFRIDDO, M. (1978), Hypergroups associated with multihomomorphisms between generalised graphs. Convegno su: Sistemi binari e loro appl., Taormina (ME), 161-174.
  • GRIFFITHS, L.W. (1938), On hypergroups, multigroups, and product systems. Am. J. Math. 60, 345–354.
  • HOSKOVA-MAYEROVA, S., MATURO, A. (2018), Algebraic hyperstructures and social relations. Ital. J. Pure Appl. Math., 39, 701–709.
  • HOSKOVA-MAYEROVA, S., MATURO, A. (2015). A. An analysis of social relations and social group behaviors with fuzzy sets and hyperstructures. Int. J. Algebraic Hyperstruct. Appl., 2, 91–99.
  • HOSKOVA-MAYEROVA, S., MATURO, A. (2013). Hyperstructures in social sciences. AWER Procedia Inf. Technol. Comput. Sci., 3, 547–552.
  • JANTOSCIAK, J. (1985), Classical geometries as hypergroups. In Proceedings of the Atti del Convegno su Ipergruppi altre Structure Multivoche et loro Applicazioni, Udine, Italy, 15–18, 93–104.
  • JANTOSCIAK, J. (1994). A brief survey of the theory of join spaces. In Proceedings of the 5th Intern. Congress on Algebraic Hyperstructures and Applications, Iasi, Romania, 4–10 July 1993; Hadronic Press: Palm Harbor, FL, USA, 109–122.
  • JOYNER, D. (2009), Adventures in Group Theory: Rubik’s Cube, Merlin’s Machine, and other Mathematical Toys. John Hopkins University Press, Second Edition, ISBN-13 978-0801890130.
  • JUN, J. (2017), Geometry of hyperfields. arXiv:1707.09348 adresinden alındı.
  • KACPRZYK, M. (2018), Eds, J., Springer International Publishing: Cham, Switzerland, 2018, 103–111.
  • KOTTWITZ, S. (2008), Example: Sudoku 3D cube. Tex example.net, http://www.texample.net/tikz/examples/sudoku-3d-cube/ 13, 14, 15, 23 adresinden alındı.
  • KRASNER, M. (1937), Sur la primitivité des corps B-adiques. Mathematica, 13, 72–191.
  • KRASNER, M. (1940), La loi de Jordan—Holder dans les hypergroupes et les suites generatrices des corps de nombres P—adiqes, (I). Duke Math. J. 6, 120–140, (II) Duke Math. J., 7, 121–135.
  • KRASNER, M. (1941), La caractérisation des hypergroupes de classes et le problème de Schreier dans ces hypergroupes. C. R. Acad. Sci. (Paris), 212, 948–950.
  • KRASNER, M. (1944), Hypergroupes moduliformes et extramoduliformes. Acad. Sci. (Paris), 219, 473–476.
  • KRASNER, M., KUNTZMANN, J. (1947), Remarques sur les hypergroupes. C.R. Acad. Sci. (Paris), 224, 525–527.
  • KRASNER, M. (1957), Approximation des Corps Valués Complets de Caractéristique p6=0 par Ceux de Caractéristique 0, Colloque d’ Algèbre Supérieure (Bruxelles, Decembre 1956), Centre Belge de Recherches Mathématiques, Établissements Ceuterick, Louvain, Librairie Gauthier-Villars, Paris., 129–206.
  • KREHLIK, S. (2020), n-Ary Cartesian Composition of Multiautomata with Internal Link for Autonomous Control of Lane Shifting. Mathematics, 8, 835.
  • KUNTZMANN, J. (1937), Opérations multiformes. Hypergroupes. C. R. Acad. Sci. (Paris), 204, 1787–1788.
  • KUNTZMANN, J. (1937), Homomorphie entre systémes multiformes. C. R. Acad. Sci. (Paris), 205, 208–210.
  • LORSCHEID, O. (2019), Tropical geometry over the tropical hyperfield. arXiv:1907.01037 adresinden alındı.
  • MARTY, F. (1934), Sur une Généralisation de la Notion de Groupe. Huitième Congrès des Mathématiciens Scand. Stockholm, 45–49.
  • MARTY, F. (1935), Rôle de la notion de hypergroupe dans l’ étude de groupes non abéliens. C. R. Acad. Sci. (Paris), 201, 636–638.
  • MARTY, F. (1936), Sur les groupes et hypergroupes attachés à une fraction rationelle. Ann. L’ Ecole Norm. 1936, 3, 83–123.
  • MASSOUROS, C.G. (1989), Hypergroups and convexity. Riv. Mat. Pura Appl., 4, 7–26.
  • MASSOUROS, C.G. (1996), Hypergroups and Geometry. Mem. Acad. Romana Math. Spec., Issue XIX, 185–191.
  • MASSOUROS, C. G. (2015), On connections between vector spaces and hypercompositional structures. Ital. J. Pure Appl. Math., 34, 133–150.
  • MASSOUROS, C.G., MITTAS, J. (1990), Languages—Automata and hypercompositional structures. In Proceedings of the 5th Intern. Congress on Algebraic Hyperstructures and Applications, Xanthi, Greece, 27–30 June 1990; World Scientific: Singapore, 199, 137–147.
  • MASSOUROS, C.G. (1993), Automata, Languages and Hypercompositional Structures. Ph.D. Thesis, National Technical University of Athens, Athens, Greece, 1993.
  • MASSOUROS, C.G. (1994), Automata and hypermoduloids. In Proceedings of the 5th Intern. Congress on Algebraic Hyperstructures and Applications, Iasi, Romania, 4–10 July 1993; Hadronic Press: Palm Harbor, FA, USA, 1994, 251–265.
  • MASSOUROS, C.G. (1994), An automaton during its operation. In Proceedings of the 5th Internation Congress on Algebraic Hyperstructures and Applications, Iasi, Romania, 4–10 July 1993; Hadronic Press: Palm Harbor, FA, USA, 1994; pp. 267–276.
  • MASSOUROS, C.G. (2003), On the attached hypergroups of the order of an automaton. J. Discrete Math. Sci. Cryptogr., 6, 207–215.
  • MASSOUROS, G.G (1994), Hypercompositional structures in the theory of languages and automata. An. Sti. Univ. Al. I. Cuza Iasi Sect. Inform., III, 65–73.
  • MASSOUROS, G.G. (2021), Hypercompositional structures from the computer theory. Ratio Math., 1999, 13, 37–42.
  • MASSOUROS, C.G., MASSOUROS, G.G. (2009), Hypergroups associated with graphs and automata. AIP Conf. Proc.,1168, 164–167.
  • MASSOUROS, C.G. (2016), On path hypercompositions in graphs and automata. MATEC Web Conf. 41, 5003.
  • MASSOUROS, G.G. (2020), Hypercompositional algebra, computer science and geometry. Mathematics 8, 1338.
  • MASSOUROS, C.G. (1988), Free and cyclic hypermodules. Ann. Mat. Pura Appl., 150, 153–166.
  • MATURO, A., HOSKOVA-MaAYEROVA, S., SOITU, D.T., KACPRZYK, J. (2017), Eds.;Studies in Systems, Decision and Control 66; Springer International Publishing: Cham, Switzerland, 2017, 211–221.
  • MITTAS, J. (1975), Espaces vectoriels sur un hypercorps. Introduction des hyperspaces affines et Euclidiens. Math. Balk., 5, 199–211.
  • MITTAS, J., MASSOUROS, C.G. (1989), Hypergroups defined from linear spaces. Bull. Greek Math. Soc., 30, 63–78.
  • NOVAK, M., KREHLIK, S., STANEK (2019), D. n-ary Cartesian composition of automata. Soft Comput., 24, 1837–1849.
  • NOVAK, M. (2008), Some remarks on constructions of strongly connected multiautomata with the input semihypergroup being a centralizer of certain transformation operators., J. Appl. Math., 2008, I, 65–72.
  • ORE, O. (1937), Structures and group theory, I. Duke Math. J., 3, 149–174.
  • PRENOWITZ, W. (1943), Projective Geometries as multigroups. Am. J. Math., 65, 235–256.
  • PRENOWITZ, W. (1946), Descriptive Geometries as multigroups. Trans. Am. Math. Soc., 59, 333–380.
  • PRENOWITZ, W. (1950), Spherical Geometries and mutigroups. Can. J. Math., 2, 100–119.
  • PRENOWITZ, W. (1961), A Contemporary Approach to Classical Geometry. Am. Math. Month., 68, 1–67.
  • PRENOWITZ, W., JANTOSCIAK, J. (1972), Geometries and Join Spaces. J. Reine Angew. Math., 257, 100–128.
  • PRENOWITZ, W., JANTOSCIAK, J. (1979), Join Geometries. A Theory of Convex Sets and Linear Geometry; Springer: Berlin/Heidelberg, Germany.
  • REYNOLDS, T. (2014), World cube Association Official Results. World Cube Organization. https://www.worldcubeassociation.org/results/regions.php 1 adresinden alındı.
  • ROKICKI, T. (2010), Twenty-Two moves suffice for Rubik’s Cube. Math Intelligencer. 32, No. 1, 33-40. 1.
  • ROKICKI, T., KOCIEMBA, H., DAVIDSON, M., DETHRIDGE, J. (2013), The diameter of the Rubik’s Cube is twenty. SIAM J. Discrete Math, 27, No. 2 , 1082-1105. 1.
  • SINGMASTER, D. (1981), Notes on Rubik’s ’Magic Cube’. Enslow Pub Inc., 1, 12.
  • VAHEDI, V., JAFARPOUR, M., AGHABOZORGI, H., CRISTEA, I. (2019), Extension of elliptic curves on Krasner hyperfields. Comm. Algebra , 47, 4806–4823.
  • VAHEDI, V., JAFARPOUR, M., CRISTEA, I. (2019), Hyperhomographies on Krasner Hyperfields. Symmetry, 11, 1442.
  • VAHEDI, V., JAFARPOUR, M., HOSKOVA-MAYEROVA, S., AGHABOZORGI, H., LEOREANU-FOTEA, V., BEKESIENE, S. (2020), Derived Hyperstructures from Hyperconics. Mathematics, 8, 429.
  • VIRO, O. (2010), Hyperfields for Tropical Geometry I. Hyperfields and dequantization. arXiv:1006.3034.
  • VIRO, O. (2011), On basic concepts of tropical geometry. Proc. Steklov Inst. Math., 273, 252–282.
  • VOUGIOUKLIS, T. (1992), Representations of hypergroups by generalized permutations, Algebra Universalis, 29, 172-183.
  • WALL, H.S. (1937), Hypergroups. Am. J. Math., 59, 77–98.
Toplam 86 adet kaynakça vardır.

Ayrıntılar

Birincil Dil Türkçe
Konular Matematik
Bölüm Matematik / Mathematics
Yazarlar

Burcu Nişancı Türkmen 0000-0001-7900-0529

Gamze Ela Kukuş 0000-0002-6890-2081

Proje Numarası 1919B012105753
Erken Görünüm Tarihi 29 Ağustos 2023
Yayımlanma Tarihi 1 Eylül 2023
Gönderilme Tarihi 21 Mart 2023
Kabul Tarihi 5 Mayıs 2023
Yayımlandığı Sayı Yıl 2023 Cilt: 13 Sayı: 3

Kaynak Göster

APA Nişancı Türkmen, B., & Kukuş, G. E. (2023). Rubik Küpün Çözümlenmesinde Hiper Grup Teori Uygulamaları. Journal of the Institute of Science and Technology, 13(3), 2068-2092. https://doi.org/10.21597/jist.1268544
AMA Nişancı Türkmen B, Kukuş GE. Rubik Küpün Çözümlenmesinde Hiper Grup Teori Uygulamaları. Iğdır Üniv. Fen Bil Enst. Der. Eylül 2023;13(3):2068-2092. doi:10.21597/jist.1268544
Chicago Nişancı Türkmen, Burcu, ve Gamze Ela Kukuş. “Rubik Küpün Çözümlenmesinde Hiper Grup Teori Uygulamaları”. Journal of the Institute of Science and Technology 13, sy. 3 (Eylül 2023): 2068-92. https://doi.org/10.21597/jist.1268544.
EndNote Nişancı Türkmen B, Kukuş GE (01 Eylül 2023) Rubik Küpün Çözümlenmesinde Hiper Grup Teori Uygulamaları. Journal of the Institute of Science and Technology 13 3 2068–2092.
IEEE B. Nişancı Türkmen ve G. E. Kukuş, “Rubik Küpün Çözümlenmesinde Hiper Grup Teori Uygulamaları”, Iğdır Üniv. Fen Bil Enst. Der., c. 13, sy. 3, ss. 2068–2092, 2023, doi: 10.21597/jist.1268544.
ISNAD Nişancı Türkmen, Burcu - Kukuş, Gamze Ela. “Rubik Küpün Çözümlenmesinde Hiper Grup Teori Uygulamaları”. Journal of the Institute of Science and Technology 13/3 (Eylül 2023), 2068-2092. https://doi.org/10.21597/jist.1268544.
JAMA Nişancı Türkmen B, Kukuş GE. Rubik Küpün Çözümlenmesinde Hiper Grup Teori Uygulamaları. Iğdır Üniv. Fen Bil Enst. Der. 2023;13:2068–2092.
MLA Nişancı Türkmen, Burcu ve Gamze Ela Kukuş. “Rubik Küpün Çözümlenmesinde Hiper Grup Teori Uygulamaları”. Journal of the Institute of Science and Technology, c. 13, sy. 3, 2023, ss. 2068-92, doi:10.21597/jist.1268544.
Vancouver Nişancı Türkmen B, Kukuş GE. Rubik Küpün Çözümlenmesinde Hiper Grup Teori Uygulamaları. Iğdır Üniv. Fen Bil Enst. Der. 2023;13(3):2068-92.
 Kapak Resmi İndir