THE TRANSPOSITION AXIOM IN HYPERCOMPOSITIONAL STRUCTURES

Ch.G. Massouros, G.G. Massouros

Abstract


The hypergroup (as defined by F. Marty), being a very general algebraic structure, was subsequently quickly enriched with additional axioms. One of these is the transposition axiom, the utilization of which led to the creation of join spaces (join hypergroups) and of transposition hypergroups. These hypergroups have numerous applications in geometry, formal languages, the
theory of automata and graph theory. This paper deals with transposition hypergroups. It also introduces the transposition axiom to weaker structures, which result from the hypergroup by the removal of certain axioms, thus defining the transposition hypergroupoid, the transposition semi-hypergroup and the transposition quasi-hypergroup. Finally, it presents hypercompositional structures with internal or external compositions and hypercompositions, in which the transposition axiom is valid. Such structures emerged during the study of formal languages and the theory of automata through the use of hypercompositional algebra.


Full Text:

PDF

References


R. Ameri, M. M. Zahedi: «Fuzzy Subhypermodules over Fuzzy Hyperrings»,

Proceedings of the 6th International Congress in Algebraic Hyperstructures and Applications, Publ. Democritus Univ. of Thrace (1997) pp. 1-14

R. Ameri: «On categories of hypergroups and hypermodules», Journal of Discrete Mathematical Sciences & Cryptography Vol. 6, No 2-3 (2003) pp. 121-123.

P. Corsini: «Binary relations and hypergroupoids», Italian J. Pure & Appl. Math. 7, (2000) 11-18.

P. Corsini: «On the hypergroups associated with binary relations», Mult. Valued Log., 5 (2000), 407-419.

J. Chvalina: «Commutative hypergroups in the sense of Marty and ordered sets», Proceedings of the Summer School on General Algebra and Ordered Sets, 1994, Olomouc, Czech Republic, pp. 1930.

J. Chvalina: «Relational Product of Join Spaces determined by quasi-orderings», Proceedings of the 6th International Congress on Algebraic Hyperstructures and Applications, Publ. Democritus Univ. of Thrace (1997), pp. 15-23.

J. Chvalina, L. Chvalinova: «State hypergroups of Automata» Acta Mathematica et Informatica Univ. Ostraviensis 4 (1996) pp. 105-120.

Chvalina, Jan, Hošková, Šárka. «Modelling of join spaces with proximities by first-order linear partial differential operators», Ital. J. Pure Appl. Math. No. 21 (2007), pp. 177-190.

I. Cristea, M. Stefanescu: «Binary relations and reduced hypergroups», Discrete Math. 308 (2008), 3537{3544, doi:10.1016/j.disc.2007.07.01

I. Cristea, M. Ştefănescu, C. Angheluţă: «About the fundamental relations defined on the hypergroupoids associated with binary relations», European Journal of Combinatorics Volume 32, Issue 1, (2011) pp. 72-81.

I. Cristea, M. Jafarpourd, S. S. Mousavic, A. Soleymanid: «Enumeration of Rosenberg hypergroups», Computers and Mathematics with Applications, 32 (2011), pp. 72-81.

S. Hoskova, J. Chvalina: «Discrete transformation hypergroups and transformation hypergroups with phase tolerance space, Discrete Math., 308 (2008), 4133{4143, doi:10.1016/j.disc.2007.08.005.

S. Hoskova: «Abelization of join spaces of affine transformation of ordered field with proximity», Appl. Gener. Topology 6, No 1 (2005), 57-65.

J. Jantosciak: «Τransposition hypergroups, Noncommutative Join Spaces», Journal of Algebra 187(1997), 97-119

J. Jantosciak, Ch. G. Massouros: «Strong Identities and fortification in Transposition hypergroups», Journal of Discrete Mathematical Sciences and Cryptography 6, No 2-3(2003), 169-193.

M. Krasner: «Approximation des corps valués complets de caractéristique p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, 1957, pp. 129-206.

M. Krasner: «A class of hyperrings and hyperfields» Internat. J. Math. & Math. Sci. vol. 6, no. 2, 1983, pp. 307-312.

F. Marty: «Sur un généralisation de la notion de groupe». Huitième Congrès des mathématiciens Scand., Stockholm 1934, pp. 45-49.

Ch. G. Massouros: «Free and cyclic hypermodules», Annali Di Mathematica Pura ed Applicata, Vol. CL. 1988, pp. 153-166.

Ch. G. Massouros: «Hypergroups and convexity», Riv. di Mat. pura ed applicata 4, (1989) pp. 7-26.

Ch. G. Massouros, «On the semi-subhypergroups of a hypergroup», Internat. J. Math. and Math. Sci. 14, No 2 (1991), pp. 293-304.

Ch. G. Massouros: «Normal homomorphisms of Fortified Join Hypergroups»

Proceedings of the 5th International Congress on Algebraic Hyperstructures and Applications, Hadronic Press (1994), pp. 133-142.

Ch. G. Massouros: «Canonical and Join Hypergroups», Analele Stiintigice Ale Universitatii “AL.I.CUZA”, Tom. XLII Matematica, fasc. 1 (1996) pp. 175-186.

Ch. G. Massouros, G. G. Massouros: «Transposition Polysymmetrical Hypergroups with Strong Identity», Journal of Basic Science Vol. 4, No. 1 (2008) pp. 85-93.

Ch. G. Massouros, G. G. Massouros: «On Join Hyperrings», Proceedings of the 10th International Congress on Algebraic Hyperstructures and Applications, Brno, Czech Republic 2008, pp. 203-216.

Ch. G. Massouros, G. G. Massouros: «Operators and Hyperoperators acting on Hypergroups», Proceedings of the International Conference on Numerical Analysis and Applied Mathematics, ICNAAM 2008, American Institute of Physics (AIP) Conference Proceedings, pp. 380-383.

Ch. G. Massouros, G. G. Massouros: «Hypergroups Associated with Graphs and Automata», Proceedings of the International Conference on Numerical Analysis and Applied Mathematics, ICNAAM 2009, American Institute of Physics (AIP) Conference Proceedings, pp. 164-167.

Ch. G. Massouros, G. G. Massouros: «Identities in Multivalued Algebraic Structures», Proceedings of the International Conference on NumericalAnalysis and Applied Mathematics, ICNAAM 2010, American Institute of Physics (AIP) Conference Proceedings, pp. 2065-2068.

Ch. G. Massouros, Ch. G. Tsitouras: «Enumeration of hypercompositional structures defined by binary relations», Italian J. Pure & Appl. Maths., to appear

Ch. G. Massouros: «Transposition hypergroups with idempotent identity» Submitted for publication.

Ch. G. Massouros, A. Dramalidis: «Transposition Hv-Groups» Submitted for publication.

G. G. Massouros, J. Mittas: «Languages, Automata and hypercompositional structures», Proceedings of the 4th International Congress in Algebraic Hyperstructures and Applications, World Scientific (1991), pp. 137-147.

G. G. Massouros: «Automata, Languages and hypercompositional structures» Doctoral Thesis, National Technical University of Athens, 1993.

G. G. Massouros: «Automata and Hypermoduloids», Proceedings of the 5th International Congress in Algebraic Hyperstructures and Applications, Hadronic Press (1994), pp. 251-266.

G. G. Massouros: «Hypercompositional Structures in the Theory of the Languages and Automata», An. stiintifice Univ. Al. I. Cuza, Iasi, Informatica, t. iii,(1994), pp. 65-73.

G. G. Massouros: «A new approach to the theory of Languages and Automata», Proceedings of the 26th Annual Iranian Mathematics Conference, Kerman, Iran, 2,(1995) pp. 237-239.

G. G. Massouros, Ch. G. Massouros, J. D. Mittas: «Fortified Join Hypergroups», Annales Matematiques Blaise Pascal, Vol 3, no 2 (1996) pp. 155-169.

G.G. Massouros: «Solving equations and systems in the environment of a Hyperringoid», Proceedings of the 6th International Congress on Algebraic Hyperstructures and Applications, Publ. Democritus Univ. of Thrace (1997),

pp. 103-113.

G. G. Massouros: «The subhypergroups of the Fortified Join Hypergroup», Italian J. Pure & Appl. Maths., no 2, (1997) pp. 51-63.

G. G. Massouros: «The Hyperringoid», Multiple Valued Logic, Vol. 3 (1998) pp. 217-234

G. G. Massouros, Ch. G. Massouros: «Homomorphic Relations on Hyperringoids and Join Hyperrings», Ratio Mathematica No. 13 (1999) pp. 61-70.

G. G. Massouros: «Hypercompositional Structures from the Computer Theory», Ratio Matematica, No 13, (1999) pp. 37-42.

G. G. Massouros: «Monogene Symmetrical Subhypergroups of the Fortified Join Hypergroup», Advances in Generalized Structures Approximate Reasoning and Applications, published by the cooperation of: University of Teramo, Romanian Society for Fuzzy Systems and BMFSA Japan (2001) pp. 45 – 53.

G. G. Massouros: «On the attached hypergroups of the order of an automaton», Journal of Discrete Mathematical Sciences & Cryptography Vol. 6, No 2-3 (2003) pp. 207-215.

G. G. Massouros, F. A. Zafiropoulos, Ch. G. Massouros: «Transposition Polysymmetrical hypergroups», Proceedings of the 8th International Congress on Algebraic Ηyperstructures and Applications, Spanidis Press (2003), pp. 191-202.

J. D. Mittas: «Hypergroupes canoniques», Mathematica Balkanica, 2 (1972) pp. 165-179.

J. D. Mittas: «Sur certaines classes de structures hypercompositionnelles»,

Proceedings of the Academy of Athens, 48, (1973), pp. 298 - 318.

W. Prenowitz: «A Contemporary Approach to Classical Geometry». Amer. Math. Month. 68, no 1, part II (1961) pp. 1-67.

W. Prenowitz , J. Jantosciak: «Join Geometries. A Theory of convex Sets and Linear Geometry», Springer - Verlag, 1979.

I. Rosenberg: «Hypergroups and join spaces determined by relations», Italian J. Pure & Appl. Math., 4, (1998) pp. 93 - 101.

Mario De Salvo, Giovanni Lo Faro: «Hypergroups and binary relations» Mult.-Valued Log. 8, No.5-6, (2002) pp. 645-657.

Mario De Salvo, Giovanni Lo Faro: «A new class of hypergroupoids associated to binary relations» J. Mult.-Val. Log. Soft Comput. 9, No. 4, (2003) pp. 361-375.

Ch. G. Tsitouras, Ch. G. Massouros: «On enumeration of hypergroups of order 3», Computers and Mathematics with Applications Vol. 59 (2010) pp. 519-523.

Ch. G. Tsitouras, Ch. G. Massouros: «Enumeration of Rosenberg type Hypercompositional structures defined by binary relations» Submitted for publication.

T. Vougiouklis: «The fundamental relation in hyperrings. The general hyperfield», Proceedings of the 4th International Congress on Algebraic Hyperstructures and Applications, World Scientific (1991), pp. 209-217.

D. Zofota: «Feeble Hypermodules over a Feeble Hyperring», Proceedings of the 5th International Congress on Algebraic Hyperstructures and Applications, Hadronic Press (1994), pp.207-214.


Refbacks

  • There are currently no refbacks.


Copyright (c) 2011 Franco Eugeni

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Ratio Mathematica - Journal of Mathematics, Statistics, and Applications. ISSN 1592-7415; e-ISSN 2282-8214.