An Axiomatic Development of Multi-Natural Numbers

  IJMTT-book-cover
 
International Journal of Mathematics Trends and Technology (IJMTT)
 
© 2017 by IJMTT Journal
Volume-48 Number-2
Year of Publication : 2017
Authors : Debjyoti Chatterjee, S. K. Samanta
  10.14445/22315373/IJMTT-V48P518

MLA

Debjyoti Chatterjee, S. K. Samanta "An Axiomatic Development of Multi-Natural Numbers", International Journal of Mathematics Trends and Technology (IJMTT). V48(2):136-146 August 2017. ISSN:2231-5373. www.ijmttjournal.org. Published by Seventh Sense Research Group.

Abstract
In this paper we define and study multi-natural number system from axiomatic point of view.

References
[1] Wayne D. Blizart, Multiset theory, Notre Dame Journal of Formal Logic 30(1), 36-66 (1989).
[2] Wayne D. Blizart, Real-valued multisets and fuzzy sets, Fuzzy Sets and Systems 33(1), 77- 97 1989.
[3] Wayne D. Blizart, Negative membership, Notre Dame Journal of Formal Logic 31(3), 346-368 (1990).
[4] Wayne D. Blizart, The development of multiset theory, Modern Logic 1(4) 319-352(1991).
[5] K. Chakraborty, Bags with interval counts, Foundation of Computing and Decision Sciences 25(1), 23-36 (2000).
[6] Chakraborty Kankana and I. Despi, kn- bags, International Journal of Intelligent Systems, 223-236 (2007).
[7] Chakraborty Kankana, R. Biswas and S. Nanda, Fuzzy Shadows, Fuzzy Sets and Systems 101(3), 413- 421 (1999).
[8] K. Chakraborty, R. Biswas and S. Nanda, On Yager’s theory of bags and fuzzy bags, Computers and Artificial Intelligence 18 -17 (1999).
[9] G. F. Clements, On multiset k-families, Discrete Mathematics 69(2), 153 -164 (1988).
[10] M. Conder, S. Marshall and Arkadii M. Slinko, Orders on multisets and discrete cones, A Journal on The Theory of Ordered Sets and Its Applications 24, 277-296 (2007).
[11] K. P. Girish, S. J. John, Multiset topologies induced by multiset relations, Information Sciences, 188(0), 298 -313 (2012).
[12] K. P. Girish, S. J. John, On multiset topologies, Theory and applications of Mathematics and Computer Science, 2(1) (2012) 37-52.
[13] K. P. Girish, S. J. John, General relations between partially ordered multisets and their chains and antichains, Mathematical Communications 14(2), 193-206 (2009).
[14] K. P. Girish, S. J. John, Relations and functions in multi set context, Inf. Sci. 179(6), 758-768 (2009).
[15] K. P. Girish, S. J. John, Rough multisets and information multisystems, Advances in Decision Sciences p. 17 pages (2011).
[16] S. P. Jena, S. K. Ghosh and B. K. Tripathy, On the theory of bags and lists, Information Sciences 132(14), 241 - 254 (2001).
[17] J. L. Petersion, Computation sequence sets, Journal of Computer and System Sciences 13(1), 1 -24 (1976).
[18] D. Singh, A note on the development of multiset theory, Modern Logic 4(4), 405-406 (1994).
[19] D. Singh, A. M. Ibrahim, T. Yohana and J. N. Singh, Complementation in multiset theory, International Mathematical Forum 6(38), 1877-1884 (2011).
[20] D. Singh, A. M. Ibrahim, T. Yohana and J. N. Singh, An overview of the applications of multisets, Novi Sad J. Math 37(2), 73-92 (2007).
[21] D. Singh, A. M. Ibrahim, T. Yohana and J. N. Singh, Some combinatorics of multisets, International Journal of Mathematical Education in Science and Technology 34(4), 489-499 (2003).
[22] D. Tokat, I. Ucok, On Soft Multi Continuous Functions, The Journal of New Theory, 1(2015) 50-58.
[23] Sk. Najmul, S. K. Samanta, On soft multi groups, 10(2), 271-285 (2015).
[24] Sk. Najmul, S. K. Samanta, On multisets and multigroups, 6(30), 643-656 (2013).
[25] Hallet M., Cantorian Set Theory and Limitation of Size, Clarendon Press, Oxford 1984.
[26] Ifrah G., From one to zero: A Universal history of Numbers, Viking Penguin, New York, 1985.
[27] Dershowitz N. & Manna Z., Proving Termination with multiset ordering, Comm. ACM 22 (1979), 465-476.
[28] Abhijit Dasgupta, Set Theory with an Introduction to Real Point Sets, ISBN: 978-1-4614-8853-8, Springer.
[29] A.G. Hamilton, Numbers, sets snd axioms The Appratus of Mathematics, ISBN: 0 521 24509 5, Cambridge University Press, Cambridge.
[30] Gary Sng Chee Hien, Denny Leung Ho-Hon, Construction of the Real Number System, Undergraduate Research Opportunity Programme in Science, Department of Mathematics, National University of Singapore.

Keywords
Multiset, Multi-natural Number, Support Successor Function, Multiplicity Successor Function.