Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ORDER RELATED CONCEPTS FOR ARBITRARY GROUPOIDS

  • Kim, Hee Sik (Research Institute for Natural Sciences Department of Mathematics Hanyang University) ;
  • Neggers, Joseph (Department of Mathematics University of Alabama) ;
  • So, Keum Sook (Department of Statistics and Financial Informatics Hallym University)
  • Received : 2016.07.15
  • Accepted : 2016.12.15
  • Published : 2017.07.31
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Abstract

In this paper, we introduce and explore suggested notions of 'above', 'below' and 'between' in general groupoids, Bin(X), as well as in more detail in several well-known classes of groupoids, including groups, semigroups, selective groupoids (digraphs), d/BCK-algebras, linear groupoids over fields and special cases, in order to illustrate the usefulness of these ideas. Additionally, for groupoid-classes (e.g., BCK-algebras) where these notions have already been accepted in a standard form, we look at connections between the several definitions which result from our introduction of these ideas as presented in this paper.

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References

  1. S. S. Ahn, Y. H. Kim, and J. Neggers, Fuzzy upper bounds in groupoids, Sci. World J. 2014 (2014), Article ID 697012, 6pages.
  2. P. J. Allen, H. S. Kim, and J. Neggers, Companion d-algebras, Math. Slovaca 57 (2007), no. 2, 93-106.
  3. P. J. Allen, H. S. Kim, and J. Neggers, Several types of groupoids induced by two-variable functions, Springer Plus 5 (2016), 1715-1725; DOI:10.1186/s40064-016-3411-y.
  4. R. H. Bruck, A Survey of Binary Systems, Springer, New York, 1971.
  5. H. F. Fayoumi, Locally-zero groupoids and the center of Bin(X), Commun. Korean Math. Soc. 26 (2011), no. 2, 163-168. https://doi.org/10.4134/CKMS.2011.26.2.163
  6. J. S. Han, H. S. Kim, and J. Neggers, The hypergroupoid semigroups as generalizations of the groupoid semigroups, J. Appl. Math. 2012 (2012), Article ID 717698, 8 pages.
  7. H. S. Kim and J. Neggers, The semigroups of binary systems and some perspectives, Bull. Korean Math. Soc. 45 (2008), no. 4, 651-661. https://doi.org/10.4134/BKMS.2008.45.4.651
  8. J. Meng and Y. B. Jun, BCK-Algebras, Kyung Moon Sa, Seoul, 1994.
  9. J. Neggers and H. S. Kim, Basic Poset, World Scientific, Singapore, 1998.
  10. J. Neggers and H. S. Kim, On d-algebras, Math. Slovaca 49 (1999), no. 1, 19-26.
  11. S. J. Shin, H. S. Kim, and J. Neggers, On Abelian and related fuzzy subsets of groupoids, Sci. World J. 2013 (2013), Article ID 476057, 5pages.
  12. S. J. Shin, H. S. Kim, and J. Neggers, The intersection between fuzzy subsets and groupoids, Sci. World J. 2014 (2014), Article ID 246285, 6pages.
  13. S. Z. Song, H. S. Kim, and Y. B. Jun, Soft saturated and dried values with applications in BCK/BCI-algebras, J. Comput. Anal. Appl. 21 (2016), no. 3, 528-544.
  14. R. Stanley, Enumerative Combinatorics. Vol. 1, Wadsworth & Brooks/Cole, Monterey, 1986.
  15. D. B. West, Introduction to Graph Theory, Prentice Hall, London, 2001.
  16. H. Yisheng, BCI-Algebra, Science Press, Beijing, 2006.
  17. J. Zhan, H. S. Kim, and J. Neggers, On (fuzzy) weak-zero groupoids and (X,N)-zero groupoids, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 78 (2016), no. 2, 99-110.