DOI QR코드

DOI QR Code

THE SEMIGROUPS OF BINARY SYSTEMS AND SOME PERSPECTIVES

  • Published : 2008.11.30

Abstract

Given binary operations "*" and "$\circ$" on a set X, define a product binary operation "$\Box$" as follows: $x{\Box}y\;:=\;(x\;{\ast}\;y)\;{\circ}\;(y\;{\ast}\;x)$. This in turn yields a binary operation on Bin(X), the set of groupoids defined on X turning it into a semigroup (Bin(X), $\Box$)with identity (x * y = x) the left zero semigroup and an analog of negative one in the right zero semigroup (x * y = y). The composition $\Box$ is a generalization of the composition of functions, modelled here as leftoids (x * y = f(x)), permitting one to study the dynamics of binary systems as well as a variety of other perspectives also of interest.

References

  1. R. L. O. Cignoli, I. M. L. D'ottaviano, and D. Mundici, Algebraic Ffoundations of Many-Valued Reasoning, Kluwer Academic Publishers, Dordrecht, 2000
  2. A. Dvurecenskij and S. Pulmannova, New Trends in Quantum Structures, Mathematics and its Applications, 516. Kluwer Academic Publishers, Dordrecht; Ister Science, Bratislava, 2000
  3. J. S. Han, H. S. Kim, and J. Neggers, Strong and ordinary d-algebras, J. Multi-Valued Logic and Soft Computing (to appear)
  4. Y. Huang, BCI-algebra, Science Press, Beijing, 2006
  5. J. Meng and Y. B. Jun, BCK-algebras, Kyung Moon Sa, Seoul, 1994
  6. L. Nebesky, An algebraic characterization of geodetic graphs, Czechoslovak Math. J. 48(123) (1998), no. 4, 701-710 https://doi.org/10.1023/A:1022435605919
  7. L. Nebesky, A tree as a finite nonempty set with a binary operation, Math. Bohem. 125 (2000), no. 4, 455-458
  8. L. Nebesky, Travel groupoids, Czechoslovak Math. J. 56(131) (2006), no. 2, 659-675. https://doi.org/10.1007/s10587-006-0046-0
  9. J. Neggers, A. Dvurecenskij, and H. S. Kim, On d-fuzzy functions in d-algebras, Found. Phys. 30 (2000), no. 10, 1807-1816 https://doi.org/10.1023/A:1026466720971
  10. J. Neggers, Y. B. Jun, and H. S. Kim, On d-ideals in d-algebras, Math. Slovaca 49 (1999), no. 3, 243-251
  11. J. Neggers and H. S. Kim, On d-algebras, Math. Slovaca 49 (1999), no. 1, 19-26

Cited by

  1. (n−1)-Step Derivations onn-Groupoids: The Casen=3 vol.2014, 2014, https://doi.org/10.1155/2014/726470
  2. LOCALLY-ZERO GROUPOIDS AND THE CENTER OF BIN(X) vol.26, pp.2, 2011, https://doi.org/10.4134/CKMS.2011.26.2.163
  3. The Hypergroupoid Semigroups as Generalizations of the Groupoid Semigroups vol.2012, 2012, https://doi.org/10.1155/2012/717698
  4. On Abelian and Related Fuzzy Subsets of Groupoids vol.2013, 2013, https://doi.org/10.1155/2013/476057
  5. Several types of groupoids induced by two-variable functions vol.5, pp.1, 2016, https://doi.org/10.1186/s40064-016-3411-y
  6. The Interaction between Fuzzy Subsets and Groupoids vol.2014, 2014, https://doi.org/10.1155/2014/246285
  7. TN-groupoids vol.63, pp.4, 2013, https://doi.org/10.2478/s12175-013-0133-2
  8. Fuzzy rank functions in the set of all binary systems vol.5, pp.1, 2016, https://doi.org/10.1186/s40064-016-3536-z
  9. Hyperfuzzy subsets and subgroupoids vol.33, pp.3, 2017, https://doi.org/10.3233/JIFS-17104
  10. Fuzzy Upper Bounds in Groupoids vol.2014, 2014, https://doi.org/10.1155/2014/697012
  11. General Implicativity in Groupoids vol.6, pp.11, 2018, https://doi.org/10.3390/math6110235
  12. A Method to Identify Simple Graphs by Special Binary Systems vol.10, pp.7, 2018, https://doi.org/10.3390/sym10070297
  13. Some Special Elements and Pseudo Inverse Functions in Groupoids vol.7, pp.2, 2019, https://doi.org/10.3390/math7020173