Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON f-DERIVATIONS FROM SEMILATTICES TO LATTICES

  • Yon, Yong Ho (Innovation Center for Engineering Education Mokwon University) ;
  • Kim, Kyung Ho (Department of Mathematics Korea National University of Transpotation)
  • Received : 2013.03.03
  • Published : 2014.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we introduce the notion of f-derivations from a semilattice S to a lattice L, as a generalization of derivation and f-derivation of lattices. Also, we define the simple f-derivation from S to L, and research the properties of them and the conditions for a lattice L to be distributive. Finally, we prove that a distributive lattice L is isomorphic to the class $SD_f(S,L)$ of all simple f-derivations on S to L for every ${\wedge}$-homomorphism $f:S{\rightarrow}L$ such that $f(x_0){\vee}f(y_0)=1$ for some $x_0,y_0{\in}S$, in particular, $$L{\simeq_-}=SD_f(S,L)$$ for every ${\wedge}$-homomorphism $f:S{\rightarrow}L$ such that $f(x_0)=1$ for some $x_0{\in}S$.

Keywords

References

  1. R. Balbes and P. Dwinger, Distributive Lattices, University of Missouri Press, Columbia, United States, 1974.
  2. G. Birkhoff, Lattice Theory, American Mathematical Society, New York, 1940.
  3. Y. Ceven, Symmetric bi-derivations of lattices, Quaest. Math. 32 (2009), no. 2, 241-245. https://doi.org/10.2989/QM.2009.32.2.6.799
  4. Y. Ceven and M. A. Ozturk, On f-derivations of lattices, Bull. Korean Math. Soc. 45 (2008), no. 4, 701-707. https://doi.org/10.4134/BKMS.2008.45.4.701
  5. L. Ferrari, On derivations of lattices, Pure Math. Appl. 12 (2001), no. 4, 365-382.
  6. S. Harmaitree and U. Leerawat, On f-derivations in lattices, Far East J. Math. Sci. 51 (2011), no. 1, 27-40.
  7. S. A. Ozbal and A. Firat, Symmetric f bi-derivations of lattices, Ars Combin. 97 (2010), 471-477.
  8. M. A. Ozturk, H. Yazarli, and K. H. Kim, Permuting tri-derivations in lattices, Quaest. Math. 32 (2009), no. 3, 415-425. https://doi.org/10.2989/QM.2009.32.3.10.911
  9. G. Pataki and A. Szaz, Characterizations of nonexpansive multipliers on partially ordered sets, Math. Slovaca 51 (2001), no. 4, 371-382.
  10. G. Szasz, Translationen der Verbande, Acta Fac. Rer. Nat. Univ. Comenianae 5 (1961), 449-453.
  11. G. Szasz, Derivations of lattices, Acta Sci. Math. (Szeged) 37 (1975), 149-154.
  12. Y. H. Yon and K. H. Kim, On expansive linear map and _-multipliers of lattices, Quaest. Math. 33 (2010), no. 4, 417-427. https://doi.org/10.2989/16073606.2010.541605