Advertisement

Hilbert Algebras in Negative Implicative BCK-Algebras

  • Qiu-na Zhang
  • Nan Ji
  • Li-nan Shi
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 219)

Abstract

The notion of BCK-algebras was formulated first in 1966 by Iséki, Japanese, and Mathematician. This notion is originated from two different ways. One of the motivations is based on set theory; another motivation is from classical and nonclassical propositional calculi. There are many classes of BCK-algebras, for example, subalgebras, bounded BCK-algebras, positive implicative BCK-algebra, implicative BCK-algebra, commutative BCK-algebra, BCK-algebras with condition (S), Griss (and semi-Brouwerian) algebras, quasicommutative BCK-algebras, direct product of BCK-algebras, and so on. The notion of positive implicative BCK-algebras was introduced by Iséki in 1975. In previous studies, scholars gave the definition of the positive implicative BCK-algebras, and its characterizations, and the relationship between other BCK-algebra, before this article I give a notion an ideal of Hilbert Algebras in BCK-algebras, as well as some propositions, so, here I will give a notion of Hilbert algebras in negative implicative BCK-algebras, and some propositions.

Keywords

BCK-algebra Hilbert algebras Negative implicative BCK-algebra 

References

  1. 1.
    Chen ZM, Wang HX (1991) On simple BCI-algebras. Math Jpn 36:627–632MATHGoogle Scholar
  2. 2.
    Fang L, Jizu L (1997) Hilbert algebras is an anti-positive implicative BCK-algebra (Chinese). Shanxi Coll Min Technol 15(2):214–217Google Scholar
  3. 3.
    Iséki K (1976) BCK-algebras. Math Semin Notes 4:77–86Google Scholar
  4. 4.
    Iséki K, Tanaka S (1978) An introduction to the theory of BCK-algebras. Math Jpn 23:1–26MATHGoogle Scholar
  5. 5.
    Meng J, Jun YB (1994) BCK-algebras. K Yung Moon Sa Co 35:146–149Google Scholar
  6. 6.
    Ahsan J, Deeba EY, Thaheem AB (1991) On prime ideals of BCK-algebras. Math Jpn 36:875–882MATHMathSciNetGoogle Scholar
  7. 7.
    Chen ZM, Wang HX (1991) On simple BCI-algebras. Math Jpn 36:627–632MATHGoogle Scholar
  8. 8.
    Huang WP (1993) On the semigroup theory of BCI-algebras. Pure Appl Math 1:28–34Google Scholar
  9. 9.
    Yi-quan Z (1999) On lattice implication algebras and BCK-algebras. Pure Appl Math 3(15):22–26Google Scholar
  10. 10.
    Qi-quan Z (2002) On implication algebras and BCK-algebras. Fuzzy Syst Math 3(16):32–38Google Scholar
  11. 11.
    Xu S-X (2003) On the semigroups description of implicative BCK-algebras. J Southwest China Norm Univ (Nat Sci) 6(28):856–858Google Scholar
  12. 12.
    Cornish WH (1980) On positive implicative BCK-algebras. Math Semin Notes 8:455–468MATHMathSciNetGoogle Scholar
  13. 13.
    Qiuna Z (2009) An ideal of Hilbert algebras in BCK-algebras. Proc 2009 Conf Commun Faculty 54:310–311Google Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  1. 1.College of ScienceHebei United UniversityHebeiChina
  2. 2.Qinggong CollegeHebei United UniversityHebeiChina

Personalised recommendations