Kim, Young-Hee;So, Keum-Sook

  • Received : 2010.08.19
  • Published : 2012.01.31


In this paper we consider pseudo-BCK/BCI-algebras. In particular, we consider properties of minimal elements ($x{\leq}a$ implies x = a) in terms of the binary relation $\leq$ which is reflexive and anti-symmetric along with several more complicated conditions. Some of the properties of minimal elements obtained bear resemblance to properties of B-algebras in case the algebraic operations $\ast$ and $\circ$ are identical, including the property $0{\circ}(0{\ast}a)$ = a. The condition $0{\ast}(0{\circ}x)=0{\circ}(0{\ast}x)=x$ all $x{\in}X$ defines the class of p-semisimple pseudo-BCK/BCI-algebras($0{\leq}x$ implies x = 0) as an interesting subclass whose further properties are also investigated below.




  1. S. S. Ahn, Y. B. Jun, H. S. Kim, and M. Kondo, Fuzzifications of pseudo-BCI-algebras, Sci. Math. Jpn. 60 (2004), no. 1, 15-19.
  2. G. Dymek, p-semisimple pseudo-BCI-algebras, submitted.
  3. G. Georgescu and A. Iorgulescu, Pseudo BCK-algebras: an extension of BCK-algebras, Combinatorics, computability and logic (Constanta, 2001), 97-114, Springer Ser. Discrete Math. Theor. Comput. Sci., Springer, London, 2001.
  4. A. Gilani and B. N. Waphare, On pseudo a-ideal of pseudo-BCI-algebras, Sci. Magna 3 (2007), no. 2, 50-59.
  5. A. Iorgulescu, Algebras of Logic as BCK-Algebras, Editura ASE, Bucharest, 2008.
  6. K. Iseki, On BCI-algebras, Math. Sem. Notes Kobe Univ. 8 (1980), no. 1, 125-130.
  7. K. Iseki and S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japon. 23 (1978/79), no. 1, 1-26.
  8. Y. B. Jun, Characterizations of pseudo BCK-algebras, Math. Japonicae Online 7 (2002), 225-230.
  9. Y. B. Jun, H. S. Kim, and J. Neggers, On pseudo-BCI ideals of pseudo BCI-algebras, Mate. Vesnik 58 (2006), no. 1-2, 39-46.
  10. J. Meng and Y. B. Jun, BCK-algebras, Kyungmoon Sa, Seoul, 1994.
  11. H. Yisheng, BCI-algebras, Science Press, Beijing, 2006.

Cited by

  1. Some properties of pseudo-BCK- and pseudo-BCI-algebras 2016,
  2. On pseudo-CI algebras pp.1433-7479, 2018,
  3. Ideals and congruences in pseudo-BCH algebras vol.69, pp.1, 2019,