Some aspects of \(d\)-units in \(d/\)BCK-algebras. (English) Zbl 1263.06008

Summary: We explore properties of the set of \(d\)-units of a \(d\)-algebra. A property of interest in the study of \(d\)-units in \(d\)-algebras is the weak associative property. It is noted that many other \(d\)-algebras, especially BCK-algebras, are in fact weakly associative. The existence of \(d/\)BCK-algebras which are not weakly associative is demonstrated. Moreover, the notions of a \(d\)-integral domain and a left-injectivity are discussed.


06F35 BCK-algebras, BCI-algebras
Full Text: DOI


[1] K. Iséki and S. Tanaka, “An introduction to the theory of BCK-algebras,” Mathematica Japonica, vol. 23, no. 1, pp. 1-26, 1978/79. · Zbl 0385.03051
[2] K. Iséki, “On BCI-algebras,” Mathematics Seminar Notes, vol. 8, no. 1, pp. 125-130, 1980. · Zbl 0473.03059
[3] J. Neggers and H. S. Kim, “On d-algebras,” Mathematica Slovaca, vol. 49, no. 1, pp. 19-26, 1999. · Zbl 0943.06012
[4] P. J. Allen, H. S. Kim, and J. Neggers, “Companion d-algebras,” Mathematica Slovaca, vol. 57, no. 2, pp. 93-106, 2007. · Zbl 1150.06323
[5] Y. B. Jun, J. Neggers, and H. S. Kim, “Fuzzy d-ideals of d-algebras,” Journal of Fuzzy Mathematics, vol. 8, no. 1, pp. 123-130, 2000. · Zbl 0979.06012
[6] Y. C. Lee and H. S. Kim, “On d-subalgebras of d-transitive d*-algebras,” Mathematica Slovaca, vol. 49, no. 1, pp. 27-33, 1999. · Zbl 0943.06011
[7] J. Neggers, A. Dvure\vcenskij, and H. S. Kim, “On d-fuzzy functions in d-algebras,” Foundations of Physics, vol. 30, no. 10, pp. 1807-1816, 2000.
[8] P. J. Allen, H. S. Kim, and J. Neggers, “Deformations of d/BCK-algebras,” Bulletin of the Korean Mathematical Society, vol. 48, no. 2, pp. 315-324, 2011. · Zbl 1236.06023
[9] J. S. Han, H. S. Kim, and J. Neggers, “Strong and ordinary d-algebras,” Journal of Multiple-Valued Logic and Soft Computing, vol. 16, no. 3-5, pp. 331-339, 2010. · Zbl 1236.06025
[10] J. Neggers, Y. B. Jun, and H. S. Kim, “On d-ideals in d-algebras,” Mathematica Slovaca, vol. 49, no. 3, pp. 243-251, 1999. · Zbl 0960.06010
[11] J. Meng and Y. B. Jun, BCK-Algebras, Kyung Moon Sa, Seoul, South Korea, 1994. · Zbl 0906.06015
[12] A. Iorgulescu, Algebras of Logic as BCK Algebras, Editura ASE, Bucharest, 2008. · Zbl 1172.03038
[13] A. Dvure\vcenskij and S. Pulmannová, New Trends in Quantum Structures, vol. 516 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.
[14] H. Yisheng, BCI-Algebras, Science Press, Beijing, China, 2006. · Zbl 1113.06016
[15] J. Neggers and H. S. Kim, Basic Posets, World Scientific Publishing, River Edge, NJ, USA, 1998. · Zbl 0928.06001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.