Quasi-Valuation Maps on BCK/BCI-Algebras

- Journal title : Kyungpook mathematical journal
- Volume 55, Issue 4, 2015, pp.859-870
- Publisher : Department of Mathematics, Kyungpook National University
- DOI : 10.5666/KMJ.2015.55.4.859

Title & Authors

Quasi-Valuation Maps on BCK/BCI-Algebras

SONG, SEOK-ZUN; ROH, EUN HWAN; JUN, YOUNG BAE;

SONG, SEOK-ZUN; ROH, EUN HWAN; JUN, YOUNG BAE;

Abstract

The notion of quasi-valuation maps based on a subalgebra and an ideal in BCK/BCI-algebras is introduced, and then several properties are investigated. Relations between a quasi-valuation map based on a subalgebra and a quasi-valuation map based on an ideal is established. In a BCI-algebra, a condition for a quasi-valuation map based on an ideal to be a quasi-valuation map based on a subalgebra is provided, and conditions for a real-valued function on a BCK/BCI-algebra to be a quasi-valuation map based on an ideal are discussed. Using the notion of a quasi-valuation map based on an ideal, (pseudo) metric spaces are constructed, and we show that the binary operation * in BCK-algebras is uniformly continuous.

Keywords

BCK/BCI-algerba;subalgebra;ideal;S-quasi-valuation map;I-quasi-valuation map;(pseudo) metric space;uniformly continuous;

Language

English

References

1.

Y. S. Huang, BCI-algebra, Science Press, China (2006).

2.

Y. Imai and K. Iseki, On axiom systems of propositional calculi. XIV, Proc. Japan Acad., 42(1966), 19-22.

4.

K. Iseki, On BCI-algebras, Math. Seminar Notes, 8(1980), 125-130.

5.

K. Iseki and S. Tanaka, An introduction to theory of BCK-algebras, Math. Japonica, 23(1978), 1-26.

6.

J. Meng, Y. B. Jun, BCK-algebras, Kyungmoon Sa Co., Seoul (1994).

7.

J. Neggers, A. Dvurecenskij and H. S. Kim, On d-fuzzy functions in d-algebras, Found. Phys., 30(2000), 1807-1816.

8.

J. Neggers, Y. B. Jun, H. S. Kim, On d-ideals in d-algebras, Math. Slovaca, 49(1999), 243-251.

9.

J. Neggers, H. S. Kim, On d-algebras, Math. Slovaca, 49(1999), 19-26.