PC-lattices: A Class of Bounded BCK-algebras

Authors

  • Sadegh Khosravi Shoar Department of Mathematics, Fasa University, Fasa, Iran
  • Rajab Ali Borzooei Department of Mathematics, Shahid Beheshti University, Tehran, Iran
  • R. Moradian Department of mathematics Farhangian University, Tehran, Iran
  • Atefe Radfar Payame Noor University, p. o. box. 19395-3697, Tehran, Iran

DOI:

https://doi.org/10.18778/0138-0680.47.1.03

Keywords:

PC-lattice, BCK-lattice, Involutory BCK-algebras, Bounded commutative BCK-algebras

Abstract

In this paper, we define the notion of PC-lattice, as a generalization of finite positive implicative BCK-algebras with condition (S) and bounded commutative BCK-algebras. We investiate some results for Pc-lattices being a new class of BCK-lattices. Specially, we prove that any Boolean lattice is a PC-lattice and we show that if X is a PC-lattice with condition S, then X is an involutory BCK-algebra if and only if X is a commutative BCK-algebra. Finally, we prove that any PC-lattice with condition (S) is a distributive BCK-algebra.  

References

[1] C. Bărbăcioru, Positive implicative BCK-algebras, Mathematica Japonica 36 (1967), pp. 11–59.
Google Scholar

[2] R. A. Borzooei, S. Khosravi Shoar, Implication Algebras are Equivalent to the Dual Implicative BCK-algebras, Scientiae Mathematicae Japonicae 633 (2006), pp. 429–431.
Google Scholar

[3] R. A. Borzooei, S. Khosravi Shoar, R. Ameri, Some new filters in MTL-algebras, Fuzzy Sets and Systems 187(1) (2012), pp. 92–102.
Google Scholar

[4] B. A. Davey, H. A. Priestley, Introduction to Lattices and Order, Cambridge University Press, 1990, 2002.
Google Scholar

[5] G. Grätzer, General Lattice Theory, Academic Press, 1978.
Google Scholar

[6] Y. Huang, BCI-algebras, Science Press, 2006.
Google Scholar

[7] Y. Huang, On involutory BCK-algebras, Soochow Journal of Mathematics 32(1) (2006), pp. 51–57.
Google Scholar

[8] Y. Imai, K. Iséki, On axioms systems of propositional calculi XIV, Proceedings of the Japan Academy 42 (1966), pp. 19–22.
Google Scholar

[9] K. Iséki, BCK-algebras with condition (S), Mathematica Japonica 24 (1979), pp. 107–119.
Google Scholar

[10] K. lséki, On positive implicative BCK-algebras with condition (S), Mathematica Japonica 24 (1979), pp. 107–119.
Google Scholar

[11] K. Iséki and S. Tanaka, An introduction to the theory of BCK-algebras, Mathematica Japonica 23 (1978), pp. 1–26.
Google Scholar

[12] J. Meng and Y. B. Jun, BCK-Algebras, Kyung Moon Sa Co, Seoul, Korea, 1994.
Google Scholar

[13] S. Tanaka, A new class of algebras, Mathematics Seminar Notes 3 (1975), pp. 37–43.
Google Scholar

[14] S. Tanaka, On ^-commutative algebras, Mathematics Seminar Notes 3 (1975), pp. 59–64.
Google Scholar

Downloads

Published

2018-03-30

How to Cite

Shoar, S. K., Borzooei, R. A., Moradian, R., & Radfar, A. (2018). PC-lattices: A Class of Bounded BCK-algebras. Bulletin of the Section of Logic, 47(1), 33–44. https://doi.org/10.18778/0138-0680.47.1.03

Issue

Vol. 47 No. 1 (2018)

Section

Research Article

License

Copyright (c) 2018 Bulletin of the Section of Logic

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Crossref
Scopus
Google Scholar
Europe PMC

Most read articles by the same author(s)

1 2 > >>